HYDRODYNAMICS OF HELIUM II*

I. M. Khalatnikov

I. HYDRODYNAMICS OF HELIUM II

1. Equations of hydrodynamics of helium II . . . . . . . . . . . . . . . . . . 70
2. Dissipative function for liquid helium II . . . . . . . . . . . . . . . . . 77
3. Sound in helium II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4. Absorption of sound in helium II . . . . . . . . . . . . . . . . . . . . . 84
5. On anomalous absorption of sound near the λ-point . . . . . . . . . . . 99

II. HYDRODYNAMICS OF SOLUTIONS

1. Equations of hydrodynamics of solutions of foreign particles in helium II . . . 103
2. Dissipative processes in solutions . . . . . . . . . . . . . . . . . . . . 112
3. Sound in solutions of foreign particles in helium II . . . . . . . . . . . 116
4. Hydrodynamics of solutions of two superfluid liquids . . . . . . . . . . . 118

III. DISCONTINUITIES AND SOUND OF LARGE AMPLITUDE IN HELIUM II

1. Discontinuities in helium II . . . . . . . . . . . . . . . . . . . . . . . 125
2. Sound of large amplitude in helium II . . . . . . . . . . . . . . . . . 130
3. On the propagation of sound in moving helium II and on the influence of heat flow on the propagation of second sound . . . . . . . . . . . . . . . 135

IV. HEAT EXCHANGE BETWEEN A SOLID BODY AND HELIUM II

1. Quantization of elastic waves . . . . . . . . . . . . . . . . . . . . . . 139
2. Emission of energy by an oscillating surface of a solid body . . . . . . 143
3. Energy exchange in collisions of rotons and phonons with a solid wall . . 145
4. Heat exchange between a solid body and liquid helium II . . . . . . . . 152
5. Passage of second sound through metallic plates. Absorption of second sound at the walls of a cylindrical vessel . . . . . . . . . . . . . . . . . 155

* The author’s first article was published in Uspekhi Fizicheskikh Nauk, 59, no. 4 (1956).

I. M. KHALATNIKOV

I. HYDRODYNAMICS OF HELIUM II

1. Equations of the hydrodynamics of helium II

The hydrodynamics of a superfluid liquid can be constructed on the basis of the fundamental conservation laws. In this way the equations of the hydrodynamics of helium II were first derived by L. Landau. It is true that in the published work¹ only equations were given that are valid for small values of the velocities of the normal and superfluid motions. Later several papers were published that contained a derivation of the hydrodynamic equations valid for the case of small velocities \(\mathbf v_n\) and \(\mathbf v_s\). However, these derivations, in contrast to that of L. Landau, suffered from a number of shortcomings. Thus, C. C. Lin² gave a derivation of the equations of the hydrodynamics of helium II by means of a variational method. The author was apparently unaware of the fact that the standard variational method is inapplicable to the derivation even of the hydrodynamic equations of ordinary liquids. The application of such a standard method makes it possible only to obtain the equations of potential motion for ordinary liquids. It is therefore not surprising that from the variational equations in Lin’s derivation there follows the condition

\[ \operatorname{rot}\left[\frac{(\mathbf v_n-\mathbf v_s)\rho_n}{\rho S}\right]=0, \]

which has no physical meaning whatever. The author did not notice this circumstance.

Nakajima, Tomita, and Usui³ used conservation laws to obtain the equations of the hydrodynamics of helium II. The expression they chose for the energy per unit volume of helium II is suitable only for small values of the velocity difference \(\mathbf v_n-\mathbf v_s\). In addition, the authors of that work ignored, in the derivation, the condition of potentiality of the superfluid motion, without which the equations of motion cannot in general be uniquely determined.

We shall begin with a derivation of the equations of the hydrodynamics of helium II that are valid for arbitrary (not small) values of the velocities of motion. In doing so we shall not for the time being take dissipative processes into account.⁴

Let us consider liquid helium II, in which two motions occur simultaneously: potential superfluid motion with velocity \(\mathbf v_s\) and normal motion with velocity \(\mathbf v_n\).

Since the momentum and the mass of the whole liquid are conserved, it may be asserted that the equations determining the change in the density \(\rho\) and in the momentum \(\mathbf j\) must have the form of continuity equations

\[ \dot{\rho}+\operatorname{div}\mathbf j=0, \tag{1,1} \]

\[ \frac{\partial}{\partial t}j_i+\frac{\partial \Pi_{ik}}{\partial x_k}=0. \tag{1,2} \]

HYDRODYNAMICS OF HELIUM II

Let us express the momentum per unit volume of the liquid \(\mathbf{j}\) and the momentum-flux tensor \(\Pi_{ik}\) in terms of their values in a reference frame in which the superfluid motion is absent. With the aid of the known transformation formulas we find\(^*\)

\[ \mathbf{j}=\rho \mathbf{v}_s+\mathbf{p}, \tag{1,3} \]

\[ \Pi_{ik}=\rho v_{si}v_{sk}+p_i v_{sk}+p_k v_{si}+\pi_{ik}. \tag{1,4} \]

Here \(\mathbf{p}\) is the momentum per unit volume of the liquid in a reference frame moving with velocity \(\mathbf{v}_s\); \(\pi_{ik}\) is the symmetric momentum-flux tensor in the same reference frame. The form of the tensor \(\pi_{ik}\) will be clarified below.

Besides the mass and momentum of the liquid, the conserved quantities are the total energy and the total entropy. Therefore one may write

\[ \dot S+\operatorname{div}\mathbf{F}=0, \tag{1,5} \]

\[ \dot E+\operatorname{div}\mathbf{Q}=0. \tag{1,6} \]

Here \(S\) is the entropy, \(E\) the energy, \(\mathbf{F}\) the entropy flux, and \(\mathbf{Q}\) the energy flux; all quantities are taken per unit volume of the liquid. We represent the entropy flux in the form

\[ \mathbf{F}=S\mathbf{v}_s+\mathbf{f}. \tag{1,7} \]

Similarly to (1,4), one can write a relation connecting the energy \(E\) with the energy per unit volume of the liquid \(\varepsilon\) in a reference frame moving with velocity \(\mathbf{v}_s\),

\[ E=\rho \frac{v_s^2}{2}+\mathbf{p}\mathbf{v}_s+\varepsilon. \tag{1,8} \]

\(^*\) The relation connecting the values of the momentum-flux tensor in a stationary reference frame \((\Pi_{ik})\) and in a frame moving with some velocity \(\mathbf{v}\) \((\pi_{ik})\) can readily be obtained in the case of classical hydrodynamics, where the form of the tensor \(\Pi_{ik}\) is known. This tensor in classical hydrodynamics is written as

\[ \Pi_{ik}=\rho u_i u_k+p\delta_{ik} \]

(\(\mathbf{u}\) is the velocity of the liquid, \(p\) the pressure). Let us express the velocity of the liquid in the stationary reference frame in terms of the velocity of the liquid \(\mathbf{u}'\) in a reference frame moving with velocity \(\mathbf{v}\):

\[ \mathbf{u}=\mathbf{u}'+\mathbf{v}. \]

After substituting this expression for \(\mathbf{u}\) into the formula given above for \(\Pi_{ik}\), we obtain

\[ \Pi_{ik}=\rho v_i v_k+\rho u'_i v_k+\rho u'_k v_i+\left(\rho u'_i u'_k+p\delta_{ik}\right). \]

Denoting the momentum per unit volume of the liquid in the moving reference frame by the letter \(\mathbf{p}\), we finally find the desired relation

\[ \Pi_{ik}=\rho v_i v_k+p_i v_k+p_k v_i+\pi_{ik}. \]

It is natural that the transformation formula obtained preserves its form also in the hydrodynamics of helium II.

For the energy \(\varepsilon\) the thermodynamic identity holds

\[ d\varepsilon=\mu\,d\rho+T\,dS+\left(\mathbf v_n-\mathbf v_s,\,d\mathbf p\right), \tag{1.9} \]

in which the temperature \(T\) and chemical potential \(\mu\) depend on all the independent variables \((\rho, S\) and \(\mathbf p)\). The third term in (1.9) may be regarded as the definition of the velocity \(\mathbf v_n\), appearing here for the first time in our formulas. We express the energy flux \(\mathbf Q\) in terms of its value \(\mathbf q\) in a reference frame moving with velocity \(\mathbf v_s\):

\[ \mathbf Q= \left( \frac{\rho v_s^2}{2}+\mathbf p\mathbf v_s+\varepsilon \right)\mathbf v_s +\frac{v_s^2}{2}\mathbf p +\pi\mathbf v_s+\mathbf q \tag{1.10} \]

\[ (\pi\mathbf v_s)_i=\pi_{ik}v_{sk}. \]

The form of the vector \(\mathbf q\) will be clarified later.

We choose the equation of superfluid motion in such a way that the condition \(\operatorname{rot}\mathbf v_s=0\) is satisfied,

\[ \dot{\mathbf v}_s+\nabla\left(\frac{v_s^2}{2}+\varphi\right)=0^*). \tag{1.11} \]

Equations (1.1), (1.2), (1.5), and (1.11), after the functions \(\pi_{ik}, \mathbf q, \mathbf f\), and \(\varphi\) have been explicitly represented in them, will give us the desired system of hydrodynamic equations for helium II. Equation (1.6), expressing the law of conservation of energy, we shall use to determine the form of the unknown functions \(\pi_{ik}, \mathbf f, \mathbf q\), and \(\varphi\). To this end we compute the time derivative \(\dot E\), expressing in it the time derivatives of the thermodynamic variables and velocities with the aid of the indicated equations. According to (1.8) and (1.9), we have

\[ \dot E= \left(\frac{v_s^2}{2}+\mu\right)\dot\rho +(\rho\mathbf v_s+\mathbf p)\dot{\mathbf v}_s +\mathbf v_n\dot{\mathbf p} +T\dot S= \]

\[ = -\left(\frac{v_s^2}{2}+\mu\right)\operatorname{div}(\rho\mathbf v_s+\mathbf p) -\mathbf v_n\mathbf p\,\operatorname{div}\mathbf v_s - \]

\[ -(\mathbf p+\rho\mathbf v_s-\rho\mathbf v_n)\nabla\varphi -T\,\operatorname{div}(S\mathbf v_s+\mathbf f) -(\rho\mathbf v_s+\mathbf p)(\mathbf v_s\nabla)\mathbf v_s - \]

\[ -\mathbf v_n(\mathbf p\nabla)\mathbf v_s -\mathbf v_n(\mathbf v_s\nabla)\mathbf p -\mathbf v_n(\nabla\pi). \tag{1.12} \]

Next we compute \(-\operatorname{div}\mathbf Q\); according to (1.10) we have

\[ -\operatorname{div}\mathbf Q = -\left\{ \frac{v_s^2}{2}\operatorname{div}(\rho\mathbf v_s+\mathbf p) +(\varepsilon+\mathbf p\mathbf v_s)\operatorname{div}\mathbf v_s + \right. \]

\[ \left. +\mathbf v_s(\rho\mathbf v_s+\mathbf p,\nabla)\mathbf v_s +\mathbf p(\mathbf v_s\nabla)\mathbf v_s +\mathbf v_n(\mathbf v_s\nabla)\mathbf p + \right. \]

\[ \left. +(\mathbf v_s\nabla\rho)\mu +(\mathbf v_s\nabla S)T +\operatorname{div}(\pi\mathbf v_s) +\operatorname{div}\mathbf q \right\}. \tag{1.13} \]

\[ {}^*)\ \text{More precisely, from equation (1.11) it follows that } \operatorname{rot}\mathbf v_s=\mathrm{const}. \]

Equating expression (1.12) for $\dot E$ to expression (1.13) for $-\operatorname{div}\mathbf Q$, after considerable cancellations we find

$$ \operatorname{div}\mathbf q=-(\boldsymbol{\pi}\nabla)\mathbf v_s+(\mathbf v_n-\mathbf v_s)(\nabla\boldsymbol{\pi})- $$

$$ -\left[\varepsilon-TS-\mu\rho-(\mathbf v_n-\mathbf v_s,\mathbf p)\right]\operatorname{div}\mathbf v_s +\left[\mathbf p-\rho(\mathbf v_n-\mathbf v_s)\right]\nabla\varphi+ $$

$$ +T\operatorname{div}\mathbf f+\mu\operatorname{div}\mathbf p+\mathbf v_n(\mathbf p\nabla)\mathbf v_s -\mathbf p(\mathbf v_s\nabla)\mathbf v_s . \tag{1.14} $$

Expression (1.14) is appreciably simplified if the tensor $\pi_{ik}$ is specified in the form:

$$ \pi_{ik}=-\left[\varepsilon-TS-\mu\rho-(\mathbf v_n-\mathbf v_s,\mathbf p)\right]\delta_{ik}+m_{ik}, \tag{1.15} $$

where $m_{ik}$ is the desired tensor. Substituting this expression for $\pi_{ik}$ into (1.14), we find

$$ \operatorname{div}\mathbf q=-(\mathbf m\nabla)\mathbf v_s+(\mathbf v_n-\mathbf v_s)(\nabla\mathbf m)+ \mathbf p(\mathbf v_n-\mathbf v_s,\nabla)(\mathbf v_n-\mathbf v_s)+ $$

$$ +\mathbf v_n(\mathbf p\nabla)\mathbf v_s-\mathbf p(\mathbf v_s\nabla)\mathbf v_s +\left[\mathbf p-\rho(\mathbf v_n-\mathbf v_s)\right]\nabla(\varphi-\mu)- $$

$$ -\nabla T\left[\mathbf f-S(\mathbf v_n-\mathbf v_s)\right] +\operatorname{div}(T\mathbf f+\mathbf p\mu). \tag{1.16} $$

In the absence of energy dissipation the quantities $m_{ik}$, $\mathbf q$, $\mathbf f$, and $\varphi$ are functions of the thermodynamic variables and velocities and do not depend on their derivatives with respect to time and coordinates. This circumstance makes it possible, from (1.16), to obtain the relations uniquely*:

$$ \left. \begin{aligned} m_{ik}&=p_i(v_{nk}-v_{sk}), & \mathbf f&=S(\mathbf v_n-\mathbf v_s), & \varphi&=\mu,\\ \mathbf q&=T\mathbf f+\mu\mathbf p-(\mathbf v_n-\mathbf v_s,\mathbf v_s)\mathbf p +(\mathbf v_n-\mathbf v_s)(\mathbf v_n\mathbf p). \end{aligned} \right\} \tag{1.17} $$

Next, with the aid of (1.15) and (1.17), we finally obtain

$$ \mathbf F=\mathbf f+S\mathbf v_s=S\mathbf v_n, \tag{1.18} $$

$$ \Pi_{ik}=\rho v_{si}v_{sk}+v_{si}p_k+v_{nk}p_i-\left[\varepsilon-TS-\mu\rho-\right. $$

$$ \left. -(\mathbf v_n-\mathbf v_s,\mathbf p)\right]\delta_{ik} \quad\text{(momentum flux);} \tag{1.19} $$

$$ \mathbf Q=\left(\mu+\frac{v_s^2}{2}\right)(\mathbf p+\rho\mathbf v_s)+ST\mathbf v_n+\mathbf v_n(\mathbf v_n\mathbf p) \quad\text{(energy flux).} \tag{1.20} $$

The expression standing in the square brackets in (1.19) is nothing other than the total pressure**)

$$ p=-\varepsilon+TS+\mu\rho+(\mathbf v_n-\mathbf v_s,\mathbf p). \tag{1.21} $$

*) In doing so one should remember that, owing to the condition $\operatorname{rot}\mathbf v_s=0$, the relation
$\mathbf v_n(\mathbf p\nabla)\mathbf v_s=\mathbf p(\mathbf v_n\nabla)\mathbf v_s$ holds.

**) The pressure, by definition, is equal to the derivative of the total energy of the liquid with respect to volume at constant values of the mass $\rho V$, total entropy $SV$, and total momentum $\mathbf pV$. According to (1.9), we have

$$ p=-\frac{\partial(\varepsilon V)}{\partial V} =-\varepsilon+TS+\mu\rho+(\mathbf v_n-\mathbf v_s,\mathbf p). $$

After substituting expressions (1.18), (1.19), and (1.20) into the corresponding equations, the complete system of hydrodynamic equations is represented in the following form:

\[ \dot{\rho}+\operatorname{div}\mathbf{j}=0, \tag{1.22} \]

\[ \frac{\partial}{\partial t}\mathbf{j}+\mathbf{v}_s\operatorname{div}\mathbf{j}+(\mathbf{j}\nabla)\mathbf{v}_s+\mathbf{p}\operatorname{div}\mathbf{v}_n+(\mathbf{v}_n\nabla)\mathbf{p}+\nabla p=0, \tag{1.23} \]

\[ \dot{S}+\operatorname{div}S\mathbf{v}_n=0, \tag{1.24} \]

\[ \dot{\mathbf{v}}_s+\nabla\left(\frac{v_s^2}{2}+\mu\right)=0. \tag{1.25} \]

Eliminating the quantity \(\mathbf{v}_s\) from equations (1.25) and (1.23), we obtain an equation replacing one of them:

\[ \dot{\mathbf{p}}+\mathbf{p}\operatorname{div}\mathbf{v}_n+(\mathbf{p}\nabla)\mathbf{v}_n+(\mathbf{v}_n\nabla)\mathbf{p}+[\mathbf{p}\cdot\operatorname{rot}\mathbf{v}_n]+S\nabla T=0. \tag{1.26} \]

From symmetry considerations it follows that the direction of the momentum \(\mathbf{p}\) coincides with the direction of the difference \(\mathbf{v}_n-\mathbf{v}_s\). We write the momentum \(\mathbf{p}\) in the form

\[ \mathbf{p}=\rho_n(\mathbf{v}_n-\mathbf{v}_s) \tag{1.27} \]

\(\rho_n\) is the density of the liquid associated with normal motion. The density of the normal part of the liquid, \(\rho_n\), is a function of the thermodynamic variables (\(\rho\) and \(S\)) and of the velocity difference \(\mathbf{v}_n-\mathbf{v}_s\). For small values of the velocity difference \(\mathbf{v}_n-\mathbf{v}_s\), the dependence of \(\rho_n\) on this difference may be neglected.

According to (1.3) and (1.27), for the total momentum \(\mathbf{j}\) we have the relation

\[ \mathbf{j}=\mathbf{p}+\rho\mathbf{v}_s=\rho_n\mathbf{v}_n+\rho_s\mathbf{v}_s; \tag{1.28} \]

\(\rho_s\) is the density of the liquid associated with superfluid motion:

\[ \rho_s=\rho-\rho_n. \tag{1.29} \]

The momentum flux tensor \(\Pi_{ik}\), taking (1.28) into account, can be rewritten in the form

\[ \Pi_{ik}=\rho_n v_{ni}v_{nk}+\rho_s v_{si}v_{sk}+p\delta_{ik}. \tag{1.30} \]

The chemical potential \(\mu\) is most simply expressed in the variables \(p\) and \(T\). According to (1.21), we have

\[ \rho d\mu=-S\,dT+dp-\mathbf{p}\,d(\mathbf{v}_n-\mathbf{v}_s). \tag{1.31} \]

In the case of small values of the velocity difference \((\mathbf{v}_n-\mathbf{v}_s)\), the expression for the potential \(\mu\) may be expanded in a series in this difference and one may restrict oneself to the first nonvanishing term. In agreement with (1.31), in this way we find

\[ \mu=\mu_0(p,T)-\frac{\rho_n}{2\rho}(\mathbf{v}_n-\mathbf{v}_s)^2. \tag{1.32} \]

The function \(\mu_0\) depends on the pressure and temperature and satisfies the thermodynamic identity

\[ \rho d\mu_0=-S dT+dp . \tag{1.33} \]

Let us now briefly consider the derivation of the hydrodynamic equations of helium II by the variational method. In order to make such a derivation consistent and to avoid the extra conditions that arose in C̆ilsel’s work\({}^{2}\), we shall use the somewhat modified method of Clebsch\({}^{5}\). We start from the Lagrangian function, which according to (1.8) can be written in the form

\[ L=\int \left(\frac{1}{2}\rho v_s^2+\mathbf{p}\mathbf{v}_s-\varepsilon'\right)\,dv\,dt . \tag{1.34} \]

Here we shall regard the internal energy \(\varepsilon'\) as a function of the density, entropy, and relative velocity \(\mathbf{v}_n-\mathbf{v}_s\). Our function \(\varepsilon'\) does not coincide with the internal energy (1.9) and differs from it by a total differential. Such a transformation, as is well known, is permissible for the Lagrangian function.

In order to obtain the complete system of hydrodynamic equations of a superfluid liquid, it is necessary to vary the Lagrangian function (1.34) with additional conditions. Two natural conditions are the continuity equations for the density \(\rho\) and entropy \(S\):

\[ \dot{\rho}+\operatorname{div}\mathbf{j}=0, \tag{1.35} \]

\[ \dot{S}+\operatorname{div}S\mathbf{v}_n=0, \tag{1.36} \]

Thus, here in these additional conditions we assume that all the entropy is contained in the normal part of the liquid and is carried by the normal motion with velocity \(\mathbf{v}_n\). However, these conditions are insufficient. Following\({}^{5}\), we supplement them with one more continuity condition for some function \(f\):

\[ \dot{f}+\operatorname{div}f\mathbf{v}_n=0. \tag{1.37} \]

The function \(f\) will not enter into the final equations. Its introduction at intermediate stages of the derivation makes it possible to avoid unreasonable consequences for quantities having physical meaning.

Let us find the conditions for an extremum of the integral (1.34) with the additional conditions (1.35)—(1.37). For this purpose we form the sum

\[ \int \left\{\left(\frac{1}{2}\rho v_s^2+\mathbf{p}\mathbf{v}_s-\varepsilon'\right)+ \alpha\left(\dot{\rho}+\operatorname{div}\mathbf{j}\right)+\right. \]

\[ \left. +\beta\left(\dot{S}+\operatorname{div}S\mathbf{v}_n\right)+ \gamma\left(\dot{f}+\operatorname{div}f\mathbf{v}_n\right)\right\}\,dv\,dt \]

\[ (\alpha,\ \beta,\ \gamma \text{ are certain functions of coordinates and time}). \]

We vary the integral obtained with respect to the variables \(\rho, S, \mathbf j, \mathbf v_s, \mathbf v_n\), and \(f\), and set the result equal to zero:

\[ \iint \left\{ \left(-\frac{v_s^2}{2}-\mu-\dot{\alpha}\right)\delta\rho +\left(\mathbf j-\rho\mathbf v_s-\frac{\partial\varepsilon}{\partial\mathbf v_s}\right)\delta\mathbf v_s +\right. \]
\[ \left. +\left(-S\nabla\beta-f\nabla\gamma-\frac{\partial\varepsilon}{\partial\mathbf v_n}\right)\delta\mathbf v_n +\left(\mathbf v_s-\nabla\alpha\right)\delta\mathbf j +\right. \]
\[ \left. +\delta S\left(-T-\dot{\beta}-\mathbf v_n\nabla\beta\right) +\delta f\left(-\dot{\gamma}-\mathbf v_n\nabla\gamma\right) \right\}\,dv\,dt=0. \tag{1,38} \]

We omit all total derivatives with respect to time and space. From (1,38) there follow the following six conditions:

\[ \frac{v_s^2}{2}+\mu+\dot{\alpha}=0, \tag{1,39} \]

\[ \mathbf j-\rho\mathbf v_s-\frac{\partial\varepsilon}{\partial\mathbf v_s}=0, \tag{1,40} \]

\[ S\nabla\beta+f\nabla\gamma+\frac{\partial\varepsilon}{\partial\mathbf v_n}=0, \tag{1,41} \]

\[ \mathbf v_s-\nabla\alpha=0, \tag{1,42} \]

\[ T+\dot{\beta}+\mathbf v_n\nabla\beta=0, \tag{1,43} \]

\[ \dot{\gamma}+\mathbf v_n\nabla\gamma=0. \tag{1,44} \]

Condition (1,42), equivalent to the condition \(\operatorname{rot}\mathbf v_s=0\), signifies the potentiality of the superfluid motion. It is easy to see that it arises only in the case when, in conditions (1,36) and (1,37), under the sign \(\operatorname{div}\) there appears the velocity \(\mathbf v_n\). This condition would be violated if the entropy flux contained a term depending on \(\mathbf v_s\). Thus condition (1,12) is closely connected with the definition of the entropy flux in the form \(S\mathbf v_n\).

Eliminating the parameter \(\alpha\) from equations (1,42) and (1,39), we obtain the equation of superfluid motion

\[ \dot{\mathbf v}_s=-\nabla\left(\mu+\frac{v_s^2}{2}\right). \tag{1,45} \]

Further, according to our definition of the function \(\varepsilon'\),

\[ \frac{\partial\varepsilon'}{\partial\mathbf v_n} = -\frac{\partial\varepsilon'}{\partial\mathbf v_s}. \]

Using this relation, we eliminate from conditions (1,40)—(1,44) the parameters \(\beta, \gamma\) and the function \(f\). In this way we obtain the equation of motion

\[ \dot{\mathbf p}+\mathbf p\,\operatorname{div}\mathbf v_n+\nabla(\mathbf p\mathbf v_n)-[\mathbf v_n\operatorname{rot}\mathbf p]+S\nabla T=0, \tag{1,46} \]

coinciding with equation (1.26). Equations (1.35), (1.36), (1.45) and (1.46) constitute the complete system of hydrodynamic equations of helium II, valid for any values of the velocities \(\mathbf v_n\) and \(\mathbf v_s\).

2. Dissipative Function for Liquid Helium II\(^6\)

The system of equations (1.22)—(1.25) describes the motion of liquid helium II in the absence of energy dissipation. Let us determine the form which the indicated equations acquire in the presence of dissipative processes. Naturally, nonequilibrium leads to the appearance of additional terms in all equations, with the exception of the continuity equation. This is explained by the fact that, in the absence of equilibrium, the concept of the flux of matter is indefinite; therefore the freedom arising in connection with this allows one to choose the vector \(\mathbf j\) in the same way as in the case when equilibrium is present. As a result the continuity equation retains its usual form

\[ \dot{\rho}+\operatorname{div}\mathbf j=0 . \tag{2,1} \]

To the remaining hydrodynamic equations (apart from the continuity equation) we shall add certain additional terms, taking account of dissipative processes, and we shall try to determine their concrete form

\[ \frac{\partial}{\partial t}j_i+\frac{\partial}{\partial x_k}\left(\Pi_{ik}+\tau_{ik}\right)=0, \tag{2,2} \]

\[ \dot{\mathbf v}_s+\nabla\left(\mu+\frac{v_s^2}{2}+h\right)=0 . \tag{2,3} \]

The additional terms \(\tau_{ik}\) and \(h\) have been included in the equations in such a way that equations (2,2) and (2,3) still preserve the form of continuity equations and, moreover, the condition \(\operatorname{rot}\mathbf v_s=0\) is fulfilled. As for the equation for the entropy, in the presence of dissipation it no longer has the form of a continuity equation. The total entropy in this case is not conserved, but increases. The rate of increase of entropy is determined by the dissipative function. We shall also determine the form of this function. We shall proceed from the fact that the total energy of the system

\[ \int E\,dv=\int\left(\frac{1}{2}\rho v_s^2+\mathbf v_s\mathbf p+\varepsilon\right)dv, \qquad \mathbf p=\mathbf j-\rho\mathbf v_s \tag{2,4} \]

is conserved. Therefore the time derivative of the energy per unit volume of the liquid, \(E\), must be equal to the divergence of some vector \(\mathbf Q'\) (the energy flux, which, naturally, is now no longer equal to \(\mathbf Q\) (1.29)).

The internal energy is determined by the identity (1.3). Differentiating expression (2.4) with respect to time, we obtain

\[ \dot E = \dot\rho\left(\mu+\frac{v_s^2}{2}\right) + (\mathbf v_s\dot{\mathbf j}) + (\mathbf v_n\dot{\mathbf p}) + T\dot S . \tag{2,5} \]

Next we express the time derivatives with the aid of equations (2,1)—(2,3); as a result of simple transformations we obtain

\[ \dot E=-\operatorname{div}\left\{ \mathbf j\left(\mu+\frac{v_s^2}{2}\right)+ST\mathbf v_n+(\mathbf v_n\mathbf p)\mathbf v_n+ \right. \]

\[ \left. +h(\mathbf j-\rho\mathbf v_n)+\tau\mathbf v_n \right\} -\frac{\partial}{\partial x_k}(v_{ni}\psi_{ik}) +T(\dot S+\operatorname{div}S\mathbf v_n)+ \]

\[ +h\,\operatorname{div}(\mathbf j-\rho\mathbf v_n) +\tau\operatorname{div}\mathbf v_n +\frac12\,\psi_{ik}\times \]

\[ \times\left( \frac{\partial v_{ni}}{\partial x_k} +\frac{\partial v_{nk}}{\partial x_i} -\frac23\,\delta_{ik}\frac{\partial v_{nl}}{\partial x_l} \right). \tag{2,6} \]

We have chosen the symmetric tensor \(\tau_{ik}\) in the form

\[ \tau_{ik}=-\tau\delta_{ik}+\mu_{ik}, \tag{2,7} \]

where \(\mu_{ik}\) is a symmetric tensor with zero trace. Under nonequilibrium conditions an additional term \(\mathbf q\) appears in the expression for the energy flux; its form we shall determine. Add and subtract the expression \(\operatorname{div}\mathbf q\) on the right-hand side of (2,6). In this way we find

\[ \dot E=-\operatorname{div}\left\{ \mathbf j\left(\mu+\frac{v_s^2}{2}\right)+ST\mathbf v_n+(\mathbf v_n\mathbf p)\mathbf v_n+h(\mathbf j-\rho\mathbf v_n)+ \right. \]

\[ \left. +\tau\mathbf v_n+\mathbf q+\boldsymbol{\nu} \right\} +T\left[\dot S+\operatorname{div}\left(S\mathbf v_n+\frac{\mathbf q}{T}\right)\right] +h\,\operatorname{div}(\mathbf j-\rho\mathbf v_n)+ \]

\[ +\tau\operatorname{div}\mathbf v_n +\frac{\mathbf q\nabla T}{T} +\frac12\,\mu_{ik}\left( \frac{\partial v_{ni}}{\partial x_k} +\frac{\partial v_{nk}}{\partial x_i} -\frac23\,\delta_{ik}\frac{\partial v_{nl}}{\partial x_l} \right). \]

\[ (\nu_k=v_{ni}\mu_{ik}). \tag{2,8} \]

From relation (2,8) there follows unambiguously the expression for the energy flux:

\[ Q'=\mathbf j\left(\mu+\frac{v_s^2}{2}\right)+ST\mathbf v_n+(\mathbf v_n\mathbf p)\mathbf v_n+ \]

\[ +h(\mathbf j-\rho\mathbf v_n)+\tau\mathbf v_n+\mathbf q+\boldsymbol{\nu} \tag{2,9} \]

and the equation determining the rate of change of entropy:

\[ T\left\{\dot S+\operatorname{div}\left(S\mathbf v_n+\frac{\mathbf q}{T}\right)\right\} =-\left\{ h\,\operatorname{div}(\mathbf j-\rho\mathbf v_n)+\tau\operatorname{div}\mathbf v_n+ \right. \]

\[ \left. +\frac{\mathbf q\nabla T}{T} +\frac12\,\mu_{ik}\left( \frac{\partial v_{ni}}{\partial x_k} +\frac{\partial v_{nk}}{\partial x_i} -\frac23\,\delta_{ik}\frac{\partial v_{nl}}{\partial x_l} \right) \right\}. \tag{2,10} \]

The expression on the right-hand side of equation (2,10) represents

represents the dissipative function for helium II:

\[ R=-\left\{h\,\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n)+\tau\,\operatorname{div}\mathbf{v}_n+\frac{\mathbf{q}\nabla T}{T} +\frac{1}{2}\mu_{ik}\left(\frac{\partial v_{ni}}{\partial x_k}+\frac{\partial v_{nk}}{\partial x_i} -\frac{2}{3}\delta_{ik}\frac{\partial v_{nl}}{\partial x_l}\right)\right\}. \tag{2,11} \]

If the spatial derivatives of the velocities and thermodynamic variables are small, then, in the first approximation, all additions in the equations \((\tau,\ \mu_{ik},\ h,\ \mathbf{q})\) are linear functions of the indicated derivatives. The dissipative function in this case, as is known, is an essentially positive quadratic form in the same derivatives. In order that this requirement be satisfied, the coefficients \(\tau,\ h,\ \mu_{ik}\) and \(\mathbf{q}\) must have the following form:

\[ \begin{aligned} \tau&=-\zeta_1\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n)-\zeta_2\operatorname{div}\mathbf{v}_n,\\ h&=-\zeta_3\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n)-\zeta_4\operatorname{div}\mathbf{v}_n,\\ \mu_{ik}&=-\eta\left(\frac{\partial v_{ni}}{\partial x_k}+\frac{\partial v_{nk}}{\partial x_i} -\frac{2}{3}\delta_{ik}\frac{\partial v_{nl}}{\partial x_l}\right),\\ \mathbf{q}&=-\chi\nabla T. \end{aligned} \tag{2,12} \]

By virtue of Onsager’s symmetry principle for the kinetic coefficients, the relation

\[ \zeta_1=\zeta_4 \tag{2,13} \]

holds.

The coefficients \(\zeta_1,\ \zeta_2,\ \zeta_3,\ \zeta_4\) have the meaning of coefficients of second viscosity. Thus in all there are three independent coefficients of second viscosity. The coefficient \(\eta\) has the meaning of a coefficient of first viscosity, and \(\chi\) is the coefficient of thermal conductivity of helium II. We now write, in final form, the equations of hydrodynamics of helium II with allowance for the dissipative terms:

\[ \begin{aligned} \frac{\partial}{\partial t}j_i+\frac{\partial\Pi_{ik}}{\partial x_k} &=\frac{\partial}{\partial x_k}\left\{\eta\left(\frac{\partial v_{ni}}{\partial x_k} +\frac{\partial v_{nk}}{\partial x_i} -\frac{2}{3}\delta_{ik}\frac{\partial v_{kl}}{\partial x_l}\right)\right\}\\ &\quad+\frac{\partial}{\partial x}\left\{\zeta_1\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) +\zeta_2\operatorname{div}\mathbf{v}_n\right\},\\ \dot{\mathbf{v}}_s+\nabla\left(\mu+\frac{v_s^2}{2}\right) &=\nabla\left\{\zeta_3\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) +\zeta_4\operatorname{div}\mathbf{v}_n\right\},\\ \dot{\rho}+\operatorname{div}\mathbf{j}&=0,\qquad \dot{S}+\operatorname{div}\left(S\mathbf{v}_n+\frac{\mathbf{q}}{T}\right)=\frac{1}{T}R. \end{aligned} \tag{2,14} \]

The dissipative function \(R\) is equal to

\[ \begin{aligned} R={}&\zeta_2(\operatorname{div}\mathbf{v}_n)^2 +\zeta_3(\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n))^2 +2\zeta_1\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n)\operatorname{div}\mathbf{v}_n\\ &+\chi\frac{(\nabla T)^2}{T} +\frac{1}{2}\eta\left(\frac{\partial v_{ni}}{\partial x_k} +\frac{\partial v_{nk}}{\partial x_i} -\frac{2}{3}\delta_{ik}\frac{\partial v_{nl}}{\partial x_l}\right)^2. \end{aligned} \tag{2,15} \]

It follows from the positivity of the function \(R\) that the viscosity coefficient \(\eta\) and the thermal conductivity \(\chi\) are essentially positive quantities. In addition, the coefficients \(\xi_2\) and \(\xi_3\) must also be positive. As for the coefficient \(\xi_1\), its sign is not determined; the magnitude of the coefficient \(\xi_1\) must satisfy the inequality

\[ \xi_1^2 \leqslant \xi_2 \xi_3 . \tag{2,16} \]

3. Sound in helium II

In a sound wave the velocities \(\mathbf v_n\) and \(\mathbf v_s\) are assumed to be small*), and the thermodynamic quantities almost equal to their equilibrium values. The propagation of sound in helium II is described by the system of hydrodynamic equations (1,22)—(1,25), which in the present case can be linearized. After linearization the indicated equations take the form

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\mathbf j=0, \tag{3,1} \]

\[ \frac{\partial \rho\sigma}{\partial t}+\rho\sigma\,\operatorname{div}\mathbf v_n=0 \quad (\rho\sigma=S), \tag{3,2} \]

\[ \frac{\partial \mathbf j}{\partial t}+\nabla p=0, \tag{3,3} \]

\[ \frac{\partial \mathbf v_s}{\partial t}+\nabla\mu=0. \tag{3,4} \]

Eliminating the momentum \(\mathbf j\) from equations (3,1) and (3,3), we obtain

\[ \frac{\partial^2\rho}{\partial t^2}=\Delta p . \tag{3,5} \]

Next, from the three equations (3,2), (3,3), and (3,4) we eliminate the velocities \(\mathbf v_n\) and \(\mathbf v_s\). To do this, we differentiate equation (3,2) with respect to time, and apply the operation \(\operatorname{div}\) to equations (3,3) and (3,4). Eliminating from the equations thus obtained the terms \(\dfrac{\partial}{\partial t}\operatorname{div}\mathbf v_n\) and \(\dfrac{\partial}{\partial t}\operatorname{div}\mathbf v_s\), we obtain

\[ \rho_s \Delta\mu-\Delta p+\frac{\rho_n}{\rho\sigma}\frac{\partial^2}{\partial t^2}(\rho\sigma)=0 . \tag{3,6} \]

Let us express in this equation the derivative \(\dfrac{\partial^2\rho}{\partial t^2}\) with the aid of equation (3,5) and use the thermodynamic identity (1,31). As a result we find

\[ \frac{\partial^2\sigma}{\partial t^2} = \frac{\rho_s}{\rho_n}\sigma^2\Delta T . \tag{3,7} \]

*) What is meant is the smallness of the velocities \(\mathbf v_n\) and \(\mathbf v_s\) in comparison with the speed of sound.

Equations (3.6) and (3.7) determine the variation of thermodynamic quantities in a sound wave.

Let us pass in the indicated equations to the independent variables \(p\) and \(T\), which we represent in the form \(p=p_0+p'\), \(T=T_0+T'\). Quantities with subscript zero are equilibrium values, and those with primes are their changes in the sound wave. As a result, equations (3.6) and (3.7) take the form

\[ \frac{\partial \rho}{\partial p}\frac{\partial^2 p'}{\partial t^2} -\Delta p' +\frac{\partial \rho}{\partial T}\frac{\partial^2 T'}{\partial t^2} =0, \tag{3.8} \]

\[ \frac{\partial \sigma}{\partial p}\frac{\partial^2 p'}{\partial t^2} +\frac{\partial \sigma}{\partial T}\frac{\partial^2 T'}{\partial t^2} -\frac{\sigma^2\rho_s}{\rho_n}\Delta T' =0. \tag{3.9} \]

We seek a solution of the system (3.8)—(3.9) representing a plane wave traveling in some direction. In such a wave the quantities \(p'\) and \(T'\) vary according to the law \(e^{i\omega(t-x/u)}\) (we choose the \(x\)-axis in the direction of propagation of the wave; \(\omega\) is the frequency, \(u\) the speed of sound). With this law of variation of the quantities \(p'\) and \(T'\), the system of equations (3.8)—(3.9) becomes

\[ \left(\frac{\partial \rho}{\partial p}u^2-1\right)p' +\frac{\partial \rho}{\partial T}u^2T' =0, \tag{3.10} \]

\[ \frac{\partial \sigma}{\partial p}u^2p' +\left(\frac{\partial \sigma}{\partial T}u^2-\frac{\sigma^2\rho_s}{\rho_n}\right)T' =0. \tag{3.11} \]

The condition for compatibility of these equations is the vanishing of their determinant. Expanding this determinant, we obtain the biquadratic equation

\[ u^4\frac{\partial(\sigma,\rho)}{\partial(T,p)} -u^2\left(\frac{\partial \sigma}{\partial T} +\sigma^2\frac{\rho_s}{\rho_n}\frac{\partial \rho}{\partial p}\right) +\frac{\rho_s}{\rho_n}\sigma^2 =0, \tag{3.12} \]

which, after simple transformations, takes the form

\[ u^4-u^2\left[ \left(\frac{\partial p}{\partial \rho}\right)_\sigma +\frac{\rho_s}{\rho_n}\sigma^2 \left(\frac{\partial T}{\partial \sigma}\right)_\rho \right] +\frac{\rho_s}{\rho_n}\sigma^2 \left(\frac{\partial T}{\partial \sigma}\right)_\rho \left(\frac{\partial p}{\partial \rho}\right)_T =0. \tag{3.13} \]

Equation (3.13) determines the two possible speeds of sound in helium II. The coefficient of thermal expansion \(\left(\dfrac{\partial \rho}{\partial T}\right)_p\) proves in practice to be very small for all bodies. In helium II the magnitude of this coefficient is anomalously small. Therefore, according to the known thermodynamic relations, the heat capacities \(c_p\) and \(c_v\) in helium II may practically be regarded as equal. But in this case the derivatives \(\left(\dfrac{\partial p}{\partial \rho}\right)_T\) and \(\left(\dfrac{\partial p}{\partial \rho}\right)_\sigma\), related by the relation

\[ \left(\frac{\partial p}{\partial \rho}\right)_\sigma = \frac{c_p}{c_v} \left(\frac{\partial p}{\partial \rho}\right)_T, \]

may also, with a high degree of accuracy, be regarded as equal. This circumstance considerably simplifies equation (3.13), its roots

in this case are equal to

\[ u_1=c=\sqrt{\left(\frac{\partial p}{\partial \rho}\right)_\sigma}, \tag{3,14} \]

\[ u_2=\sqrt{\frac{\sigma^2 \rho_s}{\rho_n\left(\frac{\partial \sigma}{\partial T}\right)}} . \tag{3,15} \]

The first root determines the velocity of ordinary (first) sound in helium II. With this velocity, according to equation (3,5), pressure (density) oscillations propagate in helium II. The second root \(u_2\) determines the velocity of the so-called second sound. According to equation (3,7), temperature (entropy) oscillations propagate in helium II with this velocity. The possibility of propagation of undamped temperature waves is a specific property of helium II. The temperature dependence of the velocity of second sound, calculated from formula (3,15), is shown graphically in Fig. 1. At the \(\lambda\)-point \(\rho_s=0\), and the velocity \(u_2\) also becomes zero. At sufficiently low temperatures (below \(0.5^\circ\mathrm{K}\)), when all thermodynamic quantities are determined only by phonons, the magnitude of the velocity \(u_2\) tends to the limit \(c/\sqrt{3}\). Experimentally, second sound in helium II was discovered by V. Peshkov\({}^{7}\). Experimental values of the velocity of second sound for various temperatures, obtained by V. Peshkov\({}^{7}\), are given in Fig. 1. There, for comparison, are also given the later data of Maurer and Herlin\({}^{8}\), who measured the velocity of second sound by a pulse method down to a temperature of \(0.86^\circ\mathrm{K}\). As is evident from the indicated figure, the theoretical and experimental values of the velocity of second sound are in good agreement. Measurements of the velocity of second sound carried out by Atkinson and Osborne\({}^{9}\) down to a temperature of \(0.2^\circ\mathrm{K}\) also gave good agreement with the theoretical predictions; the authors assert that at a temperature of \(0.2^\circ\mathrm{K}\) the velocity of second sound did indeed prove to be equal to \(c/\sqrt{3}\). However, the cited work can hardly be regarded as reliable. At such low temperatures the mean free path of phonons (there are practically no rotons in this temperature region) noticeably exceeds the wavelengths of sound waves, and therefore propagation of sound oscillations is impossible. More convincing appear the results of Kramers, van den Berg, and Gorter\({}^{10}\), who for the velocity of the wavefront of second sound near \(0.1^\circ\mathrm{K}\) obtained a value equal to \(230\ \mathrm{m/sec}\), i.e., a velocity close to that of ordinary sound. At such low temperatures the phonons emitted by the heater reach the detecting thermometer practically without collisions, and the velocity of phonons, as is known, is equal to the velocity of ordinary sound \(c\).

As is known, the existence of second sound in helium II follows not only from Landau’s theory, but also from Tisza’s theory\({}^{11}\). In

temperatures above \(1^\circ\mathrm{K}\) both theories give the same temperature dependence for the velocity of second sound. This coincidence is explained by the fact that the essentially incorrect Tisza equations in the first approximation, with a proper choice of arbitrary constants, turn out to coincide with Landau’s equations.

Fig. 1. Temperature dependence of the velocity of second sound: ● — Peshkov\(^{7}\); ○ — Maurer and Herlin\(^{8}\); — theoretical values.

In the region of temperatures lying below \(1^\circ\mathrm{K}\), the two theories differ in their predictions concerning the course of the curves for the velocity of second sound: according to Landau’s theory, near \(1^\circ\mathrm{K}\) there is a minimum, and with further lowering of the temperature the velocity \(u_2\) increases; according to Tisza, however, the sound velocity \(u_2\), beginning at \(1.6^\circ\mathrm{K}\), should decrease monotonically as the temperature is lowered. All data on the velocity of second sound fully confirm the presence of a minimum on the curve of the temperature dependence and thereby refute Tisza’s theory.

4. Sound Absorption in Helium II \({}^{12,13}\)

Relaxation times characterizing the creation of excitations in helium II

When the state of a body changes rapidly, for example when the temperature is raised or lowered, thermodynamic equilibrium in it is disturbed. The internal processes arising in connection with this tend to restore equilibrium. If the relaxation time of these processes is sufficiently small (i.e. the processes occur rapidly), then equilibrium is restored so quickly that it has time to follow the change of state. In the case where processes with large relaxation times are present, the establishment of equilibrium takes place comparatively slowly and does not keep up with the change of state. Since the processes by which equilibrium is established are irreversible, they are consequently accompanied by dissipation of energy. In helium II there are slow processes of creation and absorption of elementary excitations (phonons and rotons). Here we shall clarify the influence of these processes on the propagation of sound in helium II. The propagation of sound in helium II is described by the linearized system of hydrodynamic equations \((3,1)\)—\((3,4)\). In the equilibrium state all thermodynamic functions in equations \((3,1)\)—\((3,4)\) depend only on two variables, which we choose to be \(\rho\) and \(S\). In this case, when sound oscillations of not very high frequency propagate in helium II, one may assume that the establishment of equilibrium has time to follow the change of state. Consequently, at each instant of time the state is determined by equilibrium thermodynamic functions. A different picture occurs in the propagation of sound waves of such high frequency that equilibrium does not have time to be established and to follow the change of state. In this case the number of phonons and rotons at each instant of time is not equal to their equilibrium values, and consequently it is necessary to take into account the dependence of the thermodynamic functions not only on \(\rho\) and \(S\), but also on the number of phonons and rotons (or the corresponding chemical potentials). The system of equations \((3,1)\)—\((3,4)\) is now not complete. It must be supplemented by the equations (see \((15,5,1)\), \((15,6,1)\))*

\[ \begin{aligned} \dot N_p+\operatorname{div} N_p \mathbf v_n &=-\gamma_{pp}\mu_p+\gamma_{p\phi}\mu_\phi,\\ \dot N_\phi+\operatorname{div} N_\phi \mathbf v_n &=\gamma_{\phi p}\mu_p-\gamma_{\phi\phi}\mu_\phi, \end{aligned} \tag{4,1} \]

which describe the change in the number of phonons and rotons. Eliminating the velocities \(\mathbf v_n\) and \(\mathbf v_s\) from equations \((3,1)\)—\((3,4)\), \((4,1)\), we obtain

\[ \frac{d^2\rho}{dt^2}=\Delta p, \tag{4,2} \]

\[ \frac{d^2S}{dt^2}=\frac{S}{\rho_n}\left(\Delta p-\rho_s\Delta\mu\right), \tag{4,3} \]

* References to the author’s first paper in UFN, 59, issue 4, are denoted by the number 1.

and, moreover,

\[ \left. \begin{aligned} \frac{\partial N_p}{\partial t}-\frac{N_p}{S}\frac{dS}{dt} &=-\gamma_{pp}\mu_p+\gamma_{p\phi}\mu_\phi,\\ \frac{\partial N_\phi}{\partial t}-\frac{N_\phi}{S}\frac{dS}{dt} &=\gamma_{\phi p}\mu_p-\gamma_{\phi\phi}\mu_\phi . \end{aligned} \right\} \tag{4.4} \]

In the system (4.2)—(4.4) all thermodynamic functions can be expressed in terms of the four variables \(\rho\), \(S\), \(\mu_p\), and \(\mu_\phi\). The entropy \(S\) and the density \(\rho\) can be represented as sums of constant equilibrium values and small additions, caused by the sound wave, which we shall denote by the same letters, but with a prime (\(S'\) and \(\rho'\)). The solution of the system should be sought in the form of a plane wave, in which \(S'\), \(\rho'\), \(\mu_p\), and \(\mu_\phi\) are proportional to the factor \(e^{i\omega(t-x/u)}\) (\(u\) is the speed of sound). The condition of compatibility of our system will then be written as the equality to zero of a certain determinant of the fourth order, whose solution gives us the values of the speed of sound\(^*\).

In order not to operate with a fourth-order determinant, we shall proceed somewhat differently. Namely, let us express all functions in equations (4.2) and (4.3) in terms of the entropy \(S\) and the density \(\rho\). In doing this, in all differentiation operations one must take into account the dependence of the functions on the variables \(\mu_p\) and \(\mu_\phi\). Thus, the partial derivative of some thermodynamic function \(f\) with respect to \(\rho\) or \(S\) is expanded as follows:

\[ \left(\frac{\partial f}{\partial \rho}\right)_S = \left(\frac{\partial f}{\partial \rho}\right)_{S,\mu_p,\mu_\phi} + \frac{\partial f}{\partial \mu_p}\frac{\partial \mu_p}{\partial \rho} + \frac{\partial f}{\partial \mu_\phi}\frac{\partial \mu_\phi}{\partial \rho}, \tag{4.5} \]

\[ \left(\frac{\partial f}{\partial S}\right)_\rho = \left(\frac{\partial f}{\partial S}\right)_{\rho,\mu_p,\mu_\phi} + \frac{\partial f}{\partial \mu_p}\frac{\partial \mu_p}{\partial S} + \frac{\partial f}{\partial \mu_\phi}\frac{\partial \mu_\phi}{\partial S}. \tag{4.6} \]

The derivatives taken at constant (equal to zero) values of the chemical potentials are the equilibrium values of the corresponding derivatives. The derivatives of the chemical potentials \(\mu_p\) and \(\mu_\phi\) with respect to \(\rho\) and \(S\) that enter into expressions (4.5) and (4.6) we shall find with the aid of equations (4.3) and (4.4), which establish the connection between \(\mu_p\) and \(\mu_\phi\), on the one hand, and the small changes of the density \(\rho\) and entropy \(S\) in a plane wave, on the other hand.

Expressing, as already indicated, the small changes of \(\rho\) and \(S\) in the sound wave in the form proportional to the factor \(e^{i\omega(t-x/u)}\), we obta—

\(^*\) We draw attention to the circumstance that, although the system (4.2)—(4.4) leads to a determinant of the fourth order, nevertheless the corresponding equation determining the values of the velocity turns out to be biquadratic (quadratic with respect to \(u^2\)). This is connected with the fact that the pair of equations (4.4), which takes into account the change in the number of elementary excitations, does not contain differentiation of the variables with respect to the coordinates, and therefore, upon substituting the solutions in the form proportional to \(e^{i\omega(t-x/u)}\), the velocity does not appear in these equations. Thus the square of the velocity occurs only in the first two rows of the resulting fourth-order determinant, and the corresponding characteristic equation with respect to \(u^2\) turns out to be quadratic.

than the condition of solvability of the corresponding pair of equations (4.2) and (4.3) is the vanishing of the following second-order determinant:

\[ \left| \begin{array}{cc} u^2-\left(\dfrac{\partial p}{\partial \rho}\right)_S & \left(\dfrac{\partial p}{\partial S}\right)_\rho \\[1.2em] u^2-\rho_S\left(\dfrac{\partial \lambda}{\partial \rho}\right)_S & \dfrac{\rho_n}{S}u^2+\rho_S\left(\dfrac{\partial p}{\partial S}\right)_\rho \end{array} \right|=0. \tag{4,7} \]

The quadratic equation obtained from (4,7) contains terms of different orders. All terms containing the derivative \(\dfrac{\partial p}{\partial S}\) turn out to be small. This is connected with the fact that in helium II the heat capacities \(c_p\) and \(c_v\) are close to one another at all temperatures, while the derivative \(\left(\dfrac{\partial p}{\partial S}\right)_\rho\), according to known thermodynamic relations, is proportional to \(\sqrt{c_p-c_v}\). This circumstance makes it possible to write the roots of equation (4,7) in the very compact form

\[ u_1^2=\left(\frac{\partial p}{\partial \rho}\right)_S+ \left(\frac{\partial p}{\partial S}\right)_\rho \frac{S}{\rho}, \tag{4,8} \]

\[ u_2^2=\frac{\rho_s}{\rho_n}S\left[ \frac{1}{\rho}\left(\frac{\partial p}{\partial S}\right)_\rho- \left(\frac{\partial \lambda}{\partial S}\right)_\rho \right]. \tag{4,9} \]

From equation (4,4) one can express the chemical potentials \(\mu_p\) and \(\mu_\phi\) in terms of the small changes of entropy \(S'\) and density \(\rho'\). To determine \(\mu_p\) and \(\mu_\phi\), from (4,4) we obtain a pair of equations

\[ \left(i\omega\frac{\partial N_p}{\partial \mu_p}+\gamma_{pp}\right)\mu_p -\gamma_{p\phi}\mu_\phi = -i\omega\left( \frac{\partial N_p}{\partial \rho}\rho' +\frac{\partial N_p}{\partial S}S' -\frac{N_p}{S}S' \right), \]

\[ -\gamma_{\phi p}\mu_p+ \left(i\omega\frac{\partial N_\phi}{\partial \mu_\phi}+\gamma_{\phi\phi}\right)\mu_\phi = -i\omega\left( \frac{\partial N_\phi}{\partial \rho}\rho' +\frac{\partial N_\phi}{\partial S}S' -\frac{N_\phi}{S}S' \right). \]

Hence we easily find

\[ \mu_p=\frac{i\omega}{D(\omega)} \left\{ \left(i\omega\frac{\partial N_\phi}{\partial \mu_\phi}+\gamma_{\phi\phi}\right) \left( \frac{N_p}{S}S' -\frac{\partial N_p}{\partial S}S' -\frac{\partial N_p}{\partial \rho}\rho' \right) + \right. \]

\[ \left. +\gamma_{p\phi} \left( \frac{N_\phi}{S}S' -\frac{\partial N_\phi}{\partial S}S' -\frac{\partial N_\phi}{\partial \rho}\rho' \right) \right\}, \tag{4,10} \]

\[ \mu_\phi=\frac{i\omega}{D(\omega)} \left\{ \left(i\omega\frac{\partial N_p}{\partial \mu_p}+\gamma_{pp}\right) \left( \frac{N_\phi}{S}S' -\frac{\partial N_\phi}{\partial S}S' -\frac{\partial N_\phi}{\partial \rho}\rho' \right) + \right. \]

\[ \left. +\gamma_{\phi p} \left( \frac{N_p}{S}S' -\frac{\partial N_p}{\partial S}S' -\frac{\partial N_p}{\partial \rho}\rho' \right) \right\}, \tag{4,11} \]

where

\[ D(\omega)= \left(i\omega\frac{\partial N_p}{\partial \mu_p}+\gamma_{pp}\right) \left(i\omega\frac{\partial N_\phi}{\partial \mu_\phi}+\gamma_{\phi\phi}\right) -\gamma_{p\phi}\gamma_{\phi p}. \]

The derivatives of the chemical potentials with respect to \(\rho\) and \(S\), on which, according to (4,5) and (4,6), the expressions for the velocities \(u_1\) and \(u_2\) depend, are computed with the aid of the relations (4,10) and (4,11)

\[ \left(\frac{\partial \mu_p}{\partial \rho}\right)_S = \frac{i\omega}{D(\omega)} \left\{ -\left(i\omega\frac{\partial N_\phi}{\partial \mu_\phi} +\gamma_{\phi\phi}\right) \left(\frac{\partial N_p}{\partial \rho}\right)_S + \gamma_{p\phi} \left(\frac{\partial N_\phi}{\partial \rho}\right)_S \right\}, \]

\[ \left(\frac{\partial \mu_\phi}{\partial \rho}\right)_S = \frac{i\omega}{D(\omega)} \left\{ -\left(i\omega\frac{\partial N_p}{\partial \mu_p} +\gamma_{pp}\right) \left(\frac{\partial N_\phi}{\partial \rho}\right)_S + \gamma_{\phi p} \left(\frac{\partial N_p}{\partial \rho}\right)_S \right\}, \tag{4,12} \]

\[ \left(\frac{\partial \mu_p}{\partial S}\right)_\rho = \frac{i\omega}{D(\omega)} \left\{ \left(i\omega\frac{\partial N_\phi}{\partial \mu_\phi} +\gamma_{\phi\phi}\right) \left(\frac{N_p}{S}-\frac{\partial N_p}{\partial S}\right) - \gamma_{p\phi} \left(\frac{N_\phi}{S}-\frac{\partial N_\phi}{\partial S}\right) \right\}, \]

\[ \left(\frac{\partial \mu_\phi}{\partial S}\right)_\rho = \frac{i\omega}{D(\omega)} \left\{ \left(i\omega\frac{\partial N_p}{\partial \mu_p} +\gamma_{pp}\right) \left(\frac{N_\phi}{S}-\frac{\partial N_\phi}{\partial S}\right) - \gamma_{\phi p} \left(\frac{N_p}{S}-\frac{\partial N_p}{\partial S}\right) \right\}. \tag{4,13} \]

After the expressions (4,12)—(4,13) for the derivatives of the thermodynamic quantities have been obtained, the calculation of the velocities \(u_1\) and \(u_2\) no longer presents any difficulty. According to (4,8), (4,9), (4,12), and (4,13), we have

\[ u_1^2= \left[ \left(\frac{\partial p}{\partial \rho}\right)_S + \frac{S}{\rho} \left(\frac{\partial p}{\partial S}\right)_\rho \right]_{\mu_p=\mu_\phi=0} + \frac{i\omega}{D(\omega)} \left[ \left(i\omega\frac{\partial N_\phi}{\partial \mu_\phi} +\gamma_{\phi\phi}\right) \left(\frac{\partial p}{\partial \mu_p}\right)^2 + 2\gamma_{p\phi} \frac{\partial p}{\partial \mu_p} \frac{\partial p}{\partial \mu_\phi} + \left(i\omega\frac{\partial N_p}{\partial \mu_p} +\gamma_{pp}\right) \left(\frac{\partial p}{\partial \mu_\phi}\right)^2 \right], \tag{4,14} \]

\[ u_2^2= \frac{\rho_s}{\rho_n}S \left[ \frac{1}{\rho} \left(\frac{\partial p}{\partial S}\right)_\rho - \left(\frac{\partial \mu}{\partial S}\right)_\rho \right]_{\mu_p=\mu_\phi=0} + \]

\[ + \frac{i\omega}{D(\omega)} \left[ \left(i\omega\frac{\partial N_\phi}{\partial \mu_\phi} +\gamma_{\phi\phi}\right) \left(\frac{\partial p}{\partial \mu_p} -\rho\frac{\partial \mu}{\partial \mu_p}\right)^2 + \right. \]

\[ \left. + 2\gamma_{p\phi} \left(\frac{\partial p}{\partial \mu_p} -\rho\frac{\partial \mu}{\partial \mu_p}\right) \left(\frac{\partial p}{\partial \mu_\phi} -\rho\frac{\partial \mu}{\partial \mu_\phi}\right) + \right. \]

\[ \left. + \left(i\omega\frac{\partial N_p}{\partial \mu_p} +\gamma_{pp}\right) \left(\frac{\partial p}{\partial \mu_\phi} -\rho\frac{\partial \mu}{\partial \mu_\phi}\right)^2 \right]. \tag{4,15} \]

The values of the velocities of first and second sound for chemical potentials equal to zero are the equilibrium values, correspondi-

…ing the propagation in helium II of sound with a frequency close to zero. Denoting the indicated values by the additional subscript zero, from (4.14) and (4.15) we obtain

\[ u_{10}^{2} = \left[ \left(\frac{\partial p}{\partial \rho}\right)_{S} + \frac{S}{\rho} \left(\frac{\partial p}{\partial S}\right)_{\rho} \right] = \left(\frac{\partial p}{\partial \rho}\right)_{\sigma}, \tag{4.16} \]

\[ u_{20}^{2} = \frac{\rho_s}{\rho_n}S \left[ \frac{1}{\rho} \left(\frac{\partial p}{\partial S}\right)_{\rho} - \left(\frac{\partial \lambda}{\partial \rho}\right)_{\rho} \right] = \frac{\rho_s}{\rho\rho_n}S^2 \left(\frac{\partial T}{\partial S}\right)_{\rho} \tag{4.17} \]

(where \(\sigma=S/\rho\)). The expressions (4.16) and (4.17) coincide with the equilibrium values obtained. From expressions (4.14) and (4.15) it follows that the velocities \(u_1\) and \(u_2\) are complex quantities. Consequently, the wave vectors, related to the frequency by \(k=\omega/u\), will also be complex quantities. The complexity of the wave vector, as is well known, is a formal expression of the fact that sound absorption takes place. The real part of the wave vector determines the change of the phase of the oscillations with distance, while the imaginary part is simply the absorption coefficient.

In the case of high frequencies the values \(u_1^2\) and \(u_2^2\), according to (4.14) and (4.15), tend to certain values, respectively equal to

\[ u_{1\infty}^{2} = u_{10}^{2} + \frac{1}{\rho} \left[ \frac{ \left(\dfrac{\partial p}{\partial \mu_p}\right)^2 }{ \left(\dfrac{\partial N_p}{\partial \mu_p}\right) } + \frac{ \left(\dfrac{\partial p}{\partial \mu_\phi}\right)^2 }{ \left(\dfrac{\partial N_\phi}{\partial \mu_\phi}\right) } \right]_{\rho,S}, \tag{4.18} \]

\[ u_{2\infty}^{2} = u_{20}^{2} + \frac{\rho_s}{\rho\rho_n} \left[ \frac{ \left( \dfrac{\partial p}{\partial \mu_p} - \rho\dfrac{\partial \lambda}{\partial \mu_p} \right)^2 }{ \left(\dfrac{\partial N_p}{\partial \mu_p}\right) } + \frac{ \left( \dfrac{\partial p}{\partial \mu_\phi} - \rho\dfrac{\partial \lambda}{\partial \mu_\phi} \right)^2 }{ \left(\dfrac{\partial N_\phi}{\partial \mu_\phi}\right) } \right]_{\rho,S}. \tag{4.19} \]

With the velocities determined by expressions (4.18) and (4.19), sound would propagate with so high a frequency that the number of phonons and rotons would not have time to change and, consequently, would be constant. The kinetic coefficients \(\gamma\) occurring in the equations of the present paragraph are expressed in terms of the coefficients \(\Gamma\) by (relations 15,13,I)

\[ \gamma_{pp}=\Gamma_p+\Gamma_{\phi p},\qquad \gamma_{p\phi}=\gamma_{\phi p}=\Gamma_{\phi p},\qquad \gamma_{\phi\phi}=\Gamma_\phi+\Gamma_{\phi p}. \]

In what follows it is more convenient, instead of the kinetic coefficients \(\gamma\) and \(\Gamma\), to use relaxation times, which are defined in the following way:

\[ \frac{1}{\theta_p} = \Gamma_p \left(\frac{\partial \mu_p}{\partial N_p}\right)_{\rho,S}, \qquad \frac{1}{\theta_{p\phi}} = \Gamma_{\phi p} \left(\frac{\partial \mu_p}{\partial N_p} + \frac{\partial \mu_\phi}{\partial N_\phi}\right)_{\rho,S}, \]

\[ \frac{1}{\theta_\phi} = \Gamma_\phi \left(\frac{\partial \mu_\phi}{\partial N_\phi}\right)_{\rho,S}. \tag{4.20} \]

With allowance for (4.20), expressions (4.14) and (4.15) can be rewritten in the form

\[ \left. \begin{gathered} u_1^2=u_{10}^2+\frac{i\omega}{\rho}\frac{A_1(\omega)}{D(\omega)},\\[6pt] A_1(\omega)=\left\{ \frac{\left(i\omega+\dfrac{1}{\theta_\phi}\right) \left(\dfrac{\partial p}{\partial \varkappa_{\rm p}}\right)^2} {\dfrac{\partial N_{\rm p}}{\partial \varkappa_{\rm p}}} +\frac{1}{\theta_{\phi{\rm p}}} \frac{\left(\dfrac{\partial p}{\partial \varkappa_{\rm p}}+ \dfrac{\partial p}{\partial \varkappa_\phi}\right)^2} {\dfrac{\partial N_{\rm p}}{\partial \varkappa_{\rm p}}+ \dfrac{\partial N_\phi}{\partial \varkappa_\phi}} +\frac{\left(i\omega+\dfrac{1}{\theta_{\rm p}}\right) \left(\dfrac{\partial p}{\partial \varkappa_\phi}\right)^2} {\dfrac{\partial N_\phi}{\partial \varkappa_\phi}} \right\}_{\rho,S}, \\[8pt] D(\omega)=\left\{ \left(i\omega+\frac{1}{\theta_{\rm p}}\right) \left(i\omega+\frac{1}{\theta_\phi}\right) +\frac{1}{\theta_{{\rm p}\phi}} \left[ i\omega+ \frac{\dfrac{1}{\theta_{\rm p}}\dfrac{\partial N_{\rm p}}{\partial \varkappa_{\rm p}} +\dfrac{1}{\theta_\phi}\dfrac{\partial N_\phi}{\partial \varkappa_\phi}} {\dfrac{\partial N_{\rm p}}{\partial \varkappa_{\rm p}}+ \dfrac{\partial N_\phi}{\partial \varkappa_\phi}} \right] \right\}_{\rho,S}. \end{gathered} \right\} \tag{4.21} \]

\[ \begin{gathered} u_2^2=u_{20}^2+\frac{i\omega}{\rho}\frac{A_2(\omega)}{D(\omega)},\\[6pt] A_2(\omega)=\left\{ \frac{\left(i\omega+\dfrac{1}{\theta_\phi}\right) \left(\dfrac{\partial p}{\partial \varkappa_{\rm p}} -\rho\,\dfrac{\partial \mu}{\partial \varkappa_{\rm p}}\right)^2} {\dfrac{\partial N_{\rm p}}{\partial \varkappa_{\rm p}}} + \frac{1}{\theta_{{\rm p}\phi}} \frac{\left( \dfrac{\partial p}{\partial \varkappa_{\rm p}}+ \dfrac{\partial p}{\partial \varkappa_\phi} -\rho\,\dfrac{\partial \mu}{\partial \varkappa_{\rm p}} -\rho\,\dfrac{\partial \mu}{\partial \varkappa_\phi} \right)^2} {\dfrac{\partial N_{\rm p}}{\partial \varkappa_{\rm p}}+ \dfrac{\partial N_\phi}{\partial \varkappa_\phi}} + \frac{\left(i\omega+\dfrac{1}{\theta_{\rm p}}\right) \left(\dfrac{\partial p}{\partial \varkappa_\phi} -\rho\,\dfrac{\partial \mu}{\partial \varkappa_\phi}\right)^2} {\dfrac{\partial N_\phi}{\partial \varkappa_\phi}} \right\}_{\rho,S}. \end{gathered} \tag{4.22} \]

These expressions determine the dependence of the velocities of first and second sound on frequency, due to the slow processes of creation of elementary excitations in helium II.

Let us consider the region of low frequencies satisfying the conditions

\[ \omega\theta_\phi\ll 1,\qquad \omega\theta_{{\rm p}\phi}\ll 1. \tag{4.23} \]

In view of the smallness of the coefficient \(\Gamma_p\), we shall neglect in (4.21) and (4.22) all terms associated with the five-roton process. Taking this into account, in the indicated frequency range the expressions (4.21) and (4.22) are considerably simplified:

\[ u_1^2=u_{10}^2+\frac{i\omega}{\rho} \left[ \theta_{\phi} \left( \frac{\partial p}{\partial c_p} + \frac{\partial p}{\partial c_\phi} \right)^2 \frac{\partial \lambda_\phi}{\partial N_\phi} + \theta_{p\phi} \left( \frac{\partial p}{\partial c_p} \right)^2 \left( \frac{\partial \lambda_p}{\partial N_p} + \frac{\partial \lambda_\phi}{\partial N_\phi} \right) \right], \tag{4.24} \]

\[ u_2^2=u_{20}^2+\frac{i\omega}{\rho}\frac{\rho_s}{\rho_n} \left[ \theta_{\phi} \left( \frac{\partial p}{\partial c_\phi} + \frac{\partial p}{\partial c_p} - \rho\frac{\partial \lambda}{\partial c_p} - \rho\frac{\partial \lambda}{\partial c_\phi} \right)^2 \frac{\partial \lambda_\phi}{\partial N_\phi} + \theta_{p\phi} \left( \frac{\partial p}{\partial c_p} - \rho\frac{\partial \lambda}{\partial c_p} \right) \left( \frac{\partial \lambda_p}{\partial N_p} + \frac{\partial \lambda_\phi}{\partial N_\phi} \right) \right]. \tag{4.25} \]

The imaginary part of the wave vector, calculated with the aid of (4.24), gives the absorption coefficient \(\widetilde{\alpha}_1\) of first sound, associated with the slow processes considered in this paragraph:

\[ \widetilde{\alpha}_1 = \operatorname{Im}\left(\frac{\omega}{u_1}\right) = \frac{\omega^2}{2\rho c_1^3} \left[ \theta_{\phi} \left( \frac{\partial p}{\partial c_p} + \frac{\partial p}{\partial c_\phi} \right)^2 \frac{\partial \lambda_\phi}{\partial N_\phi} + \theta_{p\phi} \left( \frac{\partial p}{\partial c_p} \right)^2 \left( \frac{\partial \lambda_p}{\partial N_p} + \frac{\partial \lambda_\phi}{\partial N_\phi} \right) \right], \ \mathrm{s}^{*} \tag{4.26} \]

In an analogous way we find the absorption coefficient of second sound:

\[ \widetilde{\alpha}_2 = \operatorname{Im}\left(\frac{\omega}{u_2}\right) = \frac{\omega^2\rho_s}{2\rho_n\rho u_{20}^3} \left[ \theta_{\phi} \left( \frac{\partial p}{\partial c_p} + \frac{\partial p}{\partial c_\phi} - \rho\frac{\partial \lambda}{\partial \mu_p} - \rho\frac{\partial \lambda}{\partial c_\phi} \right)^2 \frac{\partial \mu_\phi}{\partial N_\phi} + \theta_{p\phi} \left( \frac{\partial p}{\partial c_p} - \rho\frac{\partial \lambda}{\partial \mu_p} \right) \left( \frac{\partial \mu_p}{\partial N_p} + \frac{\partial c_\phi}{\partial N_\phi} \right) \right], \ \mathrm{s}^{*} \tag{4.27} \]

The derivatives \(\left(\dfrac{\partial p}{\partial c_\lambda}\right)\) entering the expression for \(\widetilde{\alpha}_2\) are determined by formulas (15.31,I) and (15.32,I), while derivatives of the form \(\left(\dfrac{\partial p}{\partial c_p}-\rho\dfrac{\partial \lambda}{\partial c_p}\right)\), according to (15.22,I) and (15.23,I), are equal (in the variables \(\rho,T\)) to

\[ \left. \begin{aligned} \left( \frac{\partial p}{\partial c_p} - \rho\frac{\partial \lambda}{\partial c_p} \right) &= N_p - S\frac{\partial N_p}{\partial S} \\ &= N_p \left\{ 1-\frac{S}{C} \left( \frac{\Delta}{T} + \frac{1}{2} \right) \right\}, \\[1ex] \left( \frac{\partial p}{\partial c_\phi} - \rho\frac{\partial \mu}{\partial c_\phi} \right) &= N_\phi - S\frac{\partial N_\phi}{\partial S} = N_\phi \left\{ 1-3\frac{S}{C} \right\}. \end{aligned} \right\} \tag{4.28} \]

With the aid of experimental values of the coefficient of absorption of first sound in helium II, as has already been indicated, the coefficients \(\Gamma_\phi\) and \(\Gamma_{\phi\rho}\) can be determined.

The knowledge of the indicated coefficients makes it possible to calculate, from formulas (4.20), the relaxation times \(\theta_\phi\) and \(\theta_{\phi\rho}\). The relaxation times \(\theta_\phi\) and \(\theta_{\phi\rho}\), at temperatures that are not too low, turn out to be negligibly small, of the order of \(10^{-1}\) sec. The temperature dependence of the relaxation times \(\theta_\phi\) and \(\theta_{\phi\rho}\) is shown in Fig. 2. The extraordinary smallness of the relaxation time leads to an unusually small absorption of sound in helium II. Appreciable absorption of sound may be expected for frequencies of order \(\omega \sim 1/\theta_{\phi\rho}\) or \(1/\theta_\phi\).

Fig. 2. Temperature dependence of the relaxation times \(\theta_\phi\) and \(\theta_{\phi\rho}\).

The derivatives \(\left(\dfrac{\partial N_p}{\partial \mu_p}\right)_{\rho,S}\) and \(\left(\dfrac{\partial N_\phi}{\partial \mu_\phi}\right)_{\rho,S}\), which enter into formulas (4.20) for \(\theta_\phi\) and \(\theta_{\phi\rho}\), are most conveniently calculated in the variables \(\rho\) and \(T\). The transition to the indicated variables is carried out with the aid of the known properties of Jacobians.

\[ \left(\frac{\partial N_p}{\partial \mu_p}\right)_S = \frac{ \dfrac{\partial (N_p,S)}{\partial (\mu_p,T)} }{ \dfrac{\partial (\mu_p,S)}{\partial (\mu_p,T)} } = \left(\frac{\partial N_p}{\partial \mu_p}\right)_T - \left(\frac{\partial N_p}{\partial T}\right)_{\mu_p} \frac{ \left(\dfrac{\partial S}{\partial \mu_p}\right)_T }{ \left(\dfrac{\partial S}{\partial T}\right)_{\mu_p} }. \tag{4.29} \]

Next we use the relation that follows from the thermodynamic identity

\[ \left(\frac{\partial N_p}{\partial T}\right)_{\mu_p} = \left(\frac{\partial S}{\partial \mu_p}\right)_T \]

and rewrite (4.29) in the form

\[ \left(\frac{\partial N_p}{\partial \mu_p}\right)_S = \left(\frac{\partial N_p}{\partial \mu_p}\right)_T - \frac{ \left(\dfrac{\partial N_p}{\partial T}\right)_{\mu_p}^{2} }{ \left(\dfrac{\partial S}{\partial T}\right) }. \tag{4.30} \]

Similarly we obtain

\[ \left(\frac{\partial N_\phi}{\partial \mu_\phi}\right)_S = \left(\frac{\partial N_\phi}{\partial \mu_\phi}\right)_T - \frac{ \left(\dfrac{\partial N_\phi}{\partial T}\right)^2_{\mu_\phi} }{ \dfrac{\partial S}{\partial T} }. \tag{4,31} \]

From the distribution function for rotons (classical statistics) it follows that

\[ \left(\frac{\partial N_p}{\partial \mu_p}\right)_{T,\rho} = \frac{N_p}{kT}. \tag{4,32} \]

Using the formula for the total number of phonons in \(1\ \mathrm{cm}^3\) in a nonequilibrium state with chemical potential \(\mu_\phi\),

\[ N_\phi=(2\pi\hbar)^{-3}\int \left[ e^{\frac{cp-\mu_\phi}{kT}}-1 \right]^{-1} 4\pi p^2\,dp, \tag{4,33} \]

we find

\[ \left(\frac{\partial N_\phi}{\partial \mu_\phi}\right)_{\rho,T} \simeq \frac{N_\phi}{kT}\,\frac{5\pi^2}{36}. \tag{4,34} \]

The values of the other equilibrium derivatives are obtained from the relations

\[ \left(\frac{\partial N_p}{\partial T}\right)_{\rho,\mu_p} = \frac{N_p}{T}\left(\frac{1}{2}+\frac{\Delta}{T}\right), \qquad \left(\frac{\partial N_\phi}{\partial T}\right)_{\rho,\mu_\phi} = \frac{3N_\phi}{T}, \]

\[ \left(\frac{\partial S}{\partial T}\right)_{\rho,\mu_p\mu_\phi} = \frac{k}{T} \left[ N_p\left(\frac{\Delta^2}{T^2}+\frac{\Delta}{T}+\frac{3}{4}\right) + N_\phi\frac{\pi^4}{9} \right]. \tag{4,35} \]

We shall now use the relations obtained, (4,32), (4,34), and (4,35); according to (4,30) and (4,31) we have

\[ \left(\frac{\partial \mu_p}{\partial N_p}\right)_{\rho,S} = \frac{ \dfrac{kT}{N_p} \left[ \left(\dfrac{\Delta^2}{T^2}+\dfrac{\Delta}{T}+\dfrac{3}{4}\right) \dfrac{N_p}{N_\phi} + \dfrac{\pi^4}{9} \right] }{ \left[ \dfrac{1}{2}\dfrac{N_p}{N_\phi} + \dfrac{\pi^4}{9} \right] }, \tag{4,36} \]

\[ \left(\frac{\partial \mu_\phi}{\partial N_\phi}\right)_{\rho,S} = \frac{ \dfrac{36}{5\pi^2}\dfrac{kT}{N_\phi} \left[ \left(\dfrac{\Delta^2}{T^2}+\dfrac{\Delta}{T}+\dfrac{3}{4}\right) \dfrac{N_p}{N_\phi} + \dfrac{\pi^4}{9} \right] }{ \left[ \left(\dfrac{\Delta^2}{T^2}+\dfrac{\Delta}{T}+\dfrac{3}{4}\right) + \dfrac{\pi^4}{9} - \dfrac{324}{5\pi^2} \right] }. \tag{4,37} \]

Absorption of sound in helium II. The kinetic phenomena considered earlier—the first and second viscosities and thermal conductivity—are sources of sound absorption in helium II. Let us calculate the coefficient of sound absorption in helium II associated with the indicated dissipative processes. The linearized system of hydrodynamic equations describing the propagation of sound in helium II

HYDRODYNAMICS OF HELIUM II

taking dissipative processes into account, according to (2.14), is written in the form

\[ \dot{\rho}+\operatorname{div}\mathbf{j}=0, \tag{4.38} \]

\[ \frac{\partial}{\partial t}j_i+\frac{\partial p}{\partial x_i} = \eta \frac{\partial}{\partial x_k} \left( \frac{\partial v_{ni}}{\partial x_k} + \frac{\partial v_{nk}}{\partial x_i} - \frac{2}{3}\delta_{ik} \frac{\partial v_{nl}}{\partial x_l} \right) + \]

\[ +\frac{\partial}{\partial x_i} \left\{ \zeta_1\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) + \zeta_2\operatorname{div}\mathbf{v}_n \right\}, \tag{4.39} \]

\[ \dot{\mathbf{v}}_s+\nabla\mu = \nabla \left\{ \zeta_3\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) + \zeta_4\operatorname{div}\mathbf{v}_n \right\}, \tag{4.40} \]

\[ T\{\dot{(\sigma\rho)}+\sigma\rho\operatorname{div}\mathbf{v}_n\} = \varkappa\nabla T. \tag{4.41} \]

In a plane sound wave all quantities consist of constant equilibrium terms and small additions varying according to the law \(e^{i\omega(t-x/u)}\) (\(x\) is the direction of propagation of the wave, \(\omega\) the sound frequency, \(u\) the sound velocity). The velocities \(\mathbf{v}_n\) and \(\mathbf{v}_s\) in the sound wave are likewise small quantities varying according to this same law. Let us choose as independent variables the density \(\rho\) and the entropy \(\sigma\) (per 1 g of helium II), and eliminate the velocities \(\mathbf{v}_n\) and \(\mathbf{v}_s\) from equations (4.38)—(4.41). Restricting ourselves to terms linear in the kinetic coefficients, we obtain two equations.

In the general case these equations look very cumbersome. However, if one takes into account the anomalous smallness of the coefficient of thermal expansion of helium II, which permits one to omit all terms containing the derivatives \(\left(\frac{\partial T}{\partial \rho}\right)_\sigma\) and \(\left(\frac{\partial p}{\partial \sigma}\right)_\rho\), then the equations are considerably simplified\(^*\). In this way we obtain, first,

\[ \left(u^2-\frac{\partial p}{\partial\rho}\right)\rho' = i\omega \left\{ \left(\frac{4}{3}\eta+\zeta_2\right)\frac{\rho'}{\rho} + \right. \]

\[ \left. + \left(\frac{4}{3}\eta+\zeta_2-\rho\zeta_1\right)\frac{\sigma'}{\sigma} \right\}. \tag{4.42} \]

\(^*\) According to the thermodynamic identity

\[ d\mu=-\sigma dT+\frac{1}{\rho}\,dp \]

the derivatives \(\left(\frac{\partial T}{\partial \rho}\right)_\sigma\) and \(\left(\frac{\partial p}{\partial\sigma}\right)_\rho\) are connected by the relation

\[ \rho^2\frac{\partial T}{\partial\rho}=\frac{\partial p}{\partial\sigma}. \]

With the aid of the well-known properties of Jacobians it is easy to show that, for a small coefficient of thermal expansion \(\frac{1}{\rho}\left(\frac{\partial\rho}{\partial T}\right)_p\), the equality holds

\[ \left(\frac{\partial p}{\partial\sigma}\right)_\rho = - \frac{ \left(\frac{\partial\rho}{\partial T}\right)_p }{ \left(\frac{\partial\sigma}{\partial T}\right)_\rho \left(\frac{\partial\rho}{\partial p}\right)_T }. \]

and, secondly,

\[ \left(u^2-\frac{\partial p}{\partial \rho}\right)\frac{\rho'}{\rho} = -\left(\sigma\frac{\partial T}{\partial \sigma}-\frac{\rho_n}{\rho_s\sigma}u^2\right)\sigma' + \]

\[ +\, i\omega\left\{ \zeta_1\frac{\rho'}{\rho} +\frac{\sigma'}{\sigma}(\zeta_4-\rho\zeta_3) -\frac{\rho_n}{\rho_s}\frac{\chi}{T\rho\sigma}\frac{\partial T}{\partial \sigma}\,\sigma' \right\} \tag{4,43} \]

(\(\rho'\) and \(\sigma'\) are the variable terms in the density and energy). Excluding from the left-hand side of equation (4,43), with the aid of equation (4,42), the term containing \(\rho'\), we find that the indicated equation becomes

\[ \left(\sigma\frac{\partial T}{\partial \sigma}-\frac{\rho_n}{\rho_s\sigma}u^2\right)\sigma' = \]

\[ = i\omega\left\{ \left(\zeta_4-\frac{\zeta_3}{\rho}-\frac{4}{3}\frac{\eta}{\rho}\right)\frac{\rho'}{\rho} + \right. \]

\[ \left. +\left(\zeta_4-\rho\zeta_3-\frac{\zeta_3}{\rho}+\zeta_1\right)\frac{\sigma'}{\sigma} -\frac{\rho_n}{\rho_s}\frac{\chi}{T\rho\sigma}\frac{\partial T}{\partial \sigma}\,\sigma' \right\}. \tag{4,44} \]

The compatibility condition for the system of equations (4,42) and (4,44) is written in the form that its determinant is zero,

\[ \left| \begin{array}{cc} u^2-\dfrac{\partial p}{\partial \rho} -i\omega\dfrac{1}{\rho}\left(\dfrac{4}{3}\eta+\zeta_2\right) & -i\omega\left(\dfrac{4}{3}\eta+\zeta_2-\rho\zeta_1\right)\dfrac{1}{\sigma} \\[1.2em] i\omega\left(-\zeta_4+\dfrac{\zeta_3}{\rho}+\dfrac{4}{3}\dfrac{\eta}{\rho}\right)\dfrac{1}{\rho} & \left(\sigma\dfrac{\partial T}{\partial \sigma}-\dfrac{\rho_n}{\rho_s\sigma}u^2\right) + \\[1.2em] & \quad +\dfrac{i\omega}{\rho} \left\{ \dfrac{4}{3}\eta+\zeta_2-\rho(\zeta_1+\zeta_4)+\rho^2\zeta_3 +\dfrac{\rho_n}{\rho_s}\dfrac{\chi}{T\sigma}\dfrac{\partial T}{\partial \sigma} \right\} \end{array} \right| =0 . \tag{4,45} \]

In the region of small sound frequencies \(\omega\), equation (4,45), which determines the sound velocities in helium II, splits into two equations:

\[ u^2-\frac{\partial p}{\partial \rho} = i\omega\frac{1}{\rho}\left(\frac{4}{3}\eta+\zeta_2\right), \tag{4,46} \]

\[ \left(\sigma\frac{\partial T}{\partial \sigma}-\frac{\rho_n}{\rho_s\sigma}u^2\right) = \]

\[ = i\omega\frac{1}{\rho\sigma} \left\{ \rho(\zeta_1+\zeta_4)-\rho^2\zeta_3-\zeta_2-\frac{4}{3}\eta -\frac{\rho_n}{\rho_s}\frac{\chi}{T}\frac{\partial T}{\partial \sigma} \right\}. \tag{4,47} \]

The root of equation (4,46),

\[ u_1^2=\left(\frac{\partial p}{\partial \rho}\right)+i\omega\frac{1}{\rho}\left(\frac{4}{3}\eta+\zeta_2\right), \tag{4,48} \]

determines the velocity of first sound in helium II with allowance for kinetic phenomena. The root of equation (4,47),

\[ u_2^2 = \sigma^2\frac{\partial T}{\partial \sigma}\frac{\rho_s}{\rho_n} + \]

\[ + \frac{i\omega}{\rho}\frac{\rho_s}{\rho_n} \left\{ \zeta_2+\rho^2\zeta_3-\rho\zeta_1-\rho\zeta_4+\frac{4}{3}\eta +\frac{\rho_n}{\rho_s}\frac{\chi}{T}\frac{\partial T}{\partial \sigma} \right\} \tag{4,49} \]

determines the velocity of second sound with allowance for kinetic phenomena. The velocities \(u_1\) and \(u_2\) are complex quantities. Therefore the wave vectors \(k=\dfrac{\omega}{u}\) will also be complex quantities. The real part of the wave vector determines the change of the phase of the oscillations with distance, while the imaginary part is simply the coefficient of sound absorption.

Absorption of first sound in helium II. The imaginary part of the wave vector for first sound, according to (4,48), is equal to

\[ \alpha_1=\operatorname{Im}\left(\frac{\omega}{u_1}\right) = \frac{\omega^2}{2\rho u_1^3} \left(\frac{4}{3}\eta+\eta_2\right). \tag{4,50} \]

Thus, the absorption coefficient of first sound \(\alpha_1\) depends only on two kinetic coefficients—the coefficient of first viscosity \(\eta\) and the coefficient of second viscosity \(\zeta_2\). The fact that the other coefficients \((\zeta_1,\zeta_3\) and \(\chi)\) do not enter the expression for \(\alpha_1\) is essentially connected with neglecting the effect of thermal expansion of helium II. Thus, for example, the effect of thermal conductivity gives, in the expression for the absorption coefficient of first sound, the additional term

\[ \alpha_{1\chi}= \frac{\omega^2}{2\rho u_1^3}\, \frac{\chi}{C} \left(\frac{c_p}{c_v}-1\right). \]

The viscosity coefficient \(\eta\) can be measured experimentally independently of the phenomenon of sound propagation in helium II, for example, from the damping of torsional oscillations of a disk immersed in helium. Consequently, experiments on the absorption of first sound in helium II make it possible to determine the values of the coefficient of second viscosity \(\zeta_2\). Comparison of the measured values by Pellam and Squire\(^{14}\) with those calculated by formula (15,24, I) makes it possible to find the constants \(a\) and \(b\) entering the formulas for the kinetic coefficients \(\Gamma_\phi\) and \(\Gamma_{\mathrm{ph}}\). The magnitude of the coefficient of second viscosity \(\zeta_2\), calculated from the experimental values of the absorption coefficient of first sound in helium II, appreciably exceeds the known value of the viscosity coefficient \(\eta\) throughout the entire temperature interval from \(1.57\) to \(2^\circ\)K in which the measurements were made. Thus, the absorption of first sound is determined mainly by second viscosity. Figure 3 gives the temperature dependence of the coefficient \(\zeta_2\). The coefficient \(\zeta_2\) exceeds in magnitude the coefficient \(\eta\) over the entire temperature range shown in Fig. 3. The absorption coefficient of first sound calculated at the beginning of the paragraph,

\[ \widetilde{\alpha}_1, \]

connected with slow processes in helium II, naturally coincides with that part of \(\alpha_1\) which depends on second viscosity. This is easily verified by comparing (4,25) with (4,50), taking into account the definition of the relaxation time (4,20). The temperature depend-

the coefficient of absorption of first sound in helium II is shown in Fig. 4.

Recently, Atkins and Chase15 have carried out measurements of the absorption coefficient of ordinary sound over a wide temperature range, at substantially lower temperatures than those of Pellam and Squire, according to whom our coefficients were normalized. The authors set themselves the goal of testing the present theory. The sound frequency was 14 Mc.

Fig. 3. Temperature dependence of the coefficient of first viscosity \(\eta\) and the coefficient of second viscosity \(\zeta_2\), which determines the absorption of first sound in helium II.

Fig. 4. Temperature dependence of the absorption coefficients of first sound \(\alpha_1\) and second sound \(\alpha_2\) in helium II.

The measured values of the sound absorption coefficient proved to be in good agreement with the theoretical values over the entire temperature range (Fig. 5). Thus, the indicated experiments confirm the ideas presented here concerning the mechanism of sound absorption in helium II.

Absorption of second sound in helium II. The absorption coefficient of second sound is equal to the imaginary part of the wave vector, computed by means of expression (4.49) for the sound velocity

\[ \alpha_2=\operatorname{Im}\left(\frac{\omega}{u_2}\right) =\frac{\omega^2}{2\rho u_2^3}\frac{\rho_s}{\rho_n} \left\{\frac{4}{3}\eta+\left[\zeta_2+\rho^2\zeta_3-2\rho\zeta_1\right] +\frac{\rho_n}{\rho_s}\frac{\chi}{T}\frac{\partial T}{\partial\sigma}\right\}. \tag{4.51} \]

Thus, the absorption coefficient of second sound is composed of three parts, depending on the first viscosity, the second viscosity, and the thermal conductivity of helium II. The values of the coefficient of first viscosity and of the coefficient of thermal conductivity are known to us (see \(^{14}\) and \(^{16}\)); the effects in sound absorption that depend on these coefficients are readily calculated with the aid of the formula obtained, (4.51). The combination of second-viscosity coefficients entering formula (4.51), according to (4.24)—(4.27), is equal to

\[ \zeta_2+\rho^2\zeta_3-2\rho\zeta_1 = \frac{1}{\Gamma_\Phi} \left(N-\frac{\partial N}{\partial S}S\right)^2 + \frac{1}{\Gamma_{\Phi p}} \left(N_p-\frac{\partial N_p}{\partial S}S\right)^2 . \tag{4.52} \]

With the aid of relations (4.20), (4.22), and (4.23) it is easy to verify that expression (4.51) coincides exactly with the expression standing in the square brackets in (4.27). The latter is quite natural, since the absorption coefficient of second sound \(\tilde{\alpha}_2\) (4.27) and that part of the coefficient \(\alpha_2\) which depends on the second viscosity take into account one and the same effect, due to slow relaxation processes in helium II.

The expressions of the form \(N-\dfrac{\partial N}{\partial S}S\) entering formula (4.52) are determined by formulas (4.27). Figure 4 gives the temperature dependence of the terms standing in the braces in (4.51); at all temperatures the effect of sound absorption in helium II due to thermal conductivity plays an essential role and exceeds the viscous effects. The values, computed from formula (4.51), of the absorption coefficient of second sound in helium II are shown in Fig. 4. Owing to the presence of the factor \(\dfrac{\rho_s}{\rho_n}\) in (4.51) and to the considerably smaller value of the velocity of second sound in comparison with the velocity of first sound, the absorption coefficient of second sound \(\alpha_2\)

![Figure 5 graph]

Fig. 5. Absorption coefficient of first sound in helium II: — theoretical values, ○ — experimental values of Atkins and Chase\(^{15}\).

turns out to be several orders of magnitude larger than the absorption coefficient of first sound \(\alpha_1\), for the same values of the frequency \(\omega\). Thus, at a temperature of \(1.0^\circ\) K, appreciable absorption of second sound will be observed at a frequency \(\omega = 10^4\ \mathrm{sec}^{-1}\).

This circumstance has recently been confirmed experimentally by K. Zinov’eva\(^{16}\), who carried out measurements of the absorption of second sound in a cylindrical resonator filled with helium II. In the temperature region above \(1.2^\circ\) K, at the sound frequencies she used (less than \(10^4\ \mathrm{sec}^{-1}\)), the absorption was determined mainly by viscous effects at the walls of the vessel. In the temperature region from \(0.8\) to \(1.2^\circ\) K, however, appreciable absorption of sound in the volume of the helium was found, in full agreement with the theory.

The values obtained in these experiments for the coefficient of absorption of second sound in the volume, as well as the temperature dependence of this coefficient, agree with the theory (Fig. 6). The same figure

Fig. 6. Coefficient of absorption of second sound in helium II: ○ — K. Zinov’eva\(^{16}\); △ — Hanson and Pellam\(^{17}\), × Atkins and Hart\(^{39}\); — theoretical values.

gives the results of measurements of the absorption of second sound obtained by Hanson and Pellam\(^{17}\) and by Atkins\(^{39}\). They also agree well with the theoretical values. It is appropriate to note here that, in the experiments on the absorption of second sound, what is essentially measured is the thermal-conductivity coefficient of helium II, direct measurements of which are very difficult because of the presence of large macroscopic heat fluxes in helium II. It should also be noted that almost

over the entire range of temperatures investigated (below \(1.6^\circ\mathrm{K}\)), the phonon part of the thermal-conductivity coefficient is substantial; in calculating it, apart from the characteristics of the energy spectrum of helium II (\(\rho_0,\ \mu,\ c\)), no new constants were introduced. The characteristics of the spectrum were found from thermodynamic experiments.

In the papers of Dingle\(^{18}\) and of Kronig and Thellung\(^{19}\) an attempt was made to consider the question of sound absorption in helium II. These authors did not take into account the phenomenon of second viscosity, which has a substantial effect on the effects under consideration. In these works the question of the role of thermal conductivity was also not clarified.

5. On anomalous absorption of sound near the \(\lambda\)-point\(^{20}\)

The general theory of phase transitions makes it possible to consider the phenomenon of sound absorption near the \(\lambda\)-point. We shall show that near the \(\lambda\)-point an anomalously large absorption of sound should be observed.

Near the point of a second-order phase transition the thermodynamic potential \(\Phi\) can be represented in the form

\[ \Phi(p,T,\eta)=\Phi_0(p,T)+A(p,T)\eta^2+C(p,T)\eta^4\ldots . \tag{5.1} \]

Here \(C>0\); \(A<0\) in the low-temperature (asymmetric) phase; \(A>0\) in the high-temperature (symmetric) phase; the transition point is determined by the condition \(A(p,T)=0\). At a given pressure, near the transition point the function \(A(p,T)\) is expanded in a series in the difference \(T-T_\lambda\) (\(T_\lambda\) is the transition temperature)

\[ A(p,T)=a(p)(T-T_\lambda)\ldots . \tag{5.2} \]

The parameter \(\eta\) characterizes the degree of asymmetry. In the symmetric phase \(\eta=0\), while in the asymmetric phase the parameter \(\eta\) is different from zero. The dependence of \(\eta\) on temperature near the \(\lambda\)-point is determined from the condition that the potential \(\Phi\) be minimal. From the condition \(\dfrac{\partial \Phi}{\partial \eta}=0\) we find the nonzero value \(\eta_0^2\):

\[ \eta_0^2=-\frac{A}{2C}=\frac{A}{2C}(T_\lambda-T), \tag{5.3} \]

corresponding to a certain equilibrium state of the asymmetric phase. Let a system that is in an asymmetric state be brought out of equilibrium. The rate at which the parameter \(\eta\) approaches its equilibrium value \(\eta_0\) is determined by the kinetic equation

\[ \frac{d\eta}{dt}=-\gamma\,\frac{\partial \Phi}{\partial \eta}. \tag{5.4} \]

\(\gamma\) is a kinetic coefficient, about which we shall assume that it does not possess any peculiarities near \(T_\lambda\). Expanding

expanding the derivative \(\dfrac{d\Phi}{d\eta}\) in a series in the difference \(\eta-\eta_0\), with the aid of (5.4) we find

\[ \frac{d\eta}{dt}=2\gamma C\eta_0^2(\eta-\eta_0)=-\frac{1}{\tau}(\eta-\eta_0). \tag{5.5} \]

It follows from this that the relaxation time \(\tau\), characterizing the establishment of equilibrium in the asymmetric phase, is equal to

\[ \tau=-\frac{1}{2C\gamma\eta_0^2}. \tag{5.6} \]

Near the \(\lambda\)-point we have

\[ \tau=-\frac{1}{\gamma a(T_\lambda-T)}. \tag{5.7} \]

Consequently, the relaxation time near the transition point rapidly increases as one approaches the \(\lambda\)-point. The establishment of equilibrium, therefore, near the \(\lambda\)-point occurs extremely slowly, and this must lead to a noticeable attenuation of sound. It is quite obvious that such anomalous attenuation of sound will be observed only in the asymmetric phase, i.e. below \(T_\lambda\) (at a given pressure). In the symmetric phase, however, the parameter \(\eta\) is identically equal to zero in all states, both equilibrium and nonequilibrium; consequently, there will be no anomalies in the absorption of sound.

Let a sound wave of frequency \(\omega\) propagate in the system under consideration, i.e. let an adiabatic periodic compression and rarefaction occur in the system. According to M. Leontovich and L. Mandelstam*) the square of the velocity of sound in the low-temperature phase will be equal to

\[ c^2=\frac{1}{1-i\omega\tau}\left[c_{\mathrm{равн}}^2-i\omega\tau c_\eta^2\right]. \tag{5.8} \]

Here \(c_{\mathrm{равн}}\) is the velocity of sound for a process so slow that the system is at all times in equilibrium. This velocity in our case is, obviously, equal to the equilibrium velocity of sound \(c_{II}\) in the low-temperature phase**)

\[ c_{\mathrm{равн}}=c_{II}. \tag{5.9} \]

The velocity \(c_\eta\), however, is the velocity of sound for a process so fast that the parameter \(\eta\) remains constant during the propagation of the sound wave. Near the \(\lambda\)-point the velocity \(c_\eta\) is equal to the equilibrium velocity of sound in the high-temperature phase

\[ c_\eta=c_I. \tag{5.10} \]

*) See, for example, \({}^{21}\).

**) The values of quantities in the low-temperature phase are denoted by the Roman numeral II, and in the high-temperature phase by the numeral I.

Thus, finally, we have

\[ c^2=\frac{1}{1-i\omega\tau}\left[c_{II}^2-i\omega\tau c_I^2\right]. \tag{5,11} \]

The value of the sound absorption coefficient \(\alpha\) is equal to the imaginary part of the wave vector

\[ k=\frac{\omega}{c}=\omega\sqrt{\frac{1-i\omega\tau}{c_{II}^2-c_I^2 i\omega\tau}} . \tag{5,12} \]

Assuming the difference between the values of the velocities \(c_{II}\) and \(c_I\) to be small in comparison with \(c_I\), we find from this

\[ \alpha=\operatorname{Im} k= \frac{\omega^2\tau}{1+\omega^2\tau^2}\, \frac{1}{2c_I^3}\left(c_I^2-c_{II}^2\right). \tag{5,13} \]

The relaxation time \(\tau\) in the vicinity of the \(\lambda\)-point is determined by formula (5,7). It rapidly increases on approaching the \(\lambda\)-point. According to (5,13), on approaching the \(\lambda\)-point (at a given sound frequency \(\omega\)) the sound absorption coefficient also increases. In the immediate neighborhood of the \(\lambda\)-point (\(\omega\tau \simeq 1\)) the quantity \(\alpha\) reaches a maximum, and then begins to decrease. Such is the general picture of the phenomenon of anomalous sound absorption near the \(\lambda\)-point in the low-temperature phase. In the high-temperature phase there should be no anomalous sound absorption.

The magnitude of the jump in the sound velocity at the \(\lambda\)-point, \(c_{II}-c_I\), entering the formula for \(\alpha\), can be expressed in terms of the usually well-known magnitude of the jump in the heat capacity at the \(\lambda\)-point.

Bearing in mind the subsequent application of the results to the case of the \(\lambda\)-point in helium II, we obtain the required relation for the transition in a liquid. We start from the fact that the volume \(V\) and the temperature \(T\) at the transition point are continuous (as independent variables we choose the pressure \(p\) and the entropy \(S\)), i.e. their jumps \(\Delta V\) and \(\Delta T\) are equal to zero:

\[ \Delta V=0,\qquad \Delta T=0. \tag{5,14} \]

Differentiating these equalities with respect to the entropy along the transition curve, we have

\[ \Delta \frac{\partial V}{\partial S} +\frac{dp}{dS}\Delta \frac{\partial V}{\partial p}=0, \tag{5,15} \]

\[ \Delta \frac{\partial T}{\partial S} +\frac{dp}{dS}\Delta \frac{\partial T}{\partial p}=0. \tag{5,16} \]

Taking into account the equality of derivatives \(\dfrac{\partial V}{\partial S}=\dfrac{\partial T}{\partial p}\), we eliminate from both equalities the quantity \(\Delta\dfrac{\partial V}{\partial S}\):

\[ \Delta \frac{\partial V}{\partial p} = \Delta \frac{\partial T}{\partial S} \left(\frac{dS}{dp}\right)^2 . \tag{5,17} \]

From this we obtain the desired relation connecting the jumps in the speed of sound and in the heat capacity at the \(\lambda\)-point:

\[ \Delta \frac{1}{c^2}=-\frac{T}{V}\left(\Delta \frac{1}{c_p}\right)\left(\frac{dS}{dp}\right)^2 . \tag{5.18} \]

The derivative \(\dfrac{dS}{dp}\) along the phase-equilibrium curve can be expressed in terms of the coefficient of thermal expansion \(\dfrac{\partial V}{\partial T}\) and the derivative \(\dfrac{\partial T_\lambda}{\partial p}\),

\[ \frac{dS}{dp} = \left(\frac{dS}{dp}\right)_T + \left(\frac{dS}{dT}\right)_p \frac{dT_\lambda}{dp} = -\left(\frac{\partial V}{\partial T}\right)_p + c_p \frac{d\ln T_\lambda}{dp}. \tag{5.19} \]

The results obtained above concerning the absorption of sound near the \(\lambda\)-point are applicable in all cases of second-order phase transitions.*) We shall apply them to an analysis of the data on the absorption of first (ordinary) sound near the \(\lambda\)-point in helium II. Chase \({}^{22}\) observed an anomalous increase of the sound-absorption coefficient in helium II on approaching the \(\lambda\)-point. Above the \(\lambda\)-point, however, no anomalously large absorption of sound was observed. Using the data \({}^{23}\) for \(c_p\) and \(\dfrac{\partial V}{\partial T}\), and the data \({}^{24}\) on the dependence of \(T_\lambda\) on pressure, with the aid of (5.18) we find the magnitude of the jump in the speed of sound at the \(\lambda\)-point:

\[ c_{\mathrm{II}}-c_{\mathrm{I}}\simeq 12.5\ \text{m/sec}. \tag{5.20} \]

Next, comparing the experimental values of the absorption coefficient at different temperatures (the frequencies used in \({}^{22}\) were 2 and 12.1 megacycles per second), we find the value of the relaxation time \(\tau\). The experimentally obtained value of the relaxation time follows very well the temperature dependence given by formula (5.7):

\[ \tau=\frac{4\cdot 10^{-13}}{T_\lambda-T}\;(\text{sec}). \tag{5.21} \]

In the temperature interval in which the measurements were carried out, the quantity \(\omega\tau\) did not exceed \(10^{-2}\). Therefore the term \(\omega^2\tau^2\) in the denominator of expression (5.13) may be neglected here. This term could have an effect only at \(T_\lambda-T \simeq 3\cdot 10^{-5}\ \mathrm{K}\) (for the sound frequencies considered). Above this value of \(T\), one should expect a decrease in the value of the sound-absorption coefficient \(\alpha\).

*) In particular, they should be applicable to transitions from the normal to the superconducting state, where anomalous sound absorption near the transition point has not yet been observed experimentally by anyone.

II. HYDRODYNAMICS OF SOLUTIONS

6. Equations of Hydrodynamics of Solutions of Foreign Particles in Helium II

Behavior of foreign particles dissolved in helium II

L. Landau and I. Pomeranchuk25 showed that foreign atoms or molecules dissolved in small concentrations in helium II cannot participate in superfluid motion. Let us consider the energy spectrum of helium II containing a small number of foreign atoms. The interaction of the foreign atoms with one another may be neglected in this case. The interaction of these atoms with He\(^4\) atoms leads to the appearance in the solution of additional energy levels. Thus, in the solution, in addition to the phonon and roton branches of the energy spectrum, there arises one more branch, due to the foreign dissolved atoms. The motion of a foreign atom through helium corresponds to some momentum, on which the energy depends continuously. The dependence of the energy on the momentum may be different for atoms of different kinds. Logically, two types of energy spectra are possible: in one case the energy minimum corresponds to a momentum equal to zero; in the other case, to a momentum \(p_0\) not equal to zero. In the first case the energy near the minimum may be represented in the form

\[ \varepsilon=\varepsilon_0+\frac{p^2}{2m} \tag{6,1} \]

(\(\varepsilon_0\) is a certain constant, \(m\) is the effective mass of the foreign atom in the solution). In the second case, near the minimum we have

\[ \varepsilon=\varepsilon_0+\frac{(p-p_0)^2}{2m}. \tag{6,2} \]

Analysis of experimental data on the velocity of second sound in solutions of He\(^3\) in He\(^4\) makes it possible to choose between the two possibilities indicated.

It turned out that the experimental data agree only with the assumption of an energy spectrum of He\(^3\) in He\(^4\) of the first type. Thus, for the case in which the foreign particles are atoms of the isotope He\(^3\), the energy spectrum has the form

\[ \varepsilon=\varepsilon_0+\frac{p^2}{2m}. \tag{6,3} \]

When a foreign atom moves in helium, the latter can emit phonons. From the law of conservation of momentum and energy it follows that this is possible in the case where the velocity of the foreign atom

\[ \frac{p=p_0}{m} \]

exceeds the speed of sound. Emission of rotons is also possible; however, for this it is required that the energy of the foreign

atom exceeded \(\Delta = 8.9^\circ \mathrm{K}\). This is possible in the case when the velocity of the impurity atom exceeds some value (for \(\mathrm{He}^3\), of the order of the speed of sound), depending on the effective mass \(m\). Thus, impurity particles moving in helium II with velocities less than the speed of sound do not interact in any way with the superfluid part of the liquid. Foreign particles, colliding with phonons and rotons, will be scattered. Owing to this they will move together with the phonons and rotons and, consequently, will enter the normal part of the liquid.

It should be emphasized that the conclusion obtained concerning the participation of impurities in normal motion is in no way connected with whether the impurity atoms themselves are superfluid or non-superfluid.

The distribution of impurity atoms over energies is determined, in the region of not very low temperatures, by Boltzmann statistics. In the region of low temperatures, however, it is necessary to take into account the degeneracy of the “impurity gas” and the interaction of the impurity particles with one another.

The conditions under which deviations from classical statistics set in were established by I. Pomeranchuk \(^{26}\). For weak solutions the indicated deviations set in only at very low temperatures (of the order of \(0.10^\circ \mathrm{K}\)); in the classical region one can readily obtain the thermodynamic functions for weak solutions of foreign atoms in helium.

Normal density. The contribution of impurities to the normal density of the liquid is calculated from the general formula (4.2, I). In the case when \(p_0 \ne 0\), calculations completely analogous to those carried out for the roton (see \(^{26}\)) give

\[ \rho_{ni}=n\frac{p_0^2}{3kT} \tag{6,4} \]

(\(n\) is the number of impurity atoms in \(1\ \mathrm{cm}^3\) of helium). Thus, for \(p_0 \ne 0\) the value of the normal density of the liquid due to the impurity is inversely proportional to the temperature. In the most interesting case, however, \(p_0=0\), we obtain the obvious result

\[ \rho_{ni}=\frac{\rho \varepsilon m}{m_4} \tag{6,5} \]

(\(\varepsilon\) is the molar concentration, \(m_4\) is the mass of a \(\mathrm{He}^4\) atom). Thus, for \(p_0=0\) the contribution of the impurity to the normal part of the liquid is constant with respect to temperature.

Entropy. Application of the general formulas for the entropy of weak solutions and of the ideal gas for the case \(p_0=0\) gives the following expression, determining the entropy of \(1\ \mathrm{g}\) of solution:

\[ \sigma=\sigma_0+\frac{k\varepsilon}{m_4} \left\{ \ln\left(\frac{mkT}{2\pi\hbar^2}\right)^{3/2} \cdot \frac{(2s+1)v_0}{\varepsilon} +\frac{5}{2} \right\}, \tag{6,6} \]

where \(\sigma_0\) is the entropy of pure helium, and \(v_0\) is the volume per one helium atom.

In a similar way, for the case \(p_0 \ne 0\) we find the expression

\[ \sigma=\sigma_0+\frac{k\varepsilon}{m_4} \left\{ \ln \frac{(2s+1)v_0p_0^2}{h^3\varepsilon} \sqrt{\frac{mkT}{\pi^3}}+\frac{3}{2} \right\}. \tag{6,7} \]

Heat capacity. The heat capacity \(C\) of helium containing an admixture at \(p_0 \ne 0\) is equal to

\[ C=C_0+\frac{3}{2}\frac{k\varepsilon}{m_4}. \tag{6,8} \]

Here \(C_0\) is the heat capacity of pure \(\mathrm{He}^4\).

For \(p_0=0\) we have

\[ C=C_0+\frac{k\varepsilon}{2m_4}. \tag{6,9} \]

The contribution of the admixture to the heat capacity turns out to be less noticeable than to the entropy and to the normal density.

The hydrodynamics of weak solutions of foreign atoms in helium was considered in the already mentioned work \(^{26}\). However, the application of the equations obtained there to the question of sound propagation in helium II was carried out not quite correctly. The author did not take into account the dependence of the liquid density \(\rho\) on the concentration of the solution \(c\). This circumstance did not affect the expression for the velocity of second sound in weak solutions only because the omitted terms in the final result were multiplied by the coefficient of thermal expansion \(\left(\dfrac{\partial \rho}{\partial T}\right)\), whose value for helium is practically equal to zero.

Equations of the hydrodynamics of solutions\(^{4}\). We now proceed to the derivation of the equations of hydrodynamics for solutions of foreign particles in helium II. We shall not assume that the solutions are weak, and shall derive equations valid for arbitrary concentrations of the solution. We shall only assume that the dissolved particles obey Fermi statistics (a solution of \(\mathrm{He}^3\) in \(\mathrm{He}^4\)) and do not by themselves form a superfluid liquid\(*\).

Let us consider a solution of foreign particles in helium II whose concentration is equal to \(c\) (\(\rho c\) is the mass of dissolved particles per \(1\ \mathrm{cm}^3\) of solution). Conservation of the total mass of the liquid permits us, as before, to write the continuity equation in the form

\[ \dot{\rho}+\operatorname{div} j=0. \tag{6,10} \]

* The case of a solution of two superfluid liquids is considered in the following section.

in total differentials

\[ \left(\frac{f}{w}-S\right)dT+\left(\frac{g}{w}-\rho c\right)d\frac{Z}{\rho} - \left(\frac{p}{w}-\rho\right)d\left(\varphi-\mu+\frac{Z}{\rho}c\right)=0, \qquad w=|\mathbf v_n-\mathbf v_s|. \tag{6.20} \]

Let the helium be at absolute zero temperature. In this case equation (6.20) is simplified:

\[ \left(\frac{g}{w}-\rho c\right)d\frac{Z}{\rho} +\rho_s d\left(\varphi-\mu+\frac{Z}{\rho}c\right)=0. \tag{6.21} \]

In writing (6.21) we replace the bracket \((p/w)-\rho\) by its value

\[ \frac{p}{w}-\rho=\frac{\rho_n w}{w}-\rho=-\rho_s. \]

Introduce the notation \(\varphi-\mu+\dfrac{Z}{\rho}c=\eta,\ \dfrac{Z}{\rho}=\zeta\). It follows from (6.21) that, for \(T=0\), the function \(\eta\) depends only on \(\zeta\). Consequently, the derivative

\[ \frac{d\eta}{d\zeta} = -\frac{1}{\rho_s}\left(\frac{g}{w}-\rho c\right) =F(\zeta) \tag{6.22} \]

is also a function of the single variable \(\zeta\). We shall denote this function symbolically by \(F(\zeta)\).

As the concentration \(c\) tends to zero, the impurity flux, and consequently also the function \(F(\zeta)\), tend to zero:

\[ F(\zeta) = -\frac{1}{\rho_s}\left(\frac{g}{w}-\rho c\right) =0 \quad (c\to 0). \tag{6.23} \]

For dissolved particles obeying Fermi statistics, the potential \(\zeta\) at small values of the concentration \(c\) may be written in the form\(^*\)

\[ \zeta=A(\rho)+B(\rho)c^{2/3}+\cdots . \tag{6.24} \]

The density function \(A(\rho)\) is the potential of the pure solvent (helium II). The second term in (6.24) is due to the dissolved particles. The function \(F(\zeta)\), according to (6.23), is equal to zero at small concentrations and at all values of the density \(\rho\). But a function that is equal to zero for all values of its argument is identically equal to zero; therefore we have

\[ F(\zeta)\equiv 0. \tag{6.25} \]

According to (6.22), in this case

Consequently, at absolute zero temperature, foreign particles dissolved in helium II, at any concentrations, participate only in the normal motion.

Thus, in two limiting cases—at absolute zero temperature and arbitrary values of the impurity concentration, on the one hand, and at nonzero temperatures and small impurity concentrations, on the other hand—the relation (6.26) holds. A detailed analysis, taking into account the validity of relation (6.26) in the indicated limiting cases, makes it possible to conclude that this relation is fulfilled for all values of the temperature and concentration of the solution. Thus, the total flux of impurities in the solution is always equal to \(\rho c \mathbf v_n\), i.e., impurities at any concentrations of the solution are transported only by the normal motion. Taking relation (6.26) into account, condition (6.20) may be rewritten in the form

\[ \left(\frac{f}{w}-S\right)dT+\rho_s d\left(\varphi-\mu+\frac{Z}{\rho}c\right)=0. \]

It follows from this that the expression \(\varphi-\mu+\dfrac{Z}{\rho}c\) does not depend on the concentration \(c\). Since at small concentrations \(\dfrac{f}{w}-S=0\), \(\varphi-\mu+\dfrac{Z}{\rho}c=0\), it follows that at all concentrations the relations hold:

\[ \varphi-\mu+\frac{Z}{\rho}c=0, \tag{6.27} \]

\[ f=Sw. \tag{6.28} \]

The meaning of relation (6.28) is clear: the entropy of solutions, just as that of pure helium II, is transported only by the normal motion. The final complete system of hydrodynamic equations for solutions, taking into account (6.26), (6.27), and (6.28), has the following form:

\[ \left. \begin{aligned} \text{I.}\quad & \dot{\rho}+\operatorname{div}\mathbf j=0,\\ \text{II.}\quad & \frac{\partial}{\partial t}\mathbf j+\mathbf v_s\operatorname{div}\mathbf j+(\mathbf j\nabla)\mathbf v_s+(\mathbf v_n\nabla)\mathbf p\\ & \qquad\qquad +\mathbf p\,\operatorname{div}\mathbf v_n+\nabla p=0,\\ \text{III.}\quad & \dot S+\operatorname{div}S\mathbf v_n=0,\\ \text{IV.}\quad & (\rho c)^{\cdot}+\operatorname{div}(\rho c\mathbf v_n)=0,\\ \text{V.}\quad & \dot{\mathbf v}_s+\nabla\left(\frac{v_s^2}{2}+\mu-\frac{Z}{\rho}c\right)=0. \end{aligned} \right\} \tag{6.29} \]

Instead of equation II from (6.29), one may use the equation

\[ \dot{\mathbf p}+\mathbf p\,\operatorname{div}\mathbf v_n+\nabla(\mathbf p\mathbf v_n)-[\mathbf v_n\operatorname{rot}\mathbf p]+\rho c\nabla\frac{Z}{\rho}+S\nabla T=0, \tag{6.30} \]

which is obtained if, in || (6.29), one expresses the derivative \(v_s\) by means of equation V (6.29).

Expression (6.18) for the energy fluxes \(\mathbf Q\), taking (6.26) and (6.28) into account, is written in the form

\[ \mathbf Q=\mathbf j\left(\mu-\frac{Z}{\rho}c+\frac{v_s^2}{2}\right)+Zc\,\mathbf v_n+ST\,\mathbf v_n+(\mathbf v_n\mathbf p)\mathbf v_n. \tag{6.31} \]

Let us express the potentials \(\mu\) and \(Z\) occurring in the formulas of the present paragraph in terms of the chemical potentials \(\mu_4\) of helium II and of the dissolved particles \(\mu_3\) in the solution. From considerations of additivity, the free energy of the solution may be represented in the form

\[ F=(N_3m_3+N_4m_4)\,f\left(T,\frac{V}{N_3m_3+N_4m_4},\frac{N_3m_3}{N_3m_3+N_4m_4}\right). \tag{6.32} \]

Here \(N_3\) and \(N_4\) are the numbers of atoms of helium II and of the dissolved substance, \(m_3\) and \(m_4\) are the masses of the corresponding atoms, and \(V\) is the volume occupied by the solution.

The free energy \(F\) of a unit volume of the solution is, evidently, equal to

\[ F=\rho f. \tag{6.33} \]

The chemical potential \(\mu_4\) of helium II in the solution is equal to

\[ \mu_4=\frac{\partial F}{m_4\partial N_4}=f-\frac{\partial f}{\partial v}v-\frac{\partial f}{\partial c}c. \tag{6.34} \]

In an analogous way, for the dissolved particles we find

\[ \mu_3=\frac{1}{m_3}\frac{\partial F}{\partial N_3}=f-\frac{\partial f}{\partial v}v+(1-c)\frac{\partial f}{\partial c}; \tag{6.35} \]

\[ c=\frac{N_3m_3}{N_3m_3+N_4m_4} \quad\text{— concentration of the solution,} \]

\[ v=\frac{V}{N_3m_3+N_4m_4}=\frac{1}{\rho} \quad\text{— specific volume.} \]

Differentiating expression (6.33) and taking into account relations (6.34) and (6.35), we find the desired connection

\[ Z=\left(\frac{\partial F}{\partial c}\right)_{\rho,T}=\rho(\mu_3-\mu_4), \tag{6.36} \]

\[ \mu=\left(\frac{\partial F}{\partial \rho}\right)_{T,c}=c\mu_3+(1-c)\mu_4. \tag{6.37} \]

With the aid of (6.37) we find the expression for the entropy of unit mass of the solution:

\[ \sigma=-\left(\frac{\partial\mu}{\partial T}\right)_p=-c\frac{\partial\mu_3}{\partial T}-(1-c)\frac{\partial\mu_4}{\partial T}. \tag{6.38} \]

In the general case, owing to the presence of interaction between the particles of the dissolved substance and helium II, expressions for the chemical potentials \(\mu_3\) and \(\mu_4\) cannot be written down. The only case,

when one can write expressions for \(\mu_3\) and \(\mu_4\), is the case of an ideal solution. In this case

\[ \mu_3=\mu_{30}+\frac{kT}{m_3}\ln c, \tag{6,39} \]

\[ \mu_4=\mu_{40}+\frac{kT}{m_4}\ln(1-c) \tag{6,40} \]

(\(\mu_{30}\) and \(\mu_{40}\) are the chemical potentials of the particles of the dissolved substance and of helium II). For the entropy \(\sigma\) we have, according to (6,38),

\[ \sigma=(1-c)\sigma_{40}+c\sigma_{30}-\frac{k}{m_4}(1-c)\ln(1-c)-\frac{k}{m_3}c\ln c. \tag{6,41} \]

According to (6,12) and (1,21), for the thermodynamic potential of the solution the identity holds

\[ \rho\,d\mu=-S\,dT+dp+Z\,dc-\mathbf{p}d(\mathbf{v}_n-\mathbf{v}_s). \tag{6,42} \]

From identity (6,42), for the case of small values of the velocity difference \((\mathbf{v}_n-\mathbf{v}_s)\) (see (1,32)), there follows the integral relation

\[ \mu=\mu_c(p,T,c)-\frac{\rho_n}{2\rho}(\mathbf{v}_n-\mathbf{v}_s)^2, \tag{6,43} \]

where the potential \(\mu_c\) satisfies the identity

\[ \rho\,d\mu_c=-s\,dT+dp+z\,dc. \tag{6,44} \]

The combination of potentials \(\mu\) and \(Z\), contained in the equation of motion (6,29), according to (6,36) and (6,37), is expressed through the chemical potentials as

\[ \mu-\frac{Z}{\rho}c=c\mu_3+(1-c)\mu_4-c(\mu_3-\mu_4)=\mu_4. \tag{6,45} \]

For weak solutions and small values of the difference \(\mathbf{v}_n-\mathbf{v}_s\), according to (6,40) and (1,33), the indicated combination is equal to

\[ \mu-\frac{Z}{\rho}c=\mu_{40}-\frac{kTc}{m_4}=\mu_0(p,T)-\frac{kTc}{m_4}. \tag{6,46} \]

In this case equation V from (6,29) takes the form

\[ \dot{\mathbf{v}}_s+\nabla\left\{\mu_0(p,T)+\frac{v_s^2}{2}-\frac{\rho_n}{2\rho}(\mathbf{v}_n-\mathbf{v}_s)^2-\frac{kTc}{m_4}\right\}=0. \tag{6,47} \]

In this form this equation coincides with the equation of motion given in \(^{26}\). The first four equations (6,29) become the equations for weak solutions \(^{26}\), if the momenta \(\mathbf{j}\) and \(\mathbf{p}\) in them are expressed with the aid of the relations (1,27) and (1,29), naturally also valid for the case of solutions (in this case the densities \(\rho_n\) and \(\rho_s\) are regarded as quantities independent of the relative velocity \(\mathbf{v}_n-\mathbf{v}_s\)).

In conclusion of the present paragraph it is necessary to make the following remark. The proof given above that the constancy

particles dissolved in helium II, at all concentrations, participate only in the normal motion, depends in an essential way on the assumption that there are two motions in the solution: one superfluid, with velocity \(\mathbf v_s\), and one normal, with velocity \(\mathbf v_n\). Such a result would not obtain, for example, if one were to assume that in the solution, in addition to the normal motion, there are two superfluid motions with different velocities. However, such an assumption hardly has any meaning for a solution of nonsuperfluid particles in helium II. For a solution of a superfluid substance in helium II, however, such a possibility could be realized, i.e., two superfluid motions of both superfluid components (helium II and the dissolved substance) would be possible in the solution.

7. Dissipative processes in solutions\(^6\)

Let us dwell on dissipative processes in solutions; in clarifying this question we proceed in complete analogy with the way this was done for pure helium II. In this case we have additional terms in the equations of motion and in the continuity equation for the dissolved particles

\[ \frac{\partial}{\partial t} j_i+\frac{\partial}{\partial x_k}\left(\Pi_{ik}+\tau_{ik}\right)=0 \quad (\tau_{ik}=\tau\delta_{ik}+\mu_{ik}), \tag{7,1} \]

\[ \dot{\mathbf v}_s+\nabla\left(\mu-\frac{z}{\rho}c+\frac{v_s^2}{2}+h\right)=0, \tag{7,2} \]

\[ (\rho c)^{\cdot}+\operatorname{div}(\rho c\,\mathbf v_n+\mathbf g)=0. \tag{7,3} \]

The additional terms take into account possible dissipative processes. Further, from the law of conservation of energy we obtain the expression for the energy flux

\[ \mathbf Q=\mathbf j\left(\mu-\frac{Z}{\rho}c+\frac{v_s^2}{2}\right)+Zc\,\mathbf v_n+ST\,\mathbf v_n+ \]

\[ +(\mathbf v_n\mathbf p)\,\mathbf v_n+h(\mathbf j-\rho\mathbf v_n)+\boldsymbol{\tau}\mathbf v_n+\mathbf q+\boldsymbol{\psi} \quad (v_k=v_{ni}\psi_{ik}) \tag{7,4} \]

and the equation determining the rate of change of entropy,

\[ T\left\{\dot S+\operatorname{div}\left(S\mathbf v_n+\frac{\mathbf q}{T}-\frac{\mathbf g Z}{\rho T}\right)\right\} = -\left\{h\,\operatorname{div}(\mathbf j-\rho\mathbf v_n)+\right. \]

\[ \left. +\tau\,\operatorname{div}\mathbf v_n +\frac{1}{2}\mu_{ik}\left( \frac{\partial v_{ni}}{\partial x_k} +\frac{\partial v_{nk}}{\partial x_i} -\frac{2}{3}\delta_{ik}\frac{\partial v_{kl}}{\partial x_l} \right) +\right. \]

\[ \left. +\frac{\mathbf q\nabla T}{T} +\mathbf g\,T\nabla\frac{Z}{\rho T} \right\}. \tag{7,5} \]

HYDRODYNAMICS OF HELIUM II

Next, from the positivity of the dissipative function we conclude that the unknown coefficients must have the form

\[ \tau=-\zeta_1 \operatorname{div}(\mathbf{j}-\rho \mathbf{v}_n)-\zeta_2 \operatorname{div}\mathbf{v}_n, \tag{7,6} \]

\[ h=-\zeta_3 \operatorname{div}(\mathbf{j}-\rho \mathbf{v}_n)-\zeta_1 \operatorname{div}\mathbf{v}_n, \tag{7,7} \]

\[ \mu_{ik}=-\eta\left(\frac{\partial v_{ni}}{\partial x_k}+\frac{\partial v_{nk}}{\partial x_i} -\frac{2}{3}\delta_{ik}\frac{\partial v_{kl}}{\partial x_l}\right), \tag{7,8} \]

\[ \mathbf{g}=-\alpha \nabla \frac{Z}{\rho T}-\beta \frac{1}{T^2}\nabla T, \tag{7,9} \]

\[ \mathbf{q}=-\gamma \nabla \frac{Z}{\rho T}-\delta \frac{1}{T^2}\nabla T. \tag{7,10} \]

By virtue of the symmetry principle for kinetic coefficients, two relations hold:

\[ \zeta_1=\zeta_4,\qquad \beta=\gamma. \tag{7,11} \]

The coefficients \(\zeta_1,\ \zeta_2,\ \zeta_3,\ \zeta_4\) have the meaning of the coefficients of second viscosity of the solution; \(\eta\) is the coefficient of first viscosity of the solution.

In the expression for the heat flux \(\mathbf{q}\) it is convenient to eliminate \(\nabla \dfrac{Z}{\rho T}\), expressing this quantity through the impurity flux \(\mathbf{g}\) and \(\nabla T\)

\[ -\mathbf{q}=\frac{\gamma}{\alpha}\mathbf{g} +\left(\delta-\frac{\beta\gamma}{\alpha}\right)\frac{\nabla T}{T^2}. \tag{7,12} \]

We define the coefficient of thermal conductivity in such a way that, when the flux \(\mathbf{g}\) is equal to zero, the heat flux would be equal to \(-\varkappa\nabla T\). Thus we have

\[ \varkappa=\left(\delta-\frac{\beta\gamma}{\alpha}\right)\frac{1}{T^2}. \tag{7,13} \]

Next, analogously to how this is done in ordinary solutions, we pass in the expression for the impurity flux \(\mathbf{g}\) to the usual variables \(p,\ T\), and \(c\), and introduce the notations

\[ D=\frac{\alpha}{\rho}\frac{\partial}{\partial c}\left(\frac{Z}{\rho T}\right), \tag{7,14} \]

\[ \rho D k_T=\alpha T\frac{\partial}{\partial T}\left(\frac{Z}{\rho T}\right)+\frac{\beta}{T}, \tag{7,15} \]

\[ k_p= \frac{ p\frac{\partial}{\partial p}\left(\dfrac{Z}{\rho T}\right) }{ \frac{\partial}{\partial c}\left(\dfrac{Z}{\rho T}\right) }. \tag{7,16} \]

After this the fluxes \(\mathbf{g}\) and \(\mathbf{q}\) acquire the form

\[ -\mathbf{g}=\rho D\left(\nabla c+\frac{k_T}{T}\nabla T+\frac{k_p}{p}\nabla p\right), \tag{7,17} \]

\[ -\mathbf{q}=T^2\left(\frac{\partial}{\partial T}\frac{Z}{\rho T} -\frac{k_T}{T}\frac{\partial}{\partial c}\frac{Z}{\rho T}\right)\mathbf{g} +\varkappa\nabla T. \tag{7,18} \]

The quantity \(D\) is the diffusion coefficient, and \(k_TD\) is the thermal-diffusion coefficient. The quantity \(k_pD\) is called the barodiffusion coefficient. This latter coefficient is significant only in the presence of a pressure gradient. It does not depend on the kinetics and is completely determined by the thermodynamic properties of the solution.

Let us now determine the form of the fluxes \(\mathbf{g}\) and \(\mathbf{q}\) under stationary conditions. According to equations (6,29), and under stationary conditions (for small gradients),

\[ p=\operatorname{const}, \qquad \mu-\frac{Z}{\rho}\,c=\operatorname{const}. \tag{7,19} \]

Using the thermodynamic identity (6,42), we find the relation between the gradients of concentration and temperature under stationary conditions:

\[ \nabla c\,\frac{\partial}{\partial c}\left(\frac{Z}{\rho}\right) = -\left[ \frac{S}{\rho c} + \frac{\partial}{\partial T}\left(\frac{Z}{\rho}\right) \right]\nabla T = \nabla T\,\frac{\partial}{\partial c}\left(\frac{S}{\rho c}\right). \tag{7,20} \]

Under stationary conditions, according to the hydrodynamic equations (6,29), the total flux of dissolved particles is equal to zero,

\[ \mathbf{g}+\rho c\mathbf{v}_n=0, \tag{7,21} \]

and the total heat flux is equal to the power \(Q\) released per unit surface area of the body,

\[ Q=(Zc+ST)\mathbf{v}_n+\mathbf{q}. \tag{7,22} \]

The velocity \(\mathbf{v}_n\) under stationary conditions is not equal to zero; from equation (6,29) it follows in this case only that the total momentum is zero. According to (7,20) and (7,21), both fluxes \(\mathbf{g}\) and \(\mathbf{q}\), as well as the velocity \(\mathbf{v}_n\), under stationary conditions are proportional to the temperature gradient. We shall express them through \(\nabla T\) and substitute into formula (7,22). In this way we find

\[ Q=-\varkappa\nabla T -\rho D \frac{T}{\dfrac{\partial}{\partial c}\left(\dfrac{Z}{\rho}\right)} \times \]

\[ \times \left\{ c\frac{\partial}{\partial c}\left(\frac{S}{\rho c}\right) + \frac{k_T}{T} \frac{\partial}{\partial c}\left(\frac{Z}{\rho}\right) \right\}^{2} \nabla T. \tag{7,23} \]

It is now clear that, for characterizing solutions, it is convenient to introduce a certain effective thermal conductivity \(\varkappa_{\mathrm{eff}}\), which is a definite combination of the diffusion, thermal-diffusion, and thermal-conductivity coefficients and is equal to

\[ \varkappa_{\mathrm{eff}} = \varkappa + \rho D \frac{T}{\dfrac{\partial}{\partial c}\left(\dfrac{Z}{\rho}\right)} \left\{ c\frac{\partial}{\partial c}\left(\frac{S}{\rho c}\right) + \frac{k_T}{T} \frac{\partial}{\partial c}\left(\frac{Z}{\rho}\right) \right\}^{2}. \tag{7,24} \]

The effective thermal conductivity \(\varkappa_{\mathrm{eff}}\) relates \(\nabla T\) to the power released in the solution,

\[ Q=-\varkappa_{\mathrm{eff}}\nabla T . \tag{7.25} \]

Using formulas (6.26), (6.27), we compute \(\varkappa_{\mathrm{eff}}\) for dilute solutions:

\[ \varkappa_{\mathrm{eff}} = \varkappa+ \frac{\rho Dm_3}{kc} \left\{ \left(\sigma_0+\frac{kc}{m_3}\right) -\frac{k}{m_3}k_T \right\}^{2}. \tag{7.26} \]

(\(\sigma_0\) is the entropy per unit mass of pure helium II). For sufficiently dilute solutions, the second term in formula (7.26) is inversely proportional to the concentration of the solution and exceeds the thermal-conductivity coefficient \(\varkappa\). Let us note here that in pure helium II, in the presence of \(\nabla T\), there is no stationary solution of the hydrodynamic equations. In the stationary case, from the constancy of the pressure \(p\) and the potential \(\mu\), the constancy of the temperature in helium follows.

We now write, in final form, the hydrodynamic equations for solutions in helium II with dissipative terms:

\[ \left. \begin{aligned} &\dot{\rho}+\operatorname{div}\mathbf{j}=0,\\ &\frac{d}{dt}j_i+\frac{\partial \Pi_{ik}}{\partial x_k} = \frac{\partial}{\partial x_k} \left\{ \eta\left( \frac{\partial v_{ni}}{\partial x_k} +\frac{\partial v_{nk}}{\partial x_i} -\frac{2}{3}\delta_{ik}\frac{\partial v_{nl}}{\partial x_l} \right) \right\} + \frac{\partial}{\partial x_i} \left\{ \zeta_1\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) +\zeta_2\operatorname{div}\mathbf{v}_n \right\},\\ &(\dot{\rho c})+\operatorname{div}\rho c\mathbf{v}_n = \operatorname{div}\left\{ \rho D\left( \nabla c+\frac{k_T}{T}\nabla T+\frac{k_p}{p}\nabla p \right) \right\},\\ &\dot{\mathbf{v}}_s+\nabla\left( \mu-\frac{Z}{\rho}c+\frac{v_s^2}{2} \right) = -\nabla\left\{ \zeta_3\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) +\zeta_4\operatorname{div}\mathbf{v}_n \right\}. \end{aligned} \right\} \tag{7.27} \]

According to the same relations, the entropy-growth equation takes the form:

\[ T\left\{ \dot{S} + \operatorname{div} \left[ S\mathbf{v}_n+ \frac{\mathbf{q}-\dfrac{Z\mathbf{g}}{\rho}}{T} \right] \right\} =R, \tag{7.28} \]

\[ \begin{aligned} R={}& \zeta_3\left[\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n)\right]^2 +\zeta_2\left[\operatorname{div}\mathbf{v}_n\right]^2 +2\zeta_1\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n)\operatorname{div}\mathbf{v}_n \\ &+\frac{1}{2}\eta \left( \frac{\partial v_{ni}}{\partial x_k} +\frac{\partial v_{nk}}{\partial x_i} -\frac{2}{3}\delta_{ik}\frac{\partial v_{nl}}{\partial x_l} \right)^2 +\frac{\varkappa(\nabla T)^2}{T} \\ &+\rho D\frac{\partial}{\partial c}\left(\frac{Z}{\rho}\right) \left[ \nabla c+\frac{k_T}{T}\nabla T+\frac{k_p}{p}\nabla p \right]^2 . \end{aligned} \tag{7.29} \]

8. Sound in Solutions of Foreign Particles in Helium II

The properties of sound in weak solutions of foreign particles in helium II were elucidated by Pomeranchuk \(^{26}\). We shall use the system of equations (6.29) and obtain formulas determining the velocity of sound in solutions of arbitrary concentration. The propagation of sound in solutions is described by the system of equations (6.20), in which it is sufficient to retain only terms linear in the derivatives and velocities. The indicated system, thus linearized, is written in the form:

\[ \dot{\rho}+\operatorname{div}\mathbf{j}=0, \tag{8.1} \]

\[ \frac{\partial}{\partial t}\mathbf{j}+\nabla p=0, \tag{8.2} \]

\[ (\rho\sigma)^{\cdot}+\rho\sigma\,\operatorname{div}\mathbf{v}_{n}=0 \quad (S=\rho\sigma), \tag{8.3} \]

\[ (\rho c)^{\cdot}+\rho c\,\operatorname{div}\mathbf{v}_{n}=0, \tag{8.4} \]

\[ \dot{\mathbf{v}}_{s}+\nabla\left(\mu-\frac{Z}{\rho}c\right)=0. \tag{8.5} \]

Eliminating from the system (8.1)—(8.5) the velocities \(\mathbf{v}_{n}\) and \(\mathbf{v}_{s}\), we obtain three equations:

\[ \frac{\rho_{n}}{\rho_{s}}\frac{\ddot{\sigma}}{\sigma} = \sigma\Delta T+c\Delta\frac{Z}{\rho}, \tag{8.6} \]

\[ \ddot{\rho}=\Delta p, \tag{8.7} \]

\[ \frac{\dot{c}}{c}=\frac{\dot{\sigma}}{\sigma}. \tag{8.8} \]

Let us choose as independent variables the temperature \(T\), the pressure \(p\), and the concentration \(c\). In a sound wave these quantities may be represented as sums of constant equilibrium values and small additions, which we shall denote by the same letters with primes. We seek a solution of the system (8.6)—(8.8) corresponding to a plane wave; in this case the indicated small additions vary according to the law \(e^{i\omega(t-x/u)}\) (\(x\) is the direction of propagation of the wave, \(u\) the velocity of sound). Representing all quantities in the system (8.6)—(8.8) in this form, we obtain:

\[ \left\{ \frac{\rho_{n}}{\rho_{s}\sigma}\frac{\partial\sigma}{\partial T}u^{2} -\sigma -c\frac{\partial}{\partial T}\frac{Z}{\rho} \right\}T' + \left\{ \frac{\rho_{n}}{\rho_{s}\sigma}\frac{\partial\sigma}{\partial p}u^{2} -c\frac{\partial}{\partial p}\frac{Z}{\rho} \right\}p' + \left\{ \frac{\rho_{n}}{\rho_{s}\sigma}\frac{\partial\sigma}{\partial c}u^{2} -c\frac{\partial}{\partial c}\frac{Z}{\rho} \right\}c' =0, \tag{8.9} \]

\[ \left. \begin{aligned} \frac{\partial\rho}{\partial T}u^{2}T' + \left(\frac{\partial\rho}{\partial p}u^{2}-1\right)p' + \frac{\partial\rho}{\partial c}u^{2}c' &=0,\\ c\frac{\partial\sigma}{\partial T}T' + c\frac{\partial\sigma}{\partial p}p' + \left(c\frac{\partial\sigma}{\partial c}-\sigma\right)c' &=0. \end{aligned} \right\} \tag{8.10} \]

The compatibility condition for the system obtained gives an equation determining the velocity of sound in the solution

\[ u^4 \frac{\rho_n}{\rho_s} \left\{ \frac{d\sigma}{dT}\frac{d\rho}{dp} - \frac{d\sigma}{dp}\frac{d\rho}{dT} \right\} -u^2 \left\{ \frac{\partial \rho}{\partial p} \left[ \sigma+c\frac{\partial}{\partial T}\frac{Z}{\rho} \right]\sigma +\frac{\rho_n}{\rho_s}\frac{d\sigma}{dT} - \frac{d\rho}{dT}\frac{d}{dp}\left(\frac{Z}{\rho}\right)\sigma c \right\} +\sigma\left(\sigma+c\frac{d}{dT}\frac{Z}{\rho}\right)=0. \tag{8,11} \]

For brevity we have introduced total derivatives:

\[ \frac{df}{dT} = \frac{\partial f}{\partial T} + \frac{\partial f}{\partial c}\frac{c}{\bar{\sigma}}\frac{\partial \sigma}{\partial T}, \qquad \frac{df}{dp} = \frac{\partial f}{\partial p} + \frac{\partial f}{\partial c}\frac{c}{\bar{\sigma}}\frac{\partial \sigma}{\partial p}, \tag{8,12} \]

\[ \bar{\sigma}=\sigma-c\frac{\partial \sigma}{\partial c} \tag{8,13} \]

(\(f\) is one of the three thermodynamic functions \(\sigma,\rho,Z/\rho\)). Equation (8,11) is of little use for investigation because of its extreme cumbersomeness. We shall use a simplifying circumstance, namely that the derivative \(\partial\rho/\partial T\) practically always proves to be extremely small. We neglect in (8,11) all terms containing the indicated derivative; in addition, we use the relations following from the thermodynamic identity

\[ d\mu=\frac{1}{\rho}\,dp-\sigma\,dT+\frac{Z}{\rho}\,dc, \tag{8,14} \]

namely

\[ \frac{\partial}{\partial p}\frac{Z}{\rho} = -\frac{1}{\rho^2}\frac{\partial \rho}{\partial c}, \qquad \frac{\partial \sigma}{\partial p} = -\frac{1}{\rho^2}\frac{\partial \rho}{\partial T}, \qquad \frac{\partial \sigma}{\partial c} = -\frac{\partial}{\partial T}\frac{Z}{\rho}. \tag{8,15} \]

As a result, equation (8,11) takes the form:

\[ u^4 -u^2 \left\{ \frac{\rho_s}{\rho_n} \left[ \frac{\bar{\sigma}^2}{\dfrac{\partial \sigma}{\partial T}} + c^2\frac{\partial}{\partial c}\frac{Z}{\rho} \right] + \left[ 1+\frac{\rho_s}{\rho_n} \left(\frac{c}{\rho}\frac{\partial \rho}{\partial c}\right)^2 \right]\bigg/ \frac{\partial \rho}{\partial p} \right\} + \frac{\rho_s}{\rho_n} \left[ \frac{\bar{\sigma}^2}{\dfrac{\partial \sigma}{\partial T}} + c^2\frac{\partial}{\partial c}\frac{Z}{\rho} \right]\bigg/ \left(\frac{\partial \rho}{\partial p}\right) =0. \tag{8,16} \]

We solve this equation with respect to \(u^2\) and use the fact that one of the roots, \(u_1^2\), determining the velocity of first sound, significantly exceeds in magnitude the second, \(u_2^2\), determining the velocity of second sound; as a result we find \((u_2^2 \ll u_1^2)\)

\[ u_1^2= \left(\frac{\partial p}{\partial \rho}\right)_{c,T} \left[ 1+\frac{\rho_s}{\rho_n} \left(\frac{\partial \rho}{\partial c}\frac{c}{\rho}\right)^2 \right], \tag{8,17} \]

\[ u_2^2= \frac{\rho_s}{\rho_n} \left[ \bar{\sigma}^2\left(\frac{\partial T}{\partial \sigma}\right)_{c,p} + c^2\frac{\partial}{\partial c}\left(\frac{Z}{\rho}\right) \right]\bigg/ \left[ 1+\frac{\rho_s}{\rho_n} \left(\frac{\partial \rho}{\partial c}\frac{c}{\rho}\right)^2 \right]. \tag{8,18} \]

In the first approximation, for small values of the concentration \(c\), the velocity of first sound does not depend on the concentration*). The velocity of second sound, however, contains terms linear in the concentration \(c\), namely \(\bar{\sigma}\) and \(c^2 \dfrac{\partial}{\partial c}\dfrac{Z}{\rho}\) (the potential \(Z\) depends on the concentration \(c\) according to a logarithmic law). The quantities \(\bar{\sigma}\) and \(\dfrac{\partial}{\partial c}\dfrac{Z}{\rho}\), which determine the velocity of second sound (8,18) for an ideal solution, are equal to

\[ \bar{\sigma}=\sigma-c\frac{\partial \sigma}{\partial c} =\sigma_{40}-\frac{k}{m_4}\{c+\ln(1-c)\}+\frac{k}{m_3}c, \tag{8,19} \]

\[ c^2\frac{\partial}{\partial c}\frac{Z}{\rho} =kT\left[\frac{c^2}{m_4(1-c)}+\frac{c}{m_3}\right]. \tag{8,20} \]

For weak solutions, expressions (8,19) and (8,20) become

\[ \bar{\sigma}=\sigma_{40}+\frac{kc}{m_3},\qquad c^2\frac{\partial}{\partial c}\frac{Z}{\rho}=\frac{kTc}{m_3}, \]

and formula (8,18) becomes I. Pomeranchuk’s formula

\[ u_2^2=\frac{\rho_s}{\rho_n} \left[ \frac{\partial T}{\partial \sigma} \left(\sigma_{40}+\frac{kc}{m_3}\right)^2 +\frac{kTc}{m_3} \right]^{**)}. \tag{8,21} \]

9. Hydrodynamics of Solutions of Two Superfluid Liquids\(^{27}\)

Let us consider a mixture of two superfluid liquids; for example, liquid He\(^4\) and liquid He\(^6\). In this case it is in principle possible for three motions to occur in the liquid: one normal motion with velocity \(\mathbf{v}_n\) and two superfluid motions with velocities \(\mathbf{v}'_s\) and \(\mathbf{v}''_s\). We shall show how, from the conservation laws and the conditions of potentiality of the superfluid motions, one can obtain the equations of such three-velocity hydrodynamics. To derive the equations, we shall consider the liquid in that reference frame in which the normal (nonsuperfluid) part of the liquid is at rest. In this system the total energy of the liquid can be written in the form

\[ E=\rho\frac{v_n^2}{2}+(\rho'+\rho'')\mathbf{v}_n+\varepsilon. \tag{9.1} \]

*) Let us emphasize that this result is obtained only under the assumption that \(\dfrac{\partial \rho}{\partial T}=0\), since the exact expression for \(u_1^2\), following from equation (8,11), contains terms linear in the concentration and proportional to the product \(\dfrac{\partial \rho}{\partial T}\dfrac{\partial \rho}{\partial c}\).

**) The molar concentration \(\varepsilon=N_3/(N_3+N_4)\) is related to the concentration \(c=N_3m_3/(N_3m_3+N_4m_4)\), which appears in our formulas, by the relation

\[ \frac{1}{\varepsilon}-1=\frac{m_3}{m_4}\left(\frac{1}{c}-1\right). \]

Here the following notation has been adopted: \(\rho\) is the density of the liquid, \(\mathbf{p}'\) and \(\mathbf{p}''\) are the relative momenta of two superfluid motions, \(\varepsilon\) is the internal energy of the liquid. It is a function both of the thermodynamic variables—the densities \(\rho_1, \rho_2\) and the entropy \(S\)—and of the velocities of relative motion \(\mathbf{v}'_s-\mathbf{v}_n\) and \(\mathbf{v}''_s-\mathbf{v}_n\), and is defined by a thermodynamic identity.

The form of the thermodynamic identity is established as follows. For this purpose let us recall the expression for the total energy \(E\), used by us in deriving the hydrodynamic equations for helium II (1.8),

\[ E=\rho\frac{v_s^2}{2}+\mathbf{p}'\mathbf{v}_s+\varepsilon'. \tag{9.2} \]

Here \(\mathbf{p}'\) is the momentum of relative motion in a reference frame moving with velocity \(\mathbf{v}_s\); it is expressed in terms of the total momentum of the liquid \(\mathbf{j}\) by

\[ \mathbf{p}'=\mathbf{j}-\rho\mathbf{v}_s. \tag{9.3} \]

The energy \(\varepsilon'\) was defined by the thermodynamic identity

\[ d\varepsilon'=\mu' d\rho+T\,dS+(\mathbf{v}_n-\mathbf{v}_s,\, d\mathbf{p}'). \tag{9.4} \]

If, instead of the momentum \(\mathbf{p}'\), one introduces the momentum \(\mathbf{p}\) in a reference frame moving with velocity \(\mathbf{v}_n\),

\[ \mathbf{p}=\mathbf{j}-\rho\mathbf{v}_n=\mathbf{p}'+\rho(\mathbf{v}_s-\mathbf{v}_n), \tag{9.5} \]

then the expression for the energy \(E\) is brought to the following form:

\[ E=\rho\frac{v_n^2}{2}+\mathbf{p}\mathbf{v}_n+\varepsilon, \tag{9.6} \]

where the internal energy \(\varepsilon\) is now defined by an identity different from (9.4), namely

\[ d\varepsilon=\mu d\rho+T\,dS+\mathbf{p}\,d(\mathbf{v}_s-\mathbf{v}_n) \tag{9.7} \]

with the chemical potential \(\mu\), related to \(\mu'\) by the relation

\[ \mu=\mu'+\frac{(\mathbf{v}_n-\mathbf{v}_s)^2}{2}. \]

Thus, in the case when we pass to a reference frame associated with the normal motion, the internal energy \(\varepsilon\) should be regarded as a function of the density, entropy, and relative velocity. In this case relation (9.7) is also the definition of the momentum of relative motion \(\mathbf{p}\).

In the case of three-velocity hydrodynamics, the thermodynamic identity is written completely analogously to (9.7):

\[ d\varepsilon=\mu_1 d\rho_1+\mu_2 d\rho_2+Tds+\mathbf{p}'d(\mathbf{v}'_s-\mathbf{v}_n)+\mathbf{p}''d(\mathbf{v}''_s-\mathbf{v}_n), \tag{9.8} \]

\(\mu_1\) and \(\mu_2\) are the chemical potentials of the components of the solution. The densities \(\rho_1\) and \(\rho_2\) are expressed in terms of the concentration \(c\) of the solution as

\[ \rho_1=\rho c,\quad \rho_2=\rho(1-c). \tag{9.9} \]

Let us now write the conservation laws for the energy \(E\), the momentum of the liquid \(\mathbf{j}=\mathbf{p}' + \mathbf{p}''+\rho \mathbf{v}_n\), the amount of substance, and the entropy. For the energy \(E\) we have:

\[ \dot E+\operatorname{div}\mathbf{Q}=0, \tag{9.10} \]

where \(\mathbf{Q}\) is the energy-flux vector; its form is as yet unknown to us. The time derivatives of the momentum \(\mathbf{j}\) must be equal to the divergence of a certain tensor

\[ \frac{\partial}{\partial t}j_i+\frac{\partial \Pi_{ik}}{\partial x_k}=0. \tag{9.11} \]

The form of the symmetric momentum-flux tensor \(\Pi_{ik}\) can be established in the same way as was done in deriving the equations of hydrodynamics of a superfluid liquid (Sec. 1). Let us express the tensor \(\Pi_{ik}\) in terms of its value \(\pi_{ik}\) in a frame of reference moving with velocity \(\mathbf{v}_n\):

\[ \Pi_{ik}=\rho v_{ni}v_{nk}+(p_i'+p_i'')v_{nk}+(p_k'+p_k'')v_{ni}+\pi_{ik}. \tag{9.12} \]

The form of the tensor \(\pi_{ik}\) will be found below.

The conservation laws for matter are written in the form of continuity equations

\[ \dot\rho_1+\operatorname{div}\left(\mathbf{p}'+\rho_1\mathbf{v}_n+\mathbf{g}'\right)=0, \tag{9.13} \]

\[ \dot\rho_2+\operatorname{div}\left(\mathbf{p}''+\rho_2\mathbf{v}_n+\mathbf{g}''\right)=0, \tag{9.14} \]

which contain the unknown vectors \(\mathbf{g}'\) and \(\mathbf{g}''\). Since for the whole liquid the continuity equation holds,

\[ \dot\rho+\operatorname{div}\mathbf{j}=0, \tag{9.15} \]

there is the following relation between the vectors \(\mathbf{g}'\) and \(\mathbf{g}''\):

\[ \mathbf{g}'+\mathbf{g}''=0. \tag{9.16} \]

We write the continuity equation for the entropy in the form

\[ \dot S+\operatorname{div}\left(S\mathbf{v}_n+\mathbf{f}\right)=0 \tag{9.17} \]

with the unknown vector \(\mathbf{f}\) in the entropy flux.

We choose the equations of superfluid motion in such a way that the conditions \(\operatorname{rot}\mathbf{v}_s'=0\), \(\operatorname{rot}\mathbf{v}_s''=0\) be satisfied:

\[ \dot{\mathbf{v}}_s' + \nabla\left(\varphi_1-\frac{v_n^2}{2}+\mathbf{v}_n\mathbf{v}_s'\right)=0, \tag{9.18} \]

\[ \dot{\mathbf{v}}_s'' + \nabla\left(\varphi_2-\frac{v_n^2}{2}+\mathbf{v}_n\mathbf{v}_s''\right)=0. \tag{9.19} \]

Here \(\varphi_1\) and \(\varphi_2\) are functions still unknown.

Our problem now consists in finding the form of the unknown functions by using the conservation laws.

For this purpose we shall calculate the derivative of the energy \(E\) with respect to time and, with the aid of the hydrodynamic equations (9.13), (9.14), (9.17), (9.18), (9.19), separate out all terms that are complete divergences. According to (9.1) we have

\[ \dot E = \dot\rho\,\frac{v_n^2}{2} +\rho \mathbf v_n \dot{\mathbf v}_n +(\dot{\mathbf p}' + \dot{\mathbf p}'')\mathbf v_n +(\mathbf p' + \mathbf p'')\dot{\mathbf v}_n +\mu_1\dot\rho_1+\mu_2\dot\rho_2+T\dot S +(\dot{\mathbf p}',\,\mathbf v_s' - \mathbf v_n) +(\dot{\mathbf p}'',\,\mathbf v_s'' - \mathbf v_n) = \]

\[ = \dot\rho\,\frac{v_n^2}{2} +\dot{\mathbf j}\mathbf v_n +\mathbf p'\dot{\mathbf v}_s' +\mathbf p''\dot{\mathbf v}_s'' +\mu_1\dot\rho_1+\mu_2\dot\rho_2+T\dot S. \tag{9.20} \]

Next we express here all time derivatives with the aid of equations (9.13)—(9.19):

\[ \dot E = -\left(\mu_1-\frac{v_n^2}{2}\right) \operatorname{div}(\mathbf p' + \rho_1\mathbf v_n+\mathbf g') - \]

\[ - \left(\mu_2-\frac{v_n^2}{2}\right) \operatorname{div}(\mathbf p''+\rho_2\mathbf v_n+\mathbf g'') -\mathbf p'\nabla\left(\varphi_1-\frac{v_n^2}{2}+\mathbf v_n\mathbf v_s'\right) - \]

\[ - \mathbf p''\nabla\left(\varphi_2-\frac{v_n^2}{2}+\mathbf v_n\mathbf v_s''\right) -v_{ni}\frac{\partial \Pi_{ik}}{\partial x_k} -T\,\operatorname{div}(S\mathbf v_n+\mathbf f). \tag{9.21} \]

Separating out from this the complete divergences, after simple calculations we obtain:

\[ \dot E = -\operatorname{div}\left\{ (\mathbf p'+\rho_1\mathbf v_n+\mathbf g') \left(\mu_1-\frac{v_n^2}{2}\right) + \right. \]

\[ \left. +(\mathbf p''+\rho_2\mathbf v_n+\mathbf g'') \left(\mu_2-\frac{v_n^2}{2}\right) +T(S\mathbf v_n+\mathbf f) \right\} + \]

\[ + \left\{ -\rho\mathbf v_n\nabla\frac{v_n^2}{2} +\mathbf v_n(\rho_1\nabla\mu_1+\rho_2\nabla\mu_2+S\nabla T) -v_{ni}\frac{\partial \Pi_{ik}}{\partial x_k} - \right. \]

\[ \left. -\mathbf p'\nabla(\mathbf v_n\mathbf v_s') -\mathbf p''\nabla(\mathbf v_n\mathbf v_s'') \right\} + \left\{ \mathbf p'\nabla(\mu_1-\varphi_1) +\mathbf p''\nabla(\mu_2-\varphi_2) + \right. \]

\[ \left. +\mathbf g'\nabla(\mu_1-\mu_2) +\mathbf f\nabla T \right\}. \tag{9.22} \]

Now let us transform the second curly bracket in (9.22). For this purpose it is convenient to choose the tensor \(\pi_{ik}\) in the following way (\(\mathfrak M_{ik}\) is an unknown tensor):

\[ \pi_{ik}=(-\varepsilon+\mu_1\rho_1+\mu_2\rho_2+TS)\delta_{ik}+\mathfrak M_{ik}. \tag{9.23} \]

After substituting \(\Pi_{ik}\) in the form (9.12) with \(\pi_{ik}\) so chosen, the second brace is reduced to the form

\[ -\operatorname{div}\{(\mathbf j\mathbf v_n)\mathbf v_n+\mathbf p'(\mathbf v_n\mathbf v_s')+\mathbf p''(\mathbf v_n\mathbf v_s'')\}- \]

\[ -\,v_{ni}\frac{\partial}{\partial x_k} \left\{\mathfrak M_{ik}-p_k'(v_{si}'-v_{ni})-p_k''(v_{si}''-v_{ni})\right\}. \]

Using this result, we obtain the following expression for \(\dot E\):

\[ \begin{aligned} \dot E={}&-\operatorname{div}\Bigg\{ (\mathbf p'+\rho_1\mathbf v_n+\mathbf g') \left(\mu_1-\frac{v_n^2}{2}\right)+ \\ &\quad +(\mathbf p''+\rho_2\mathbf v_n+\mathbf g'') \left(\mu_2-\frac{v_n^2}{2}\right) +\mathbf v_n(\mathbf j\mathbf v_n)+\mathbf p'(\mathbf v_n\mathbf v_s')+ \\ &\quad +\mathbf p''(\mathbf v_n\mathbf v_s'') \Bigg\} +\Bigg\{ \mathbf p'\nabla(\mu_1-\varphi_1) +\mathbf p''\nabla(\mu_2-\varphi_2) +\mathbf g''\nabla(\mu_1-\mu_2)+ \\ &\quad +\mathbf f\nabla T +v_{ni}\frac{\partial}{\partial x_k} \left[p_k'(v_{si}'-v_{ni})+p_k''(v_{si}''-v_{ni})-\mathfrak M_{ik}\right] \Bigg\}. \end{aligned} \tag{9.24} \]

The quantities \(\varphi_1\), \(\varphi_2\), \(\mathbf f\), \(\mathbf g''\), and \(\mathfrak M_{ik}\), in the absence of dissipation, may depend only on the thermodynamic variables and velocities, and cannot depend on their derivatives with respect to time and coordinates. By examining various limiting cases, such as: temperature equal to zero, small concentrations of one of the components, etc., one can determine unambiguously the form of the unknown functions. To simplify the problem, let us invoke certain physical considerations. Namely, in a superfluid the entire entropy is contained in the thermal excitations, which participate only in the normal motion. Therefore the entropy flux is equal to the product \(S\mathbf v_n\), and the vector \(\mathbf f\), which we introduced into the flux, should simply be set equal to zero,

\[ \mathbf f=0. \tag{9.25} \]

As for the tensor \(\mathfrak M_{ik}\), we also know its form to a certain extent. Indeed, if we introduce the densities of the liquid associated with the normal \(\rho_n\) and the two superfluid motions \((\rho_s'\) and \(\rho_s'')\), then the tensor \(\Pi_{ik}\) can be written in the form

\[ \Pi_{ik}=\rho_n v_{ni}v_{nk}+\rho_s'v_{si}'v_{sk}'+\rho_s''v_{si}''v_{sk}''+p\delta_{ik} \tag{9.26} \]

(\(p\) is the pressure).

The sum \(\rho_n+\rho_s'+\rho_s''\) is equal to the total density of the liquid \(\rho\). The momenta of the relative motion \(\mathbf p'\) and \(\mathbf p''\) are then equal to

\[ \mathbf p'=\rho_s'(\mathbf v_s'-\mathbf v_n),\qquad \mathbf p''=\rho_s''(\mathbf v_s''-\mathbf v_n). \tag{9.27} \]

Taking into account (9.26) and (9.27), we find that the tensor \(\mathfrak{M}_{ik}\) is equal to\(^*\)

\[ \mathfrak{M}_{ik}=p'_k(v'_{si}-v_{ni})+p''_k(v''_{si}-v_{ni}). \tag{9.28} \]

It now remains only to consider the combination of the remaining nondivergent-type terms in (9.24).

For this purpose we use limiting cases in which the concentration of one of the components is close to zero. Then, as is known\(^2\), this component will participate entirely in the normal motion, i.e. one of the momenta of the superfluid motion \((p')\) is strictly equal to zero; moreover, in this case \(\mu_2=\varphi_2\) and \(\mathbf{g}'=-\mathbf{g}''=0\). Since the vector \(\mathbf{g}'\) is equal to zero in this case for all values of the component densities, by a simple argument one may conclude that this condition will always be satisfied. In a similar way we find

\[ \mu_1=\varphi_1,\qquad \mu_2=\varphi_2,\qquad \mathbf{g}'=0. \tag{9.29} \]

Taking into account (9.25), (9.28), and (9.29), we write, in final form, the hydrodynamic equations for a solution of two superfluid liquids.

The continuity equations:

\[ \dot{\rho}'_1+\operatorname{div}(\mathbf{p}'+\rho_1\mathbf{v}_n)=0, \tag{9.30} \]

\[ \dot{\rho}'_2+\operatorname{div}(\mathbf{p}''+\rho_2\mathbf{v}_n)=0. \tag{9.31} \]

One of them may be replaced by the continuity equation for the whole liquid

\[ \dot{\rho}+\operatorname{div}\mathbf{j}=0,\qquad \mathbf{j}=\rho\mathbf{v}_n+\mathbf{p}'+\mathbf{p}''. \tag{9.32} \]

The continuity equation for the entropy

\[ \dot{S}+\operatorname{div}S\mathbf{v}_n=0. \tag{9.33} \]

The equations of the superfluid motions:

\[ \left. \begin{aligned} \dot{\mathbf{v}}'_s+\nabla\left(\mu_1-\frac{v_n^2}{2}+\mathbf{v}_n\mathbf{v}'_s\right)&=0,\\ \dot{\mathbf{v}}''_s+\nabla\left(\mu_2-\frac{v_n^2}{2}+\mathbf{v}_n\mathbf{v}''_s\right)&=0. \end{aligned} \right\} \tag{9.34} \]

The equation of motion of the whole liquid as a whole

\[ \frac{\partial}{\partial t}j_i+\frac{\partial\Pi_{ik}}{\partial x_k}=0, \tag{9.35} \]

\(^*\) It is easy to see that, owing to (9.27), the tensor \(\mathfrak{M}_{ik}\) is symmetric.

where the momentum-flux tensor \(\Pi_{ik}\) is equal to

\[ \Pi_{ik}=\rho v_{ni}v_{nk}+\left(p'_k v'_{si}+v_{nk}p'_i\right)+\left(p''_k v''_{si}+v_{nk}p''_i\right)+p\delta_{ik}, \tag{9,36} \]

and the pressure \(p\) is equal to

\[ p=-\varepsilon+TS+\mu_1\rho_1+\mu_2\rho_2 . \tag{9,37} \]

Taking into account the thermodynamic identity (9,7), we obtain from this

\[ dp=\rho_1 d\mu_1+\rho_2 d\mu_2+S\,dT-p'\,d\left(\mathbf{v}'_s-\mathbf{v}_n\right)-p''\,d\left(\mathbf{v}''_s-\mathbf{v}_n\right). \tag{9,38} \]

Finally, the law of conservation of energy takes the form

\[ \left. \begin{gathered} \dot{E}+\operatorname{div}\mathbf{Q}=0,\\ \mathbf{Q}=(p'+\rho_1\mathbf{v}_n)\left(\mu_1-\frac{v_n^2}{2}\right) +(p''+\rho_2\mathbf{v}_n)\left(\mu_2-\frac{v_n^2}{2}\right)+\\ +\mathbf{v}_n(\mathbf{j}\mathbf{v}_n)+p'\left(\mathbf{v}_n\mathbf{v}'_s\right)+p''\left(\mathbf{v}_n\mathbf{v}''_s\right). \end{gathered} \right\} \tag{9,39} \]

For small velocities, from (9,38) we find the dependence of the chemical potentials on the relative velocities

\[ \mu_1=\mu_{10}+\frac{\rho'_s}{2\rho}\left(\mathbf{v}'_s-\mathbf{v}_n\right)^2,\qquad \mu_2=\mu_{20}+\frac{\rho''_s}{2}\left(\mathbf{v}''_s-\mathbf{v}_n\right)^2. \tag{9,40} \]

Here \(\mu_{10}\) and \(\mu_{20}\) are parts of the chemical potentials independent of the velocities.

The presence of three motions in solutions of superfluid liquids may lead to a number of peculiar phenomena. Thus, for example, in such solutions the propagation of sound oscillations of three types with different velocities is possible. In addition to ordinary sound waves, the propagation of two types of waves is possible, in which oscillations of the temperature and of the concentration of the solution occur.

In conclusion, let us note that at present only two superfluid liquids are known to us—liquid He\(^4\) and liquid He\(^6\). Unfortunately, the isotope He\(^6\) is short-lived (half-life \(0.8\) sec), and this circumstance, naturally, hampers the possibility of experiments with solutions of these two liquids.

III. DISCONTINUITIES AND SOUND OF LARGE AMPLITUDE IN HELIUM II

Osborne\({}^{28}\) experimentally proved the possibility of observing temperature discontinuities in helium II. The attempt made in this connection by Temkin to give a theoretical analysis of this question\({}^{29}\) cannot be recognized as satisfactory, since the indicated work is based on an incorrect, contradictory system of hydrodynamic equations for helium II. The author also does not take into account the difference between the local sound velocity and its equilibrium value. Moreover, he uses an erroneous proposition of the theory, contrary to experiment,

Tisza’s1 assertion concerning the proportionality of the entropy and the density of the normal part of helium II.

In the present section we shall consider the phenomenon of the formation of discontinuities in a superfluid liquid and the related phenomenon of the propagation of sound of large amplitude.

10. Discontinuities in helium II2

Let us determine the conditions that must be satisfied on the surface of a discontinuity in helium II. To this end we consider a certain element of the discontinuity surface and choose a coordinate system moving together with this element. We choose the \(x\)-axis in the direction of the normal to this element. On the discontinuity surface the following must be continuous:

a) the \(x\)-component of the flux of liquid*)

\[ [j_x] = [\rho_s v_{s x} + \rho_n v_{n x}] = 0 \tag{10,1} \]

(\(\rho_n\) and \(\rho_s\) are the densities, \(v_n\) and \(v_s\) the velocities of the normal and superfluid parts of the liquid).

b) the \(x\)-components of the momentum flux

\[ [p+\rho_s v_{s x}^{2}+\rho_n v_{n x}^{2}] = 0, \]

\[ [\rho_s v_{s x}v_{s y}+\rho_n v_{n x}v_{n y}] = 0,\quad [\rho_s v_{s x}v_{s z}+\rho_n v_{n x}v_{n z}] = 0 \tag{10,2} \]

(\(p\) is the pressure).

c) the \(y\)- and \(z\)-components of the velocity \(\mathbf v_s\)

\[ [v_{s y}] = 0,\qquad [v_{s z}] = 0. \tag{10,3} \]

This follows directly from the condition \(\operatorname{rot}\mathbf v_s = 0\).

d) the \(x\)-component of the energy flux

\[ \left[ j_x\left(\widetilde{\mu}+\frac{v_s^2}{2}\right) +\widetilde{\rho}\,\widetilde{\sigma}T v_{n x} +\rho_n(\mathbf v_n-\mathbf v_s,\mathbf v_n)v_{n x} \right]=0. \tag{10,4} \]

\(\widetilde{\mu}\) is the thermodynamic potential depending on the relative velocity \(\mathbf v_n-\mathbf v_s\),

\[ \widetilde{\mu}=\mu(p,T)-\frac{\rho_n}{2\rho}(\mathbf v_n-\mathbf v_s)^2. \tag{10,5} \]

From the thermodynamic identity

\[ d\widetilde{\mu} = -\widetilde{\sigma}\,dT +\frac{1}{\widetilde{\rho}}\,dp -\frac{\rho_n}{\rho_s}(\mathbf v_n-\mathbf v_s)d(\mathbf v_n-\mathbf v_s) \tag{10,6} \]

it follows that the density $\tilde{\rho}$ and the entropy $\tilde{\sigma}$ are also functions of the difference $\mathbf{v}_n-\mathbf{v}_s$:

$$ \tilde{\sigma}=\sigma(\rho,T)+\frac{1}{2}\frac{\partial}{\partial T}\left(\frac{\rho_n}{\rho}\right)(\mathbf{v}_n-\mathbf{v}_s)^2, \tag{10,7} $$

$$ \frac{1}{\tilde{\rho}}=\frac{1}{\rho}-\frac{1}{2}\frac{\partial}{\partial p}\left(\frac{\rho_n}{\rho}\right)(\mathbf{v}_n-\mathbf{v}_s)^2. \tag{10,8} $$

d) Finally, the forces acting on the superfluid part of the liquid on both sides of the discontinuity surface must be equal, i.e. the quantity

$$ \left[\tilde{\mu}+\frac{v_s^2}{2}\right]=0. \tag{10,9} $$

must be continuous.

This condition follows directly from the equation of superfluid motion

$$ \dot{\mathbf{v}}_s+\nabla\left(\tilde{\mu}+\frac{v_s^2}{2}\right)=0. $$

With the aid of (10,1) and (10,9), instead of condition (10,4) on the discontinuity surface one may formulate a simpler condition, namely,

$$ \left[\tilde{\rho}\tilde{\sigma}T v_{nx}+\rho_n(\mathbf{v}_n-\mathbf{v}_s,\mathbf{v}_n)v_{nx}\right]=0. \tag{10,10} $$

Temerly, as one of the conditions on the discontinuity surface, requires continuity of the entropy flux $\rho\sigma v_{nx}$. Such a condition is not satisfied for the same reasons as in ordinary hydrodynamics.

The expressions written above for the thermodynamic functions $\tilde{\mu}, \tilde{\rho}, \tilde{\sigma}$ are valid up to terms of third order with respect to the difference $(\mathbf{v}_n-\mathbf{v}_s)$. Therefore, in all transformations we shall restrict ourselves*) to retaining terms of order no higher than the second with respect to $(\mathbf{v}_n-\mathbf{v}_s)$. In what follows we shall consider normal discontinuities in helium, usually called shock waves. It is natural that the restriction indicated above permits consideration only of shock waves of small strength.

We shall denote all quantities in front of the shock wave in the undisturbed liquid by the subscript 0. The values of the functions behind the shock wave will be denoted by the same letters, but without the subscript. Let the velocity of the shock wave relative to the undisturbed liquid be equal to $u$; we shall also introduce the velocity $\mathbf{v}$, connected with the flux $\mathbf{j}$ by the relation $\mathbf{j}=\rho\mathbf{v}$; finally, we shall denote the velocity difference $\mathbf{v}_n-\mathbf{v}_s$ by a single letter $\mathbf{w}$.

*) In this approximation one may disregard the dependence of the densities $\rho_n$ and $\rho_s$ on the relative velocity $\mathbf{v}_n-\mathbf{v}_s$. A tilde denotes thermodynamic quantities depending on the velocities $\mathbf{v}_n$ and $\mathbf{v}_s$.

HYDRODYNAMICS OF HELIUM II

Consider the case of one-dimensional motion of the liquid; in this case we shall omit the indices on vector quantities. In the stationary coordinate system the conditions (10.1), (10.2), (10.9), and (10.10) are rewritten in the form

\[ \rho_0 u=\tilde{\rho}(u-v), \tag{10.1} \]

\[ p_0+\rho_0 u^2=p+\tilde{\rho}(u-v)^2+\rho_s\rho_n\frac{w^2}{\rho}, \tag{10.2} \]

\[ \mu_0+\frac{u^2}{2}=\mu+\frac{1}{2}\left(u-v+\rho_n\frac{w}{\tilde{\rho}}\right)^2, \tag{10.9} \]

\[ \rho_0 T_0\sigma_0 u=\tilde{\rho}\,\tilde{\sigma}T \left(u-v-\rho_s\frac{w}{\tilde{\rho}}\right) +\rho_n w\left(u-v-\rho_s\frac{w}{\tilde{\rho}}\right)^2. \tag{10.10} \]

Eliminating the velocity from the equations given above with the aid of (10.1), and expressing the quantities \(\tilde{\sigma}\) and \(\tilde{\rho}\) with the aid of (10.7) and (10.8), we obtain

\[ p-p_0-\frac{\rho_0}{\rho}u^2(\rho-\rho_0) +\rho_0 w^2\left[ \frac{\rho_s\rho_n}{\rho^2} -u^2\rho_0\frac{1}{2}\frac{\partial}{\partial p} \left(\frac{\rho_n}{\rho}\right) \right]=0, \tag{10.11} \]

\[ \mu-\mu_0-\frac{1}{2}\frac{u^2}{\rho^2} (\rho^2-\rho_0^2) +\frac{\rho_0\rho_n}{\rho^2}uw - w^2\left[ \frac{\rho_n\rho_s}{2\rho^2} +\frac{1}{2}\rho_0 u^2 \frac{\partial}{\partial p}\frac{\rho_n}{\rho} \right]=0, \tag{10.12} \]

\[ \rho_0 u(T\sigma-T_0\sigma_0) -w\left(\sigma T\rho_s+\rho_n\frac{\rho_0^2}{\rho^2}u^2\right) + \rho_0 w^2u\left[ \frac{2\rho_n\rho_s}{\rho^2} +\frac{1}{2}T\frac{\partial}{\partial T} \left(\frac{\rho_n}{\rho}\right) \right]=0. \tag{10.13} \]

The system (10.11)—(10.13) obtained makes it possible to find the velocity of the shock wave and the jumps of the thermodynamic quantities and velocities on the wave surface. As independent variables we shall choose the pressure \(p\) and the temperature \(T\). As is known, in helium II the value of the derivative \(\left(\dfrac{\partial \rho}{\partial T}\right)_p\) proves to be anomalously small. This circumstance makes it possible to neglect everywhere the dependence of the density \(\rho\) on the temperature. In the system (10.11)—(10.13) let us expand all quantities in series in powers of \(\Delta p=p-p_0\) and \(\Delta T=T-T_0\), restricting ourselves to terms of order no higher than the second. The latter restriction follows directly from the fact that the quantity \(w\), according to the equations of hydrodynamics, is a quantity of the first order with respect to the differences \((p-p_0)\) and \((T-T_0)\). Above, however, we agreed to retain terms no higher than second order in \(w\). After the indicated

After expansion, the system (10.10)—(10.13) takes the form

\[ \Delta p\left(1-u^2\frac{\partial \rho}{\partial p}\right) +(\Delta p)^2 u^2\left[\frac{1}{\rho}\left(\frac{\partial \rho}{\partial p}\right)^2-\frac{1}{2}\frac{\partial^2\rho}{\partial p^2}\right]+ \]

\[ +w^2\left[\frac{\rho_s\rho_n}{\rho} -\frac{1}{2}\rho^2 u^2\frac{\partial}{\partial p}\frac{\rho_n}{\rho}\right]=0, \tag{10.14} \]

\[ \frac{1}{\rho}\Delta p\left(1-u^2\frac{\partial \rho}{\partial p}\right) -\sigma\Delta T+\frac{\rho_n}{\rho}uw+ \]

\[ +(\Delta p)^2\left[-\frac{1}{2\rho^2}\frac{\partial \rho}{\partial p} +\frac{3}{2\rho^2}\left(\frac{\partial \rho}{\partial p}\right)^2u^2 -\frac{1}{2\rho}u^2\frac{\partial^2\rho}{\partial p^2}\right]+ \]

\[ +\Delta p\,w\,\frac{1}{\rho}\left[\frac{\partial \rho_n}{\partial p} -2\frac{\rho_n}{\rho}\frac{\partial \rho}{\partial p}\right] +w^2\left[-\frac{\rho_n\rho_s}{2\rho^2} -\frac{1}{2}\rho u^2\frac{\partial}{\partial p}\frac{\rho_n}{\rho}\right]- \]

\[ -\frac{1}{2}(\Delta T)^2\frac{\partial \sigma}{\partial T} +\Delta T\,wu\,\frac{\partial}{\partial T}\frac{\rho_n}{\rho}=0, \tag{10.15} \]

\[ \Delta T\rho u\left(\sigma+T\frac{\partial \sigma}{\partial T}\right) -w(\sigma T\rho_s+\rho_n u^2)+ \]

\[ +(\Delta T)^2\rho u\left[\frac{\partial \sigma}{\partial T} +\frac{1}{2}T\frac{\partial^2\sigma}{\partial T^2}\right] -w\Delta T\left[u^2\frac{\partial \rho_n}{\partial T} +\sigma\rho_s+T\frac{\partial}{\partial T}(\sigma\rho_s)\right]- \]

\[ -w\Delta p\left[-\frac{\rho_n}{\rho}\frac{\partial \rho}{\partial p}u^2 +\frac{\partial \rho_n}{\partial p}u^2 +T\sigma\frac{\partial \rho_s}{\partial p}\right]+ \]

\[ +w^2u\left(T\frac{\partial \rho_n}{\partial T} +2\rho_s\rho_n\right)=0. \tag{10.16} \]

In the linear approximation, the resulting system of equations is considerably simplified. Equating to zero the determinant of this system (the condition for its compatibility), we find the equation

\[ \left(1-u^2\frac{\partial \rho}{\partial p}\right) \left(\frac{\partial \sigma}{\partial T}u^2\rho_n-\sigma^2\rho_s\right)=0, \tag{10.17} \]

which expresses, in the first approximation, the value of the velocity \(u\). The roots of equation (10.17) are

\[ u_{10}^{2}=\frac{\partial p}{\partial \rho}, \tag{10.18} \]

\[ u_{20}^{2}=\frac{\rho_s}{\rho_n}\sigma^2\frac{\partial T}{\partial \sigma}. \tag{10.19} \]

The system of equations (10.14)—(10.16), after substitution of the obtained values of the velocity \(u\), makes it possible to clarify the relation between the jumps \(\Delta p\), \(\Delta T\), and \(w\) at the discontinuity. In this way we verify that to the first root there correspond jumps \(\Delta T\) and \(w\) of higher order than first relative to \(\Delta p\). To the second root, on the other hand, there correspond jumps \(\Delta p\) of order higher than first relative to \(\Delta T\) and \(w\). In addition, in this case the following relation holds between \(\Delta T\) and \(w\):

\[ \Delta T=w u_2\frac{\rho_n}{\rho\sigma}. \tag{10.20} \]

The first root corresponds to a pressure discontinuity, analogous to shock waves in ordinary media. The second root corresponds to a temperature discontinuity (a temperature discontinuity).

Pressure discontinuities (shock waves). Let us neglect, in equations (10,11), terms higher than second order; as a result we obtain the equation

\[ \left(1-u^2\frac{\partial \rho}{\partial p}\right) +\left[\frac{1}{\rho}\left(\frac{\partial \rho}{\partial p}\right)^2 -\frac{1}{2}\frac{\partial^2 \rho}{\partial p^2}\right]u^2\Delta p=0, \tag{10,21} \]

which determines the value of the velocity of the pressure discontinuity in the second approximation,

\[ u_1=-u_{10}\left\{1+\Delta p\,\frac{\partial}{\partial p}\ln(\rho u_{10})\right\}. \tag{10,22} \]

For small values of the jump in velocity at the discontinuity surface, according to (10,1) we have

\[ \Delta p=\rho u_{10}v. \]

Taking this relation into account, we rewrite formula (10,22) in the form

\[ u_1=-u_{10}\left\{1+\frac{v}{2}\frac{\partial}{\partial p}(\rho u_{10})\right\}. \tag{10,22′} \]

Solution (10,22′) coincides with the expression for the velocity of discontinuities in ordinary media. Substituting the obtained value of \(u^2\) into the other two equations, we find that in this case the jumps \(\Delta T\) and \(w\) are of higher than second order relative to the pressure jump \(\Delta p\).

Thus, pressure discontinuities in helium II are completely analogous to shock waves in ordinary hydrodynamics. The sign of the expression

\[ \frac{\partial}{\partial p}\ln(\rho u_{10}) \]

in helium II is also positive, as in ordinary media. Consequently, shock waves of this type can only be compression waves. The first term in (10,22) is equal to the velocity of first sound in helium II; in the first approximation a pressure discontinuity propagates with a velocity equal to the velocity of first sound.

Temperature discontinuities. To the second root \(u_{20}\) there correspond, in the first approximation, jumps \(\Delta T\) and \(w\) different from zero. Let us neglect, in equations (10,14)—(10,16), terms of order higher than the second. Equation (10,14) makes it possible to express the pressure jump \(\Delta p\) through the value of the velocity \(w\); the indicated jump turns out to be of second order of smallness relative to \(w\) (or \(\Delta T\)):

\[ \Delta p=-w^2\left[\frac{\rho_s\rho_n}{\rho} -\frac{1}{2}\rho^2u_{20}^2\frac{\partial}{\partial p}\left(\frac{\rho_n}{\rho}\right)\right]. \tag{10,23} \]

Substitute the obtained value of \(\Delta p\) into equation (10,15) and eliminate the velocity \(w\) from equations (10,15) and (10,16). As a result

we find the equation

\[ \rho\left(-\sigma^2 \rho_s+u^2\frac{\partial \sigma}{\partial T}\rho_n\right)\Delta T +\rho_s\rho\sigma(\Delta T)^2 \left\{ -3\frac{\partial \sigma}{\partial T} +\frac{3}{2}\sigma -\frac{\rho}{\rho_n\rho_s}\frac{\partial\rho_n}{\partial T} -\frac{1}{2}\sigma\frac{\partial T}{\partial \sigma}\frac{\partial^2\sigma}{\partial T^2} \right\}=0, \tag{10,24} \]

which determines the value of the velocity \(u_2\) in the second approximation. Solving equation (10,24) with respect to \(u\) and taking into account definition (10,19), we obtain

\[ u_2=u_{20}\left\{1+\frac{1}{2}\frac{\partial}{\partial T} \ln\left(u_{20}^3\frac{\partial\sigma}{\partial T}\right)\Delta T\right\}. \tag{10,25} \]

The formula obtained can also be rewritten in the form (cf. (10,10))

\[ u_2=u_{20}\left\{1+\frac{1}{2}\frac{\rho_n u_{20}^2}{\rho\sigma}\,\omega \frac{\partial}{\partial T}\ln\left(u_{20}^3\frac{\partial\sigma}{\partial T}\right)\right\}. \tag{10,25′} \]

Temperature discontinuities are a specific phenomenon characteristic only of a superfluid liquid. Formula (10,25) determines the velocity of temperature discontinuities in helium. In the first approximation, temperature discontinuities move with a velocity equal to the velocity of second sound \(u_{20}\). The sign of the derivative
\[ \frac{\partial}{\partial T}\left[\ln\left(u_{20}^3\frac{\partial\sigma}{\partial T}\right)\right] \]
changes depending on the temperature. As will be shown in the next paragraph, this circumstance leads to peculiar phenomena.

11. Sound of large amplitude in helium II\({}^{31}\)

Let us consider the problem of the propagation of sound oscillations in helium II in the second approximation. Here by the first approximation we mean the solution obtained from the linearized system of hydrodynamic equations. The complete system of hydrodynamic equations for helium II, according to (1,22)—(1,25), has the form

\[ \frac{\partial \widetilde{\rho}}{\partial t}+\operatorname{div}\mathbf{j}=0, \qquad \mathbf{j}=\rho_s\mathbf{v}_s+\rho_n\mathbf{v}_n, \tag{11,1} \]

\[ \frac{\partial j_i}{\partial t}+\frac{\partial \Pi_{ik}}{\partial x_k}=0, \qquad \Pi_{ik}=p\delta_{ik}+\rho_n v_{ni}v_{nk}+\rho_s v_{si}v_{sk}, \tag{11,2} \]

\[ \frac{\partial(\widetilde{\rho}\,\widetilde{\sigma})}{\partial t} +\operatorname{div}\,\widetilde{\rho}\,\widetilde{\sigma}\,\mathbf{v}_n=0, \tag{11,3} \]

\[ \frac{\partial \mathbf{v}_s}{\partial t} +\nabla\left(\widetilde{\mu}+\frac{v_s^2}{2}\right)=0. \tag{11,4} \]

The thermodynamic potential \(\widetilde{\mu}\), the entropy \(\widetilde{\sigma}\), and the density \(\widetilde{\rho}\) are functions of the relative velocity \((\mathbf{v}_n-\mathbf{v}_s)\) (cf. (10,5), (10,7), (10,8)).

Introduce, for convenience, as in the preceding subsection, the velocity \(\mathbf v\) associated with the flux \(\mathbf j\) by the relation \(\mathbf j=\tilde\rho \mathbf v\), and denote the difference of the velocities \(\mathbf v_n-\mathbf v_s\) by a single letter \(\mathbf w\). We shall seek such a solution of the system (11,1)—(11,4) as would describe a wave traveling in one direction. In doing so we shall regard the quantity \(\mathbf w\) as small. As for the velocity \(\mathbf v\), we impose no restrictions on its magnitude. In practice only the case of small values of \(\mathbf w\) is of interest, since for large values of \(\mathbf w\) a violation of superfluidity is possible. Consider a sound wave propagating along the \(x\)-axis. In a traveling wave all quantities can be expressed in terms of one another; in other words, all quantities are functions of a certain parameter, which we denote by the letter \(l\). The coordinate \(x\) and the time \(t\) are functions of the same parameter. The velocity of the points of the profile in the wave will therefore be equal to the derivative \(u=\left(\dfrac{dx}{dt}\right)_l=-\dfrac{\partial l}{\partial t}\big/\dfrac{\partial l}{\partial x}\), taken at a certain value of this parameter. We shall denote derivatives of functions with respect to the indicated parameter by a prime and, retaining only terms linear in the quantity \(w\), rewrite the system (11,1)—(11,4) in the form*)

\[ (v-u)\frac{\partial \rho}{\partial T}T' +(v-u)\frac{\partial \rho}{\partial p}p' + (v-u)\rho^2\frac{\partial}{\partial p} \left(\frac{\rho_n}{\rho}\right)ww' +\rho v'=0, \tag{11,5} \]

\[ v(v-u)\frac{\partial \rho}{\partial T}T' + \left[1+v(v-u)\frac{\partial \rho}{\partial p}\right]p' + \left[ v(v-u)\rho^2\frac{\partial}{\partial p} \left(\frac{\rho_n}{\rho}\right) +\frac{2\rho_s\rho_n}{\rho} \right]ww' + \rho(2v-u)v'=0, \tag{11,6} \]

\[ \left[ \rho(v-u)\frac{\partial \sigma}{\partial T} +w\frac{\partial}{\partial T}(\rho_s\sigma) \right]T' + \left[ \rho(v-u)\frac{\partial \sigma}{\partial p} +w\frac{\partial}{\partial p}(\rho_s\sigma) \right]p' + \left[ \rho_s\sigma+\rho(v-u)\frac{\partial}{\partial T} \left(\frac{\rho_n}{\rho}\right)w \right]w'=0, \tag{11,7} \]

\[ \left[ -\sigma-(v-u)\frac{\partial}{\partial T} \left(\frac{\rho_n}{\rho}\right)w \right]T' + \left[ \frac{1}{\rho} - \frac{\partial}{\partial p} \left(\frac{\rho_n}{\rho}\right)(v-u)w \right]p' + \left[ -\frac{\rho_n}{\rho}(v-u) -\frac{\rho_s\rho_n}{\rho^2}w \right]w' + \left[ (v-u)-\frac{\rho_n}{\rho}w \right]v'=0. \tag{11,8} \]

*) For vector quantities we omit the index \(x\). We likewise omit the subscript \(0\) on the quantities \(\sigma\) and \(\rho\).

The compatibility condition for the system obtained in this way is that its determinant be equal to zero. Expanding the indicated determinant and neglecting all terms containing the derivative \(\dfrac{\partial \rho}{\partial T}\), after simple calculations we obtain the equation

\[ \left\{(u-v)^2 \frac{\partial \rho}{\partial p}-1\right\} \left\{ \left[\rho_n \rho (u-v)^2 \frac{\partial \sigma}{\partial T} -\sigma^2\rho_s\rho\right] - w(u-v)\left[ 4\rho_s\rho_n\frac{\partial \sigma}{\partial T} -2\rho\sigma\frac{\partial \rho_n}{\partial T} \right] \right\}=0, \tag{11,9} \]

which determines the velocity \(u\). The equation obtained splits into two equations. The first of them is

\[ (u-v)^2\frac{\partial \rho}{\partial p}-1=0. \tag{11,10} \]

It determines the Riemann solution well known from ordinary hydrodynamics. Solving equation (11,10), we find

\[ u_1=c_1+v,\qquad c_1^2=\frac{\partial p}{\partial \rho}. \tag{11,11} \]

Equating the second bracket in (11,9) to zero, we obtain the equation

\[ \rho_n\rho(v-u)^2\frac{\partial \sigma}{\partial T} -\sigma^2\rho_s\rho - w(u-v)\left[ 4\rho_s\rho_n\frac{\partial \sigma}{\partial T} -2\rho\sigma\frac{\partial \rho_n}{\partial T} \right]=0, \]

whose root is equal to

\[ u_2=c_2+v+w\left( \frac{2\rho_s}{\rho} -\frac{\sigma}{\rho_n}\frac{\partial \rho_n}{\partial T}\frac{\partial T}{\partial \sigma} \right). \tag{11,12} \]

Expression (11,12) determines the velocity of the points of the profile of the wave of second sound in the second approximation.

First sound is a wave of compression and rarefaction. In such a wave the deviation of the pressure from the equilibrium value, \(\delta p\), according to equation (11,5), is related to the velocity \(v\) in the first approximation by the relation

\[ \delta p=\rho\frac{v}{c_1}\frac{\partial p}{\partial \rho}. \tag{11,13} \]

The velocity \(c_1\) is the local sound velocity, changing from point to point of the profile with the change in pressure. Let us expand in (11,12) the expression for \(c_1\) in a series in the quantity \(\delta p\), restricting ourselves to the linear term; we have

\[ u_1=u_{10}+\frac{dc_1}{dp}\,\delta p+v =u_{10} + v\left(1+\frac{\rho}{u_{10}}\frac{du_{10}}{dp}\frac{\partial p}{\partial \rho}\right) = u_{10}+v u_{10}\frac{\partial}{\partial p}(\rho u_{10}) \tag{11,14} \]

(\(u_{10}\) is the value of the first-sound velocity for equilibrium values of the pressure and temperature). The fact that different points of the profile move with different velocities leads, as is known, to a distortion of the profile and to the appearance of discontinuities. The quantity

\[ \alpha_1=u_{10}\frac{\partial}{\partial p}(\rho u_{10}) \]

has a positive sign for all known media. Therefore regions of increased pressure (cf. (11,13)) move with a velocity greater than \(u_{10}\), while regions of rarefaction move with a velocity less than \(u_{10}\). As a result, the discontinuity is formed at the leading front of the wave. The velocity of the discontinuity that arises is, according to (10,22), equal to

\[ u_1=u_{10}+\frac{1}{2}\alpha_1(v_1+v_2) \tag{11,15} \]

(\(v_1\) and \(v_2\) are the velocities of the medium on both sides of the discontinuity). Thus the discontinuity surface moves with a velocity equal to the half-sum of the velocities on both sides of the discontinuity.*)

Second sound is a property characteristic only of helium II. It is a temperature wave. In such a wave, in the absence of coupling between second and first sound, the velocity \(v\) is equal to zero, while the deviation of the temperature from its equilibrium value, \(\delta T\), according to (11,7), is related to the quantity \(w\) by the relation

\[ \delta T=\frac{\rho_s\sigma}{\rho c_p}\frac{\partial T}{\partial\sigma}\,w. \tag{11,16} \]

As was already indicated, formula (11,12) determines the velocity of the points of the profile in second sound. The velocity \(c_2\) entering into this formula is the velocity of second sound, varying from point to point of the wave profile with the change in the quantity \(\delta T\). In order to obtain the profile velocity in the second approximation of interest to us, it is necessary to expand \(c_2\) up to terms linear in the quantity \(\delta T\); as a result, with the aid of (11,16), we find

\[ u_2=u_{20}+\frac{\partial c_2}{\partial T}\delta T +w\left(\frac{2\rho_s}{\rho}-\frac{\sigma}{\rho_n}\frac{\partial\rho_n}{\partial T}\frac{\partial T}{\partial\sigma}\right)= \]

\[ = u_{20}+w\frac{\rho_s\sigma}{\rho}\frac{\partial T}{\partial\sigma}\frac{\partial}{\partial T} \ln\left(u_{20}^{3}\frac{\partial\sigma}{\partial T}\right) \tag{11,17} \]

\[ \left( u_{20}=\sqrt{\frac{\rho_s}{\rho_n}\sigma^2\frac{\partial T}{\partial\sigma}} \text{ is the velocity of second sound as a function of the equilibrium values of pressure and temperature} \right). \]

The points of the profile of a second-sound wave, according to (11,17), move with different velocities. This leads to a change in the form

*) In deriving formula (10,22), the velocity \(v\) on one side of the discontinuity was assumed to be zero. In the general case the result indicated in the text is obtained.

of the wave profile with time. At the moment when the shape of the profile becomes a multivalued function of the coordinate, discontinuities arise (in the present case, temperature discontinuities). The velocities of the points of the wave profile on the surface of the discontinuity change discontinuously. The velocity of the surface of the discontinuity that arises then depends on the indicated jump in velocity. According to formula (10.24) it is equal to (see the note on p. 133)

\[ u_2=u_{20}+\frac{1}{2}(w_1+w_2)\frac{\rho_s\sigma}{\rho}\frac{\partial T}{\partial\sigma}\, \frac{\partial}{\partial T}\ln\left(u_{20}^{3}\frac{\partial\sigma}{\partial T}\right) \tag{11.18} \]

(\(w_1\) and \(w_2\) are the values of the relative velocity on the two sides of the surface of discontinuity).

In second sound the velocity \(w\) is related to the velocity of normal motion by the relation \(w=v_n\,\frac{\rho}{\rho_s}\). Expressing the quantity \(w\) in terms of \(v_n\), we rewrite (11.18) in the form

\[ u_2=u_{20}+\frac{1}{2}\alpha_2(v_{n1}+v_{n2}), \tag{11.19} \]

where

\[ \alpha_2=\frac{\sigma T}{C}\frac{\partial}{\partial T}\ln\left(u_{20}^{3}\frac{C}{T}\right), \]

\[ C=T\frac{\partial\sigma}{\partial T} \tag{11.20} \]

(\(C\) is the heat capacity of 1 g of helium II).

Fig. 7. Temperature dependence of the coefficient \(\alpha_2\), which determines the velocity of temperature discontinuities in helium II.

The coefficient \(\alpha_2\) changes sign with a change in temperature. Its dependence on temperature is shown graphically in Fig. 7. At temperatures above \(2.00^\circ\mathrm{K}\) and in the interval \(0.4\text{--}0.9^\circ\mathrm{K}\) the coefficient \(\alpha<0\); in the remaining region \(\alpha>0\). In the temperature region where \(\alpha<0\), surfaces of discontinuity arise on the rear front of the wave, whereas in the temperature region where \(\alpha>0\), on the contrary, they arise on the front front of the wave. The occurrence of a discontinuity on the rear front of the wave is a specific property of second sound in helium II, unknown for ordinary sound. It should be noted that, strictly speaking, from the moment a discontinuity arises the acoustic approximation becomes inapplicable. However, as follows from (10.24), the velocity of a normal discontinuity of small intensity coincides with that given by the second-sound approximation. The situation is analogous to that which occurs in ordinary hydrodynamics.

Osborne \(^{28}\) observed a change in the shape of a rectangular temperature pulse propagating in helium II. The indicated pulse gradually assumed the form of a triangular tooth, one side of which was vertical. This indicates that in these experiments the formation of discontinuities was observed.

The velocity of the surface of discontinuity near the \(\lambda\)-point turned out, in accordance with our results, to be greater than the velocity of second sound, while at low temperatures (a little above \(1^\circ\text{K}\)), on the contrary, it was less than the velocity of second sound. At a temperature of \(1.05^\circ\text{K}\) and a pulse power up to \(1.3\ W/\text{cm}\), the velocity of the discontinuity (shock wave) exceeded the velocity of second sound by \(1.8\ \text{m/sec}\). The calculated value of the entropy at this temperature is equal to \(0.5\cdot 10^{-3}\ \text{cal}/\text{g}\cdot\text{deg}\). The velocity \(v_n\) at the tooth was equal to \(q/\rho T\sigma=4.15\ \text{m/sec}\). Multiplying the half-sum of the values of \(v_n\) at the tooth and at the base of the tooth (zero) by the value \(a_2\) taken from the graph, we obtain \(u_2-u_{20}=1.5\ \text{m/sec}\), in sufficient agreement with the results of the measurements.

12. On the propagation of sound in moving helium II and on the influence of heat flow on the propagation of second sound \(^{32}\)

It is well known from ordinary hydrodynamics that in a moving fluid there occurs a “drift” of sound. An analogous phenomenon must take place in the hydrodynamics of a superfluid liquid. Since in a superfluid liquid two motions are possible (normal with velocity \(\mathbf{v}_n\) and superfluid with velocity \(\mathbf{v}_s\)), as well as two types of sound oscillations propagating with different velocities, it is natural that the picture of sound propagation in a moving superfluid liquid will differ from the corresponding phenomenon in classical hydrodynamics.

Let sound oscillations of frequency \(w\) propagate in helium II, in which there are normal and superfluid motions with constant velocities \(\mathbf{v}_n\) and \(\mathbf{v}_s\), in a direction characterized by a unit vector \(\mathbf{n}\) (along the axis \(x\)). The wave vector \(\mathbf{k}\) is equal to \(\dfrac{w}{u}\mathbf{n}\), where \(u\) is the velocity of the sound oscillations. Here we shall find the velocity of the sound oscillations \(u\) (for first and second sounds) in moving helium II, under the assumption that the motion takes place with velocities small in comparison with the velocity of sound. We shall seek additions to the value of the sound velocity in a liquid at rest that are linear in the velocities \(\mathbf{v}_n\) and \(\mathbf{v}_s\). For this it is necessary to rewrite the hydrodynamic equations with accuracy up to terms quadratic in the velocities.

The problem is in many respects analogous to the preceding one. We shall denote the projections of vectors onto the direction of the wave vector by the subscript \(k\), and onto the plane perpendicular to \(\mathbf{k}\) by the subscript \(\perp\).

The hydrodynamic equations take the form \((U=u-v_k)\)

\[ -\,U\,\frac{\partial \rho}{\partial T}\,T' -\left(U\frac{\partial \rho}{\partial p}\,p' -U\rho^2\frac{\partial}{\partial p}\left(\frac{\rho_n}{\rho}\right)\right) \]

\[ \times\left(w_k w_k' + w_\perp w_\perp'\right)+\rho v_k'=0, \tag{12,1} \]

\[ p'+2\frac{\rho_s\rho_n}{\rho}\,w_k w_k' - \rho U v_k'=0, \tag{12,2} \]

\[ \frac{\rho_s\rho_n}{\rho}\left(w_k w_\perp' + w_\perp w_k'\right)-\rho U v_\perp'=0, \tag{12,3} \]

\[ \left[\rho U\frac{\partial \sigma}{\partial t} +w_k\frac{\partial}{\partial T}(\rho_s\sigma)\right]T' + \]

\[ +\left[-\rho U\frac{\partial \sigma}{\partial p} +w_k\frac{\partial}{\partial p}(\rho_s\sigma)\right]p' +\rho_s\sigma w_k'=0, \tag{12,4} \]

\[ \left[-\sigma-Uw_k\frac{\partial}{\partial T}\left(\frac{\rho_n}{\rho}\right)\right]T' -\frac{\partial}{\partial p}\left(\frac{\rho_n}{\rho}\right)Uw_k p' + \]

\[ +\left[\frac{\rho_n}{\rho}U-\frac{3\rho_n^2 s}{\rho^2}w_k\right]w_k' +\left(-\frac{\rho_n}{\rho}v_\perp-\frac{\rho_s\rho_n}{\rho^2}w_\perp\right)w_\perp' - \]

\[ -\frac{\rho_n}{\rho}w_k v_k' - \frac{\rho_n}{\rho}w_\perp v_\perp'=0, \tag{12,5} \]

\[ v_\perp' - \left(\frac{\rho_n}{\rho}w_\perp\right)'=0. \tag{12,6} \]

The compatibility conditions of the system obtained give equations determining the possible values of the speed of sound \(u\). The projections of the equations of motion onto the plane perpendicular to the wave vector (eqs. (12,3), (12,6)) separate from the remaining equations. This is explained by the fact that the components \(v_\perp\) and \(w_\perp\) enter equations (12,1), (12,2), (12,4), (12,5) only in terms of second order*). In this way we find

\[ \left(U^2\frac{\partial \rho}{\partial p}-1\right) \left\{\left(\rho_n\rho U^2\frac{\partial \sigma}{\partial T}-\sigma^2\rho\rho_s\right)-\right. \]

\[ \left. -\,Uw_k\left(4\rho_s\rho_n\frac{\partial \sigma}{\partial T} -2\sigma\rho\frac{\partial \rho_n}{\partial T}\right)\right\}=0, \tag{12,7} \]

\[ U-\frac{\rho_s}{\rho}w_k=0. \tag{12,8} \]

The roots of equation (12,7) determine the velocities of first and second sounds in moving helium II. For first sound we find a solution coinciding with the known solution in ordinary hydrodynamics

\[ u_1=c_1+v_k,\qquad c_1^2=\frac{\partial p}{\partial \rho}. \tag{12,9} \]

\[ \text{*) In this case equation (12,6) is a consequence of the condition of potentiality of the superfluid motion.} \]

For the velocity of second sound we have

\[ u_2=c_2+v_k+w_k\left(\frac{2\rho_s}{\rho}-\frac{\sigma}{\rho_n}\frac{\partial \rho_n}{\partial T}\frac{\partial T}{\partial \sigma}\right), \qquad c_2^2=\frac{\rho_s}{\rho_n}\sigma^2\frac{\partial T}{\partial \sigma}. \tag{12.10} \]

Oscillations of the transverse components of the velocities \(v_\perp\) and \(w_\perp\) in the plane perpendicular to the direction of the wave vector propagate with the phase velocity

\[ u=v_k+\frac{\rho_s}{\rho}w_k=\frac{v_n k}{k}. \tag{12.11} \]

In formulas (12.10) and (12.11), the velocities \(c_1\) and \(c_2\) are the velocities of sound in helium in the absence of motions with constant velocities. If the amplitudes of the velocities in the sound wave are small in comparison with the constant components of the velocities, then one may neglect the nonlinear effect caused by the variability of \(c_1\) and \(c_2\) along the profile of the sound wave (for more detail see § 11). In this case \(v_k\) and \(w_k\) in formulas (12.10) and (12.11) are the projections of the constant velocities on the direction of the wave vector \(\mathbf{k}\).

It follows from formula (12.10) that, in the presence of a constant heat flux in helium II, second sound will be carried along by this flux. In a heat flux \(\mathbf{j}=0\), and consequently also \(\mathbf{v}=0\). Further, we have

\[ \mathbf{w}=\mathbf{v}_n-\mathbf{v}_s= \]

\[ =\mathbf{v}_n+\frac{\rho_n}{\rho_s}\mathbf{v}_n=\frac{\rho}{\rho_s}\mathbf{v}_n. \]

Formula (12.10) in this case takes the form

\[ u_2=c_2+v_{nk}\frac{\rho}{\rho_s} \left( \frac{2\rho_s}{\rho} -\frac{\sigma}{\rho_n}\frac{\partial \rho_n}{\partial T}\frac{\partial T}{\partial \sigma} \right) = \]

\[ =c_2+\gamma v_{nk}. \tag{12.12} \]

The quantity \(v_n\) is related to the density of the heat flux \(\mathbf{q}\) by

\[ \mathbf{q}=\rho\sigma T\mathbf{v}_n. \tag{12.13} \]

Fig. 8. Temperature dependence of the coefficient \(\gamma\).

The coefficient \(\gamma\) multiplying \(v_{nk}\) in formula (12.12) is a function of temperature; its magnitude is of order unity. In Fig. 8 the dependence of \(\gamma\) on temperature is shown. In the temperature region above \(1^\circ\mathrm{K}\), where only the roton parts of the thermodynamic

of the quantities, we have approximately

\[ \rho_n \sim \sigma \qquad \text{and} \qquad \gamma=-\frac{\rho}{\rho_s}\left(\frac{2\rho_s}{\rho}-1\right)=\frac{\rho_s-\rho_n}{\rho}. \]

From the fact that \(\gamma\) changes sign at certain temperatures, it follows that the “drift” of second sound may occur both in the direction of the heat flow and against the flow.

Finally, let us make a remark on the influence of a heat flow on the velocity of standing waves of second sound. From the symmetry of standing waves it follows that this effect will be quadratic with respect to the ratio of the drift velocity to the velocity of second sound. Even at velocities \(v_n\) of the order of \(1\ \mathrm{m/sec}\), the influence on the velocity of second sound in a standing wave will be of the order of fractions of a percent.

IV. HEAT EXCHANGE BETWEEN A SOLID BODY AND HELIUM II

The unusual character of heat transfer in helium II is responsible for a number of peculiar phenomena occurring at the boundary between a solid body and liquid helium II. P. L. Kapitsa\(^{33}\) discovered that, when heat is liberated in a solid body in contact with liquid helium II, a constant temperature difference arises between the solid body and helium II. The magnitude of this difference proves to be proportional to the flux of liberated heat. In the same experiments it was shown that the jump is localized in a narrow near-wall layer of liquid whose thickness is less than \(10^{-3}\ \mathrm{cm}\). The magnitude of the thermal resistance, equal to the ratio of the temperature difference to the flux of liberated heat, in Kapitsa’s experiments increased with decreasing temperature according to a cubic law. This very interesting phenomenon, usually called in the literature Kapitsa’s temperature jump, has recently been investigated by various authors. Qualitatively the picture obtained by all authors agrees with that observed by Kapitsa. As for the quantitative laws, however, they differed substantially among the various authors. Thus, E. Andronikashvili and G. Mirskaya\(^{34}\) obtained values for the thermal resistance of the solid-body—liquid-helium-II boundary noticeably larger than Kapitsa’s.

White, Gonzales, and Johnston\(^{35}\) obtained a temperature dependence of the thermal-resistance magnitude close to that obtained by Kapitsa. In their experiments the thermal resistance proved to be inversely proportional to the temperature to the power \(2.6\).

The most recent detailed experiments of Fairbank and Wilks\(^{36}\) give higher values for the thermal resistance than Kapitsa’s; for the temperature dependence of the thermal resistance they obtain the law \(T^{-2}\).

In this section we shall discuss possible mechanisms of heat exchange between a solid body and helium II.

For this purpose let us consider a solid isotropic body filling one half of space and bordering on helium II, which fills

HYDRODYNAMICS OF HELIUM II

the other half of space. Let the temperature of the solid be \(T'\), and the temperature of helium II be \(T\). Our problem is to determine the heat flux arising between the indicated bodies. Heat exchange between the solid and helium II can occur in two ways: (a) by the transfer of energy in collisions of phonons and rotons with the solid wall; (b) by radiation of energy from the vibrating surface of the solid (radiation of phonons). The difficulty of heat exchange between a solid and helium II can be understood qualitatively if one takes into account the considerable difference in the speeds of sound in the solid and in liquid helium II (the speed of sound in helium II is an order of magnitude smaller than the speed of sound in solids). Owing to this difference, the momenta of phonons in the solid and in liquid helium II at the same temperature will differ appreciably. The latter circumstance does not allow phonons from the solid to pass into liquid helium II (and conversely), since in such a transformation it is impossible to satisfy simultaneously the laws of conservation of momentum and energy.

We shall show that the basic mechanism determining heat exchange between a solid and liquid helium II is the radiation (and absorption) of energy by the boundary of the solid body, which performs thermal vibrations. The energy flux arising at the boundary between the solid and helium II proves to be proportional to the difference of the fourth powers of the temperatures of the indicated bodies.

In order to proceed to the calculation of the heat-exchange effects, it is necessary first to quantize the elastic waves in the solid occupying a half-space.

1. Quantization of elastic waves\(^{37}\). In a solid three types of elastic vibrations are possible: longitudinal, transverse, and surface.

Longitudinal waves. Liquid helium II has a very small density in comparison with a solid. Therefore the boundary solid body—liquid helium II, in its properties, will not differ from the boundary solid body—vacuum. Upon reflection from the boundary solid body—vacuum of a longitudinal wave incident in the plane \(XZ\), two reflected waves arise: a longitudinal and a transverse one. From symmetry considerations it follows that the displacement vector in the reflected transverse wave, as well as the wave itself, will lie in the plane \(XZ\). The total displacement in the solid is equal to

\[ \mathbf{n}=\left(A_0\mathbf{n}_0 e^{i\mathbf{k}_0\mathbf{r}}+A_l\mathbf{n}_l e^{i\mathbf{k}_l\mathbf{r}}+A_t[\mathbf{a}\mathbf{n}_t] e^{i\mathbf{k}_t\mathbf{r}}\right)e^{-i\omega t}, \tag{13,1} \]

\(A_0\), \(A_l\), \(A_t\) and \(\mathbf{n}_0\), \(\mathbf{n}_l\), \(\mathbf{n}_t\) are, respectively, the amplitudes and unit vectors along the directions of the incident longitudinal and the reflected longitudinal and transverse waves; \(\mathbf{a}\) is the unit vector perpendicular to the plane \(XZ\), \(k_0=k_l=\omega/c_l\); \(k_t=\omega/c_t\). The angle of incidence \(\theta_0\) and the angles of reflection of the longitudinal \(\theta_l\) and transverse \(\theta_t\) waves are related by the relations:

\[ \theta_0=\theta_l,\qquad c_t\sin\theta_0=c_l\sin\theta_t. \tag{13,2} \]

The components of the displacement vector $\mathbf{u}$ are equal to:

\[ u_z=\left(\cos\theta_0\,A_0 e^{i\mathbf{k}_0\mathbf{r}}-\cos\theta_l\,A_l e^{i\mathbf{k}_l'\mathbf{r}}-A_t\sin\theta_t\,e^{i\mathbf{k}_t'\mathbf{r}}\right)e^{-i\omega t}, \tag{13.3} \]

\[ u_x=\left(\sin\theta_0\,A_0 e^{i\mathbf{k}_0\mathbf{r}}+\sin\theta_l\,A_l e^{i\mathbf{k}_l'\mathbf{r}}-A_t\cos\theta_t\,e^{i\mathbf{k}_t'\mathbf{r}}\right)e^{-i\omega t}. \tag{13.4} \]

Next we calculate the components of the strain tensor $u_{ik}$ and of the stress tensor

\[ \sigma_{ik}=2\rho c_t^2 u_{ik}+\rho\left(c_l^2-2c_t^2\right)u_{ll}\delta_{ik}. \]

($\rho$ is the density of the body).

The boundary conditions at the solid–vacuum boundary give two conditions, $\sigma_{zz}=\sigma_{xz}=0$, relating the amplitudes $A_l$, $A_t$ to the amplitude $A_0$. Thus we find

\[ A_l=A_0\frac{c_t^2\sin2\theta_t\sin2\theta_0-c_l^2\cos^2 2\theta_t} {c_t^2\sin2\theta_t\sin2\theta_0+c_l^2\cos^2 2\theta_t} \tag{13.5} \]

and

\[ A_t=-A_0\frac{2c_lc_t\sin2\theta_0\cos2\theta_t} {c_t^2\sin2\theta_t\sin2\theta_0+c_l^2\cos^2 2\theta_t}. \tag{13.6} \]

Substituting (13.5) and (13.6) into (13.3), we find the magnitude of the $z$-component of the displacement vector on the surface separating the solid and vacuum:

\[ u_{zl}=A_0\frac{2c_l^2\cos\theta_0\cos2\theta_t} {c_t^2\sin2\theta_t\sin2\theta_0+c_l^2\cos^2 2\theta_t}\,e^{-i\omega t}. \tag{13.7} \]

Transverse waves. In reflection from a solid–vacuum boundary of a transverse wave incident in the plane $xz$, two reflected waves arise: a transverse wave and a longitudinal wave. Two cases are possible: either the directions of vibration in the incident transverse wave lie in the plane of incidence $xz$, or these vibrations occur in a direction perpendicular to this plane. Let the direction of vibration of the transverse wave lie in the plane of incidence; then in the reflected transverse wave as well, as follows from symmetry considerations, the direction of vibration lies in the same plane. In this case the displacement vector is equal to

\[ \mathbf{u}=\left([\mathbf{a}\mathbf{n}_0]A_0e^{i\mathbf{k}_0\mathbf{r}}+A_l\mathbf{n}_l'e^{i\mathbf{k}_l'\mathbf{r}}+A_t[\mathbf{a}\mathbf{n}_t]e^{i\mathbf{k}_t'\mathbf{r}}\right)e^{-i\omega t}, \tag{13.8} \]

($\mathbf{n}_0$ is a unit vector in the direction of the incident transverse wave, and $A_0$ is the corresponding amplitude).

The angle of incidence $\theta_0$ and the angles of reflection of the longitudinal wave $\theta_l$ and of the transverse wave $\theta_t$ are related by

\[ \theta_0=\theta_t,\qquad c_t\sin\theta_l=c_l\sin\theta_0. \tag{13.9} \]

Just as in the case of an incident longitudinal wave, the boundary conditions allow the amplitudes $A_l$ and $A_t$ to be expressed in terms of the amplitude $A_0$;

we have

\[ A_t=A_0\,\frac{c_t^2\sin 2\theta_l\sin 2\theta_0-c_l^2\cos^2 2\theta_0} {c_t^2\sin 2\theta_l\sin 2\theta_0+c_l^2\cos^2 2\theta_0}, \tag{13,10} \]

\[ A_l=A_0\,\frac{2c_lc_t\sin 2\theta_0\cos 2\theta_0} {c_t^2\sin 2\theta_l\sin 2\theta_0+c_l^2\cos^2 2\theta_0}. \tag{13,11} \]

At the solid–vacuum interface, the \(z\)-component of the displacement vector in this case is equal to

\[ u_{zl}=-A_0\,\frac{2c_t^2\cos\theta_0\sin 2\theta_l} {c_t^2\sin 2\theta_l\sin 2\theta_0+c_l^2\cos^2 2\theta_0}\,e^{-i\omega t}. \tag{13,12} \]

Transverse waves whose direction of oscillation is perpendicular to the plane of incidence will have no influence on the phenomenon we are considering. Indeed, as will be seen below, the effect of interest to us is determined by the magnitude of the normal \(z\)-component of the displacement vector of the solid–vacuum interface. In transverse oscillations of this type, however, the \(z\)-component of the displacement vector at the interface is zero both in the incident and in the reflected waves*).

Expressions (13,7) and (13,12) contain, as a factor, the amplitude of the incident wave \(A_0\). The magnitude of this amplitude in the case of a single phonon is found from the condition, which reduces to the requirement that the total energy contained in the incident wave of frequency \(\omega\) be equal to \(\hbar\omega\). The total energy of the oscillations is equal to twice the kinetic (or potential) energy; thus we have (\(\rho_T\) is the density of the solid)

\[ \varepsilon=\int \rho_T|\dot{\mathbf{u}}|^2\,dV =\rho_T|A_0|^2\omega^2V=\hbar\omega, \tag{13,13} \]

whence we find

\[ |A_0|=\sqrt{\frac{\hbar}{\rho_T\omega V}}. \tag{13,14} \]

Taking this result into account, we write the expression for the moduli of the amplitudes of the \(z\)-component of the velocity at the surface of the body (\(z=0\)):

\[ |\dot u_{zl}|=\sqrt{\frac{\hbar\omega}{\rho_TV}}\left| \frac{2c_l^2\cos\theta_0\cos 2\theta_t} {c_t^2\sin 2\theta_t\sin 2\theta_0+c_l^2\cos^2 2\theta_t} \right|, \tag{13,15} \]

\[ |\dot u_{zt}|=\sqrt{\frac{\hbar\omega}{\rho_TV}}\left| \frac{2c_t^2\cos\theta_0\sin 2\theta_l} {c_t^2\sin 2\theta_l\sin 2\theta_0+c_l^2\sin^2 2\theta_0} \right|. \tag{13,16} \]

Surface waves. The components of the displacement vector in a surface wave propagating in the direction of the \(x\)-axis

*) In this case, as is clear from symmetry considerations, no longitudinal reflected wave arises.

are equal to

\[ u_z=\left(a\chi_t e^{ikx+\chi_t z}+bk e^{ikx+\chi_l z}\right)e^{-i\omega t}, \tag{13,17} \]

\[ u_x=-i\left(ak e^{ikx+\chi_t z}+b\chi_l e^{ikx+\chi_l z}\right)e^{-i\omega t}. \tag{13,18} \]

Here \(k=\omega/c\), \(c=\xi c_t\) is the propagation velocity of the surface wave; \(\xi\) is a function of the ratio \(c_l/c_t\);

\[ \chi_t=\omega\sqrt{c^{-2}-c_t^{-2}}, \qquad \chi_l=\omega\sqrt{c^{-2}-c_l^{-2}}; \]

\(a\) and \(b\) are certain constants.

Using expressions (13,17) and (13,18), we compute the components of the strain tensor \(u_{ik}\) and the stress tensor \(\sigma_{ik}\).

The boundary conditions on the surface of the solid \((z=0)\): \(\sigma_{ik}n_k=0\), give two equations:

\[ a\left(\chi_t^2+k^2\right)+2\chi_l kb=0, \tag{13,19} \]

\[ 2\chi_t ka+\left(\chi_t^2+k^2\right)b=0. \tag{13,20} \]

The compatibility condition for the system (13,19)—(13,20) has the form

\[ 4k^2\chi_l\chi_t=\left(\chi_t^2+k^2\right)^2 \tag{13,21} \]

and relates the surface-wave velocity \(c\) to the velocities of bulk waves—the transverse \(c_t\) and the longitudinal \(c_l\). Equations (13,19)—(13,20) thus give one relation between the quantities \(a\) and \(b\). The quantity \(a\) (or \(b\)) is determined by the corresponding normalization, which amounts to requiring that the total energy contained in the surface wave be equal to \(\hbar\omega\). Since for oscillatory motion the potential energy is equal to the kinetic energy, and since in the present case the expression for the kinetic energy can be obtained in a simpler way, we shall find the total energy of the surface wave by doubling the expression for the kinetic energy. Thus, the total energy of a surface wave of frequency \(\omega\) is equal to

\[ \varepsilon=\int_0^\infty \rho_T\left(|\dot u_x|^2+|\dot u_z|^2\right)s\,dz=\hbar\omega \tag{13,22} \]

(\(\rho_T\) is the density of the solid, \(s\) its surface area).

Next, using (13,17) and (13,18), we compute the time derivatives of the components of the displacement vector \(u\) and express the coefficient \(b\) in terms of the coefficient \(a\) according to (13,19). After substituting the resulting expressions for \(\dot u_x\) and \(\dot u_z\) into (13,22) and integrating over \(z\), we obtain

\[ s\rho_T\left\{ \frac{k^2+\chi_t^2}{2\chi_t} +\frac{k^2+\chi_l^2}{2\chi_l}\cdot\frac{\chi_t}{\chi_l} -\frac{k^2+\chi_t^2}{\chi_l} \right\}\omega^2|a|^2=\hbar\omega . \tag{13,23} \]

From this we find the expression for the modulus of the coefficient

\[ |a|=\left|\sqrt{\frac{\hbar}{\rho_T \omega S}}\sqrt{\frac{1}{f}}\right|, \qquad f=\frac{k^2+x_t^2}{2x_t}+ \frac{k^2+x_l^2}{2x_l}\frac{x_t}{x_l} -\frac{x_t^2+k^2}{x_l}, \tag{13,24} \]

The magnitude of the \(z\)-component of the displacement vector \(\mathbf u\) at the surface of the body, according to (13,18) and (13,19), is equal to

\[ u_z=-i\,\frac{k^2-x_t^2}{2k}\sqrt{\frac{\hbar\omega}{\rho_T S}}\frac{1}{\sqrt f}. \tag{13,25} \]

The quantity that will interest us below, the modulus of the \(z\)-component of the velocity \(\dot u_z\), is equal to

\[ |\dot u_z|=\frac{k^2-x_t^2}{2k}\sqrt{\frac{\hbar\omega}{\rho_T S}}\frac{1}{\sqrt f}. \tag{13,26} \]

14. Radiation of Energy by an Oscillating Surface of a Solid Body[^37]

A solid body whose surface performs an oscillatory motion with frequency \(\omega\) will radiate sound into the surrounding liquid medium. One can calculate the energy carried away by the sound per unit time. It is equal to

\[ Q=\rho_{\mathrm{liq}}c_{\mathrm{liq}}\int |\dot u_n|^2\,ds^{*}), \tag{14,1} \]

where \(\rho_{\mathrm{liq}}\) is the density of liquid helium II, \(c_{\mathrm{liq}}\) is the speed of sound in helium II, and \(\dot u_n\) is the component of the velocity of surface motion normal to the surface of the body. The integration is carried out over the surface of the body. In the case of a plane boundary of the body, the integration is simplified, and expression (14,1) can be rewritten in the form

\[ q(\omega)=\frac{Q}{S}=\rho_{\mathrm{liq}}c_{\mathrm{liq}}|\dot u_z|^2, \tag{14,2} \]

where \(q(\omega)\) is the energy flux.

Along with the radiation of sound (phonons) from the surface, the reverse process will also occur: absorption of phonons incident from helium II on the surface of the body.

As for the possible process of absorption and emission of rotons by the surface of a solid body, it is easy to see that such

* Expression (14,1) applies in the case when the dimensions of the body exceed the wavelength of the emitted sound. In the case considered here of a body filling half of space, this condition is, of course, satisfied.

the process cannot play a significant role. Indeed, a phonon of the solid can be transformed into a roton only if its energy is equal*) to or greater than \(\Delta\). But \(\Delta \gg kT\), and therefore the number of such phonons in the solid, distributed according to Planck’s law, is exponentially small. Consequently, the indicated process is unlikely.

In order to find the energy flux exchanged by the solid with liquid helium II, it is necessary to calculate the difference of the energy fluxes directed from the solid to the liquid and from the liquid to the solid. This difference, integrated over all frequencies \(\omega\), gives the desired flux \(\Delta W\)

\[ \Delta W=\int q(\omega)\{n(T)[n(T')+1]-[n(T)+1]n(T')\}\,d\tau_{\omega}= \]

\[ =\int q(\omega)\{n(T)-n(T')\}\,d\tau_{\omega}=W(T)-W(T'); \tag{14,3} \]

\(d\tau_{\omega}\) is an element of phase volume, \(n(T)\) is the Planck distribution function

\[ n(T)=\left(e^{\frac{\hbar\omega}{kT}}-1\right)^{-1}. \tag{14,4} \]

Thus the desired flux is equal to the difference of the values of the integrals \(W\), obtained by integrating \(q(\omega)\) over the Planck distribution, taken at the temperatures of the solid, \(T'\), and of liquid helium II, \(T\). Taking into account all possible types of oscillations, according to (14,3) we have

\[ W=\int \rho_{\mathrm{ж}}c_{\mathrm{ж}} \left(e^{\frac{\hbar\omega}{kT}}-1\right)^{-1} \left\{ \int |\dot u_{zt}|^2\frac{\omega^2\,do\,V}{2(2\pi c_t)^3} +\right. \]

\[ \left. +\int |\dot u_{zl}|^2\frac{\omega^2\,do\,V}{2(2\pi c_l)^3} +|\dot u_{zs}|^2\frac{2\pi\omega s}{(2\pi c)^2} \right\}d\omega, \tag{14,5} \]

\(do=2\pi\sin\theta_0\,d\theta_0\) is an element of solid angle.

In the expressions for the phase volumes of transverse and longitudinal oscillations, the additional factor \(1/2\) takes into account the possibility of integration over half the solid angle \(do\). Substituting into (14,5) the expressions for the corresponding \(z\)-components of the velocity \(\dot u_z\), determined by formulas (13,15), (13,16), and (13,26),

\[ W=\rho_{\mathrm{ж}}c_{\mathrm{ж}}\int \left(e^{\frac{\hbar\omega}{kT}}-1\right)^{-1} \left\{ \frac{1}{2c_t^3} \left| \frac{2c_t^2\cos\theta_0\sin2\theta_l} {c_t^2\sin2\theta_0\sin2\theta_l+c_l^2\cos^2 2\theta_0} \right|^2 +\right. \]

\[ \left. +\frac{1}{2c_l^3} \left| \frac{2c_l^2\cos\theta_0\cos2\theta_t} {c_t^2\sin2\theta_0\sin2\theta_t+c_l^2\cos^2 2\theta_t} \right|^2 +\frac{\pi(k^2-\chi^2)^2}{4k^3c^3f} \right\} \times \frac{\hbar\omega}{\rho_T}\, \frac{\omega^2\,d\omega\,do}{(2\pi)^3}. \tag{14,6} \]

*) The minimum energy of a roton is equal to \(\Delta/k=8.9^\circ\mathrm{K}\).

Integration over the frequencies \(\omega\) is carried out in an elementary way. The integrals over angles require a special analysis, making it possible to determine the regions of integration. After integration over frequencies and simple transformations of the angular variables, we obtain the final expression for the heat flux \(W\)

\[ W=\frac{\rho_{\mathrm{ж}}}{\rho_{\mathrm{т}}}\,c_{\mathrm{ж}}\, \frac{4\pi^5}{15}\, \frac{(kT)^4}{(2\pi h c_t)^3}\,F(\eta), \tag{14,7} \]

where \(F(\eta)\) is a certain function of the elastic constants of the solid
\((\eta=c_l/c_t,\ \xi=c/c_t,\ c\) is the velocity of surface waves)

\[ \begin{aligned} F(\eta)=&\;2\int_0^1 \frac{4y\sqrt{1-y}\,(y\eta^2-1)\,dy} {\eta^2(1-2y)^4+16(1-y)(y\eta^2-1)} \\ &+2\int_0^{1/\eta^2} \frac{\eta(1-2y)^2\sqrt{1-y\eta^2}} {\eta^2(1-2y)^4+16y^2(1-y)(y\eta^2-1)} \\ &+\frac{\pi}{4}\left\{ \left(\frac{2}{\xi}-\xi\right) \left(\frac{1}{2\sqrt{1-\xi^2}}-\frac{1}{\sqrt{1+\dfrac{\xi^2}{\eta^2}}}\right)\right. \\ &\left.\qquad\qquad +\frac{1}{\xi}\sqrt{1-\xi^2}\, \frac{2-\dfrac{\xi^2}{\eta^2}}{2\left(1-\dfrac{\xi^2}{\eta^2}\right)} \right\}^{-1}. \tag{14,8} \end{aligned} \]

The function \(F(\eta)\) is of order unity. It was tabulated for two values
\(\eta=1.71\) (glass), \(\eta=2.22\) (platinum)

\[ F(1.71)\simeq 2.5,\qquad F(2.22)\simeq 2.0 . \]

The energy flux transferred between the solid and liquid helium II, according to (14,3) and (14,8), is equal to

\[ \Delta W_{\mathrm{зв}}=W(T)-W(T') =\frac{\rho_{\mathrm{ж}}}{\rho_{\mathrm{т}}}\,c_{\mathrm{ж}}\, \frac{16\pi^5}{15}\, \frac{1}{(2\pi h c_t)^3}\times \]

\[ \times F\{(kT)^4-(kT')^4\}. \tag{14,9} \]

Thus the indicated flux is proportional to the difference of the fourth powers of the temperatures of the solid and of liquid helium II. For small temperature differences, formula (14,9) can be rewritten in the form

\[ \Delta W_{\mathrm{зв}}= \frac{\rho_{\mathrm{ж}}}{\rho_{\mathrm{т}}}\,c_{\mathrm{ж}}\, \frac{16\pi^5}{15}\, \frac{(kT)^3}{(2\pi h c_t)^3}\,Fk(T-T'). \tag{14,10} \]

The energy radiated by the solid wall propagates in helium II in the form of phonons. The excess of phonons arising in connection with this,

however, decay very rapidly. The probability of a phonon transforming into a roton is comparatively large; the corresponding time characterizing this process was found by us earlier. At a temperature of the order of \(1.5\text{--}2^\circ\mathrm{K}\) it is equal to \(10^{-9}\text{--}10^{-10}\) sec. Thus the excess (over the equilibrium number) phonons transform into rotons comparatively quickly, at distances of the order of \(10^{-4}\) cm from the wall. On the other hand, heat transfer from the solid wall to helium II (or conversely) takes place very slowly, and the heat flux arising between them is negligibly small. Thus one may assume that heat exchange takes place between the solid and helium II that are in equilibrium.

15. Energy exchange in collisions of rotons and phonons with a solid wall

Energy exchange between a solid body and liquid helium II is possible in collisions of rotons (phonons) with a solid wall. A roton (phonon), upon reflection from a solid wall, may acquire or lose a certain energy. Such a change in energy is accompanied by the absorption or emission of a phonon in the solid. The energy of the emitted phonon \(\varepsilon\) and the angle \(\theta'\) at which it is emitted are determined by two conservation laws: the law of conservation of energy

\[ E = E_1 + \varepsilon \qquad (\varepsilon = h\omega) \tag{15,1} \]

and the conservation law for the components of momentum tangent to the interface:

\[ p \sin\theta = \]

\[ = p_1 \sin\theta_1 + p' \sin\theta', \tag{15,2} \]

Fig. 9.

\(E, p\) and \(E_1, p_1\) are the energy and momentum of the incident and reflected roton (or phonon), respectively; \(\varepsilon, p'\) are the energy and momentum of the phonon emitted in the solid. The meaning of the angles is clear from Fig. 9.

The speed of sound in the solid is considerably greater than the speed of sound (of the phonon) in helium II. Therefore, when the temperatures of the solid and of helium II do not differ greatly, the momenta of phonons in helium II will considerably exceed the momenta of phonons in the solid. The momenta of rotons, whose magnitude is close to \(P_0 = 2 \cdot 10^{-19}\ \mathrm{g\,cm\,sec^{-1}}\), considerably exceed the momenta of phonons in helium II, and consequently also the momenta of phonons in the solid. This circumstance

allows us henceforth to neglect the momentum of the phonon in the solid in the right-hand side of relation (15.2).

It is necessary to note that relation (15.2), strictly speaking, is applicable only to the case of collisions of phonons with a solid wall. The wavelength of phonons considerably exceeds interatomic distances. Therefore, for phonons the microstructure of the wall is immaterial; the wall may be regarded as smooth, and consequently relation (15.2) holds. Rotons have a wavelength of the order of \(h/P_0 = \tfrac{1}{2}\cdot 10^{-8}\,\text{cm}\), comparable with interatomic distances. Therefore, in the case of a roton colliding with a wall, the wall may be regarded as smooth only approximately. In what follows we shall nevertheless assume relation (15.2) to be valid also for rotons. For this reason the results for rotons are of an approximate character. The influence of the microstructure of the surface can be effectively taken into account by a certain modification of the surface of the solid. Thus the formulae determining the heat exchange associated with rotons are valid only up to a certain factor of order unity. Anticipating what follows, we point out that such an approximation does not affect the final results, since this mechanism of heat exchange plays an insignificant role in the total energy balance. The surface of the solid performs small oscillations. The magnitude of the velocity of the oscillating surface \(\mathbf{u}\) was determined in the preceding subsection. A roton (or phonon) incident on a solid wall may be considered as a certain particle located in an oscillating medium (in the phonon field). In such a treatment the internal structure of the roton and phonon is immaterial. The energy of the elementary excitation incident on the solid wall, in the nonmoving frame of reference, is equal to

\[ H' = H - (\mathbf{p}\dot{\mathbf{u}}), \tag{15.3} \]

(\(H\) is the energy of the elementary excitation in the frame of reference moving together with the surface). It follows from this that the energy of interaction of the elementary excitation with the solid wall is equal to \(-(\mathbf{p}\dot{\mathbf{u}})\). The corresponding quantum Hermitian operator describing the interaction of the elementary excitation with the solid wall is

\[ V = -\frac{1}{2}(\mathbf{p}\dot{\mathbf{u}}+\dot{\mathbf{u}}\mathbf{p}), \tag{15.4} \]

where \(\mathbf{p}=-i\hbar\nabla\) is the momentum operator.

After these general considerations, let us turn to the case of a roton colliding with a solid wall. The probability of a transition upon collision from a state with energy \(E\) to a state with energy \(E_1\) is equal to

\[ W=\frac{2\pi}{h}|M|^2, \tag{15.5} \]

where \(M\) is the matrix element of the interaction energy \(V\), taken between

wave functions: for the initial state, normalized to unit flux, and for the final state, normalized to energy. The wave function of the incident roton, normalized to unit flux and satisfying the condition \(\psi=0\) at \(z=0\), is

\[ \psi=\frac{2}{\sqrt{v\cos\theta}}\sin\left(\frac{p\cos\theta}{h}z\right) \tag{15,6} \]

\[ \left(v=\frac{\partial E}{\partial p}\text{ is the velocity of the incident roton}\right). \]

For the final state we similarly have

\[ \psi_1=\frac{2}{\sqrt{\pi h v_1\cos\theta_1}}\sin\left(\frac{p_1\cos\theta_1}{h}z\right). \tag{15,7} \]

\[ \left(v_1=\frac{\partial E_1}{\partial p_1}\right). \]

Let us calculate the transition matrix element \(M\); we have

\[ M=\int_0^\infty \psi_1 V\psi\,dz= \]

\[ =\int_0^\infty \frac{4ip\cos\theta\,u_z}{\sqrt{\pi h v v_1\cos\theta\cos\theta_1}} \cos\frac{pz\cos\theta}{h}\, \cos\frac{p_1z\cos\theta_1}{h}\,dz. \tag{15,8} \]

Here \(\dot u_z\) is the amplitude of the velocity of oscillations of the surface of the solid body, determined for the various oscillations by formulas (13,15), (13,16), and (13,26). After integration with respect to \(z\) in (15,8) we find

\[ M= \frac{4ihp\cos\theta\,p_1\cos\theta_1} {\left(p_1^2\cos^2\theta_1-p^2\cos^2\theta\right) \sqrt{v v_1\cos\theta\cos\theta_1\pi h}}. \tag{15,9} \]

Substituting this result into (15,5), we obtain the probability of the transition under consideration

\[ w=\frac{2\pi}{h}\, \frac{16h p^2p_1^2\cos\theta\cos\theta_1|\dot u_z|^2} {\left(p_1^2\cos^2\theta_1-p^2\cos^2\theta\right)^2 v v_1} = \]

\[ = \frac{32\pi p^2p_1^2\cos\theta\cos\theta_1|\dot u_z|^2} {\left(p_1^2-p^2\right)^2 v v_1}. \tag{15,10} \]

In doing so we took into account that, according to (15,2),

\[ p_1^2\cos^2\theta_1-p^2\cos^2\theta=p_1^2-p^2. \tag{15,11} \]

The energy flux carried between the solid body and liquid helium II is equal to

\[ \Delta W_p=\iint w\{n(n'+1)-n'n_1\}\,h\omega\,d\tau_\omega\,v\cos\theta\,d\tau_p. \tag{15,12} \]

Here \(d\tau_\omega\) is the element of phase volume corresponding to the frequency of the emitted phonon; \(d\tau_p\) is the element of phase volume of the incident roton; \(n\) and \(n_1\) are the Boltzmann distribution functions for the incident and reflected rotons*):

\[ n=e^{-\frac{E}{kT}},\qquad n_1=e^{-\frac{E_1}{kT}}; \]

\(n'\) is the Planck distribution function for phonons in a solid,

\[ n'=\left(e^{\frac{\hbar\omega}{kT'}}-1\right)^{-1}. \]

The expression in braces in (15,12), after substitution of the corresponding distribution functions, for small temperature differences is reduced to the form

\[ \{n(n'+1)-n'n_1\}= \]

\[ = e^{-\frac{\Delta}{kT}}\, \frac{\hbar\omega\,(T'-T)} {kT^2\left[\exp\left\{\frac{(p-p_0)^2}{2\mu kT}\right\} -\exp\left\{\frac{(p_1-p_0)^2}{2\mu kT'}\right\}\right]} . \tag{15,13} \]

The product \(|u_z|^2 d\tau_\omega\), contained in the integral, taking into account all three possible types of surface vibrations, according to (15,5) and (15,6), is transformed as follows:

\[ |u_z|^2 d\tau_\omega \to \frac{\hbar\omega}{\rho_T}\, \frac{4\pi\omega^2 d\omega}{(2\pi c_t)^3}\,F, \tag{15,14} \]

where \(F(c_t/c_l)\) is determined by formula (15,8).

Substituting (15,10), (15,13), and (15,14) into (15,12) and passing from the variable of integration \(\omega\) to the variable \(p_1\) \((\hbar d\omega=-v_1dp_1)\), we obtain

\[ \Delta W_p = \left\{ \int \frac{ 256\pi^2(\hbar\omega)^2 p^4 dp\,p_1^2 dp_1\,\cos^2\theta\,\cos\theta_1\,d\cos\theta }{ (2\pi\hbar)^3 c_t^3 (p^2-p_1^2)^2 \left[ \exp\left\{\frac{(p-p_0)^2}{2\mu kT}\right\} -\exp\left\{\frac{(p_1-p_0)^2}{2\mu kT'}\right\} \right] } \times \right. \]

\[ \left. \times \frac{e^{-\frac{\Delta}{kT}}(T-T')}{kT^2}\,F, \qquad \hbar\omega=\frac{(p-p_0)^2}{2\mu}-\frac{(p_1-p_0)^2}{2\mu}. \right\} \tag{15,15} \]

The momenta of the rotons \(p\) are, in magnitude, close to \(p_0\). Therefore, in the integral (15,15), everywhere that the difference \(p_1-p\) is not contained in an essential way, \(p\) and \(p_1\) may be replaced by \(p_0\). The angular

*) Rotons obey classical statistics.

the integral is then calculated in the usual way (integration over a hemisphere)

\[ \int_0^1 \cos^2\theta \cos\theta_1\, d\cos\theta \simeq \int_0^1 \cos^3\theta\, d\cos\theta=\frac14. \]

Introduce the notation

\[ (p-p_0)^2/2\mu kT=x, \]

\[ (p_1-p_0)^2/2\mu kT=y. \]

We finally obtain

\[ \Delta W_p= \frac{p_0^4(kT)^4 e^{-\frac{\lambda}{kT}}}{4\pi^4\rho_T h^6 c_t^3} FG_p\,\frac{T-T'}{T}, \tag{15,16} \]

where

\[ G_p=\int_0^\infty dx\int_0^x \frac{(\sqrt{x}+\sqrt{y})^2(x-y)^3\,dy} {\sqrt{xy}\,(e^x-e^y)} \simeq 50. \tag{15,17} \]

Let us carry out analogous calculations for the case of phonon collision with a solid wall. For phonons the analog of the wave function is the velocity potential \(\psi\), satisfying the wave equation. At a solid wall the potential \(\psi\) satisfies the boundary condition

\[ \left(\frac{\partial\psi}{\partial z}\right)_{z=0}=0. \tag{15,18} \]

The \(\psi\)-function of the incident phonon, normalized to unit flux and satisfying this boundary condition, is equal to

\[ \psi=\frac{2}{\sqrt{c_{\mathrm{ж}}\cos\theta}}\cos\frac{(p\cos\theta)z}{h}. \tag{15,19} \]

For the final-state \(\psi\)-function normalized to energy, we have the expression

\[ \psi_1=\frac{2}{\sqrt{\tau h c_{\mathrm{ж}}\cos\theta_1}} \cos\frac{p_1 z\cos\theta_1}{h}. \tag{15,20} \]

The matrix element of the interaction energy of a phonon with a solid wall for the transition under consideration is equal to

\[ M=\int_0^\infty \psi_1 V\psi\,dz = -\frac{4i u_z p\cos\theta}{c_{\mathrm{ж}}\sqrt{\tau h\cos\theta\cos\theta_1}} \int_0^\infty \cos\frac{p_1 z\cos\theta_1}{h} \sin\frac{pz\cos\theta}{h}\,dz = \]

\[ = \frac{4ihp^2\cos^2\theta} {(p^2\cos^2\theta-p_1^2\cos^2\theta_1)c_{\mathrm{ж}}\sqrt{\tau h\cos\theta\cos\theta_1}}. \tag{15,21} \]

Further, for the probability of the transition under consideration we have

\[ w=\frac{2\pi}{h}|M|^2=\frac{32p^4\cos^3\theta\,|\dot u_z|^2}{\cos\theta_1\,(p_1^2-p^2)^2c_{\text{ж}}^2}. \tag{15,22} \]

The energy flux thereby transferred between the solid body and liquid helium II is equal to

\[ \Delta W=\iint w\{n(n'+1)(n_1+1)- \]

\[ -n'n_1(n+1)\}\,c_{\text{ж}}\cos\theta\,d\tau_p\,h\omega\,d\tau_\omega . \tag{15,23} \]

Here \(n\), \(n_1\), and \(n'\) are the Planck distribution functions, respectively, for the incident, reflected, and emitted phonons in the solid body:

\[ n=\left(e^{\frac{E}{kT}}-1\right)^{-1},\qquad n_1=\left(e^{\frac{E_1}{kT}}-1\right)^{-1},\qquad n'=\left(e^{\frac{\varepsilon}{kT'}}-1\right)^{-1}. \]

The expression standing in braces in the integral (15,23), after substitution of the indicated distribution functions, for small temperature differences takes the form:

\[ \{n(n_1+1)(n'+1)-n_1n'(n+1)\}= \]

\[ =\frac{T'-T}{kT^2}\, \frac{E-E_1} {\left(e^{\frac{E}{kT}}-e^{\frac{E_1}{kT}}\right) \left(1-e^{-\frac{E}{kT}}\right) \left(1-e^{-\frac{E_1}{kT}}\right)} . \tag{15,24} \]

Substitute (15,22) and (15,24) into (15,23) and make use of (15,15); as a result we obtain

\[ \Delta W= \frac{256\pi^2 F\,(T'-T)} {kT^2(2\pi c_l)^3(2\pi h)^3\rho_T c_{\text{ж}}} \times \]

\[ \times \iint \frac{(h\omega)^3\omega^2d\omega\,p^6dp} {(p^2-p_1^2)^2 \left(e^{\frac{E}{kT}}-e^{\frac{E_1}{kT}}\right) \left(1-e^{-\frac{E}{kT}}\right) \left(1-e^{-\frac{E_1}{kT}}\right)} \times \]

\[ \times\int_0^1\frac{\cos^4\theta}{\cos\theta_1}\,d\cos\theta . \]

Next we introduce a new variable of integration \(p_1\) instead of \(\omega\) and the notations

\[ E/kT=x,\qquad E_1/kT=y,\qquad h\omega=kT(x-y). \]

Finally, the required energy flux \(\Delta W_{\phi}\) can be written in the form

\[ \Delta W_{\phi}= \frac{4(kT)^8FG_{\phi}(T-T')} {\pi^4h^6\rho_Tc_{\text{ж}}^3c_l^4T}, \tag{15,25} \]

where

\[ G_{\phi}=\int\limits_{0}^{\infty} dx \int\limits_{0}^{x} \frac{x^6(x-y)^3\,dy}{(x+y)(e^x-e^y)(1-e^{-x})(1-e^{-y})} \times \left\{ \frac{3}{16}\frac{y}{x}\left(1-\frac{y^2}{x^2}\right)\ln\frac{x+y}{x-y} +\frac{5y^2}{8x^2} -\frac{3}{8} -\frac{y^4}{x^4} \right\} \simeq \frac{1}{2}\,7! \tag{15.26} \]

16. Heat exchange between a solid body and liquid helium II

Let us determine the relative role of the three fluxes computed above, \(\Delta W_{\mathrm{зв}}\) (14.13), \(\Delta W_{\mathrm{p}}\) (15.16), and \(\Delta W_{\phi}\) (15.25), in the process of heat transfer between a solid body and liquid helium II. For this purpose we shall find the ratios

\[ \frac{\Delta W_{\phi}}{\Delta W_{\mathrm{p}}} \quad \text{and} \quad \frac{\Delta W_{\mathrm{зв}}}{\Delta W_{\phi}}. \]

According to (15.16) and (15.25), we have

\[ \frac{\Delta W_{\phi}}{\Delta W_{\mathrm{p}}} = \frac{16(kT)^4}{c_{\text{ж}}^4 \rho_0^4} \frac{G_{\phi}}{G_{\mathrm{p}}} e^{-\frac{\Delta}{kT}} \simeq 8\cdot 10^{-4}T^4 e^{-\frac{\Delta}{kT}}. \]

This ratio is close to unity at \(T=2^\circ\mathrm{K}\), and then increases, reaching 10 at \(T=1^\circ\mathrm{K}\). Further, according to (14.8) and (15.25), we have

\[ \frac{\Delta W_{\mathrm{зв}}}{\Delta W_{\phi}} = \frac{\pi^6 \rho_{\text{ж}} c_{\text{ж}} h^3 F}{30G_{\phi}} \left(\frac{50}{kT/c_{\text{ж}}}\right)^4 \simeq \frac{50}{T^4}. \]

This ratio is close to three at \(T=2^\circ\mathrm{K}\), and then, as the temperature is lowered, grows rapidly, reaching 50 at \(T=1^\circ\mathrm{K}\).

Thus, at all temperatures below the \(\lambda\)-point, heat transfer between a solid body and liquid helium II is effected mainly by the radiation (and absorption) of sound from the vibrating surface of the solid body. The magnitude of the resulting heat flux between a solid body at temperature \(T'\) and liquid helium II at temperature \(T\), in the first approximation, is proportional to \(T-T'\). The coefficient of proportionality, according to (14.8), varies with temperature according to a cubic law.

These results agree well with Kapitza’s experimental data. In particular, the coincidence of the temperature laws for the magnitude of the thermal resistance should be emphasized. The cubic law discovered by Kapitza also follows from the theory.

A detailed comparison of the theoretical data with the experimental ones is somewhat difficult, since formula (14.8), which determines the heat flux \(\Delta W_{\mathrm{зв}}\), substantially contains the elastic

constants of the solid body \(c_t\) and \(F(c_t/c_1)\). In Kapitza’s experiments \(^{33}\) the heat radiation came from a glass surface coated with a thin layer of platinum. The surface was, of course, not sufficiently smooth, and the flux could not be calculated exactly.

Calculations for platinum by formula (14.8) give

\[ \frac{1}{T-T'}\,\Delta W_{\mathrm{зв}}P_t =1.2\cdot 10^{-2}T^3 W\,\mathrm{deg}^{-1}\mathrm{cm}^{-2}. \]

The experimental value of this ratio, obtained by Kapitza, exceeds the theoretical one by several times. That is, in Kapitza’s experiments temperature jumps smaller than those predicted by the theory were observed.

The later experiments of Fairbank and Wilks \(^{36}\) give, for the thermal resistance, values larger than Kapitza’s and in good agreement with those given by the theory. However, these authors obtain for the temperature dependence of the thermal resistance the law \(T^{-2}\) instead of \(T^{-3}\), which follows from the theory. For the ratio \(\Delta W/(T-T')\) the authors give the value \(2.2\cdot 10^{-2}T^2\).

This discrepancy may be explained by a number of causes: a) in calculating the flux in work \(^{33}\), the roughness of the heat-radiating surface was not taken into account (allowance for this circumstance may reduce the flux by a factor of \(1.5\)—\(2\)); b) in the indicated experiments there may have been unaccounted parasitic heat fluxes; c) our calculations were carried out with values of the elastic constants corresponding to bulk specimens, whereas the experiment is performed on very thin specimens obtained by depositing platinum on glass; d) in comparing theoretical and experimental results we neglect the effect of heat exchange in collisions of phonons and rotons with the solid wall. This may be done at temperatures not very close to the \(\lambda\)-point (\(2.19^\circ\mathrm{K}\)). In the region of temperatures close to the \(\lambda\)-point, allowance for this effect increases the above values of the flux by a factor of \(1.5\)—\(2\).

Andronikashvili and Mdivskaya \(^{34}\) also measured the temperature difference arising between a constantan wire immersed in liquid helium II and the helium in the bath when a certain power was liberated in the wire. The experiment was arranged in such a way that only the lower boundary of the arising temperature difference could be determined. The lower value of the indicated temperature difference obtained in their experiments agrees with the predictions of the theory.

From what has been said above it follows that heat exchange between a solid body and liquid helium II is effected with great difficulty. One can calculate the maximum velocity of normal motion \(v_n\) which arises in liquid helium II when a solid body is immersed in it. The convective heat flux in helium II is equal to

\[ \rho_{\mathrm{ж}}T\sigma v_n \]

(\(\sigma\) is the entropy per unit mass of helium II). The maximum heat flux

arises in the case when the solid is at absolute zero temperature. In this case the flux \(W_{\text{зв}}\), according to (14,8), is equal to

\[ \frac{\rho_{\text{ж}}}{\rho_T}\,c_{\text{ж}}\,\frac{4\pi^5}{15}\, \frac{(kT)^4}{(2\pi h c_t)^3}\,F \]

(\(T\) is the temperature of helium II). Equating the two fluxes, we find the maximum value of \(v_n\):

\[ v_{n\,\max}= \frac{4\pi^5 c_{\text{ж}}(kT)^4F}{\rho T\sigma\,15(2\pi h c_t)^3}. \tag{16,1} \]

The value of \(v_{n\,\max}\) thus obtained turns out to be of the order of tenths of a \(\text{cm}/\text{sec}\) at a temperature of \(2^\circ\mathrm{K}\). At lower temperatures \(v_{n\,\max}\) increases, tending to a constant limiting value (at low temperatures the entropy \(\sigma\) varies with temperature according to a cubic law*),

\[ v_{n\,\max}=c_{\text{ж}}\frac{\rho_{\text{ж}}}{\rho_T} \left(\frac{c_{\text{ж}}}{c_t}\right)^3F . \tag{16,2} \]

The boundary between the solid and liquid helium possesses a large thermal resistance. Under certain conditions this resistance exceeds the thermal resistance of the solid itself. According to (14,8), the heat flux between the solid and helium II \(\Delta W\) is equal to

\[ \begin{aligned} \Delta W&=K\left(T^4-T'^4\right)\simeq 4KT^3\left(T-T'\right),\\ K&=c_{\text{ж}}\frac{\rho_{\text{ж}}}{\rho_T}\frac{4\pi^5}{15} \frac{k^4}{(2\pi h c_t)^3}. \end{aligned} \tag{16,3} \]

Let \(l\) be the thickness of the solid, and \(\varkappa\) its thermal conductivity. It is evident that for sufficiently small dimensions \(l\), when the condition

\[ \frac{\varkappa}{l}\gg 4kT^3, \tag{16,4} \]

is satisfied, the thermal resistance of the boundary exceeds the thermal resistance of the solid. In this case let us determine the temperature \(T\) of a solid surrounded by liquid helium II, having on one side of the solid the temperature \(T_1\), and on the other \(T_2\). Since the thermal resistance of the body itself is relatively small, the temperature of the body will be constant. The value of this temperature is determined from the condition of equality of the heat fluxes at the left and right boundaries of the solid. According to (16,3) we have

\[ K\left(T_1^4-T^4\right)=K\left(T^4-T_2^4\right), \]

*) At sufficiently low temperatures the phonon part of the entropy is significant, \(\sigma_\phi\), equal to

\[ \sigma_\phi=\frac{16\pi^5}{45\rho_{\text{ж}}}\, \frac{k(kT)^3}{(2\pi h c_{\text{ж}})^3}. \]

whence we find

\[ T^{4}=\frac{1}{2}\left(T_{1}^{4}+T_{2}^{4}\right). \tag{16,5} \]

It is also appropriate to point out that all the analysis carried out in the present work is applicable only to the case of a boundary between a solid and a quantum liquid. In ordinary liquids one cannot speak of separate excitations. In the case of heat exchange between a solid and an ordinary liquid there are no reasons that would restrict the exchange of energy. Therefore, at the indicated boundary there is no temperature discontinuity. Essential in this case is also the circumstance that ordinary liquids possess a finite thermal conductivity. The thermal resistance of the liquid in the volume exceeds the thermal resistance of the boundary with the solid; this leads to a distribution of the temperature difference between the solid and the liquid throughout the entire volume of the liquid; helium II, however, possesses a practically infinite thermal conductivity, and the thermal resistance of helium in the volume is equal to zero (the temperature along the liquid helium remains constant). Therefore the temperature difference between the solid and helium II is realized in the form of a jump at the boundary.

17. Passage of second sound through metallic plates. Absorption of second sound on the walls of a cylindrical vessel

The large thermal resistance of the boundary solid body—liquid helium II is responsible for the anomalously small value of the coefficient of transmission of second sound through thin metallic foils, which was observed by Osborne[^28]. We shall assume condition (16,4) to be fulfilled.

Let a plane wave of second sound be incident normally on a plate. The energy flux in second sound \(J\) to the left of the plate may be represented in the form

\[ J_l=\left(J_1 e^{ikx}-J_2 e^{-ikx}\right)e^{-i\omega t}, \tag{17,1} \]

where \(J_1\) and \(J_2\) are the amplitudes of the energy flux in the incident and reflected waves, \(u_2\) is the velocity of second sound.

Fig. 10.

According to the equation of continuity for entropy, the temperature oscillations in helium II to the left of the plate are determined by the expression

\[ T_l=\frac{1}{C u_2}\left(J_1 e^{ikx}+J_2 e^{-ikx}\right)e^{-i\omega t} \tag{17,2} \]

(\(C\) is the heat capacity per unit volume of helium II).

In an analogous way, to the right of the plate the energy flux is equal to

\[ J_n=J_3 e^{ikx-i\omega t}, \tag{17,3} \]

and the temperature oscillations are

\[ T_n=\frac{1}{C u_2}\,J_3 e^{ikx-i\omega t}. \tag{17,4} \]

When second sound passes through the heat-conducting plate, its temperature, if condition (16,4) is satisfied, will remain constant along the plate, but will differ from the temperature in the absence of sound by an amount \(T'\). The conditions of continuity of the energy fluxes at the left and right boundaries will be written, respectively, in the form:

\[ J_1-J_2=4KT^3\left(T_{\ell\,(x=0)}-T'\right) =J_3 e^{ikl}=4KT^3\left(T'-T_{n\,(x=l)}\right) \tag{17,5} \]

(\(T\) is the temperature of helium II).

The smallness of the plate thickness makes it possible to neglect the change of phase when second sound passes through the plate (\(kl\ll 1\)). Substituting into (17,5) the expressions (17,2) and (17,4) for \(T_\ell\) and \(T_n\), while neglecting the change of phase, we obtain

\[ J_1-J_2=4KT^3\left(\frac{J_1+J_2}{C u_2}-T'\right) =J_3=4KT^3\left(T-\frac{J_3}{C u_2}\right). \tag{17,6} \]

From this it is easy to find the quantities \(J_2\) and \(J_3\)

\[ J_2=J_1\,\frac{1}{1+\dfrac{4KT^3}{C u_2}}, \tag{17,7} \]

\[ J_3=J_1\,\frac{4KT^3}{C u_2}\, \frac{1}{1+\dfrac{4KT^3}{C u_2}}. \tag{17,8} \]

The coefficient of transmission of second sound, \(J_3/J_1\), is therefore equal to

\[ \frac{J_3}{J_1}= \frac{4KT^3}{C u_2}\, \frac{1}{1+\dfrac{4KT^3}{C u_2}}. \tag{17,9} \]

The ratio \(4KT^3/Cu_2\) proves to be considerably less than unity at all temperatures. At extremely low temperatures, when the heat capacity \(C\) of helium II is determined only by phonons, the magnitude of the coefficient of transmission of second sound tends to a constant limit independent of temperature. In this case we have

\[ C=C_\phi=4\frac{4\pi^5}{15}\frac{k\,(kT)^3}{(2\pi\hbar c_{\mathrm{ж}})^3}, \qquad u_2=\frac{C_m}{\sqrt{3}}. \]

Substituting these expressions into (17.9), we obtain

\[ \left(\frac{J_3}{J_1}\right)_{T\to 0} = \frac{\rho_n}{\rho_T}\sqrt{3}\left(\frac{c_{\mathrm{ж}}}{c_T}\right)^2 F \tag{17.10} \]

(for platinum this ratio is equal to \(2.5\cdot 10^{-4}\)).

The presence of a temperature discontinuity at the boundary between a solid and liquid helium II has a substantial effect on the phenomenon of reflection of second sound from a solid wall. Let a plane wave of second sound be normally incident on the plane boundary between helium II and a solid occupying a half-space. We choose the origin of coordinates at the indicated boundary, and the \(x\)-axis along the normal to the plane of the boundary. The energy flux in helium II and the temperature distribution \(T\) are determined, respectively, by expressions (17.1) and (17.2). The temperature distribution in the solid is determined by the heat-conduction equation

\[ C_T \frac{\partial T}{\partial t}=\chi \frac{\partial^2 T}{\partial x^2}. \tag{17.11} \]

Denoting the temperature of the solid wall by \(T_0\), from (17.11) we find without difficulty

\[ T=T_0 \exp\left[-\sqrt{C_T i\omega/\chi}\cdot x\right] e^{i\omega t}. \tag{17.12} \]

The requirement of continuity of the energy fluxes at the boundary between helium and the solid gives two conditions, namely:

\[ J_1-J_2=4KT^3\left\{\frac{J_1+J_2}{Cu_2}-T_0\right\} = -\chi \operatorname{Re}\left(\frac{\partial T}{\partial x}\right)_{x=0}. \tag{17.13} \]

The expression standing in (17.13) between the two equality signs is the energy flux due to the temperature discontinuity at the boundary. Substituting into (17.13) expression (17.12) for \(T\) and eliminating the quantity \(T_0\) from our conditions, we thus find the relation connecting the fluxes \(J_2\) and \(J_1\)

\[ J_2 = J_1 \left( 1-\frac{4KT^3/Cu_2}{1+4KT^3\sqrt{2/C_T\chi\omega}} \right) \Bigg/ \left( 1+\frac{4KT^3/Cu_2}{1+4KT^3\sqrt{2/C_T\chi\omega}} \right). \tag{17.14} \]

As already noted, the inequality \(Cu_2/4KT^3\gg 1\) holds. Taking this circumstance into account, from (17.14) we find the expressions for the reflection coefficient and the transmission coefficient into the solid for second sound:

\[ \frac{J_2}{J_1} = 1- \frac{2}{Cu_2\left(\dfrac{1}{4KT^3}+\sqrt{\dfrac{2}{C_T\chi\omega}}\right)} \tag{17.15} \]

(the reflection coefficient),

\[ \frac{J_1-J_2}{J_1} = \frac{2}{Cu_2\left(\dfrac{1}{4KT^3}+\sqrt{\dfrac{2}{C_T\chi\omega}}\right)} \tag{17.16} \]

(the transmission coefficient).

In the denominator of formula (17.16) there is the sum of the thermal resistances of the boundary, \(1/4KT^3\), and of the solid body, \(\sqrt{2/C_T \chi \omega}\). If the solid body has a high thermal conductivity, then the reflection of second sound is determined mainly by the thermal resistance of the boundary. Conversely, in the case of low thermal conductivity of the solid body, reflection is determined by the magnitude of the thermal resistance of the solid body. Owing to the condition \(C u_2/4KT^3 \gg 1\), the reflection coefficient of second sound differs little from unity.

With the aid of formula (17.16) one can easily calculate the absorption coefficient of second sound at the walls of a cylindrical vessel. In the case when the sound propagates along the axis of a tube of radius \(r\), from formula (17.16) there follows the following expression for the sound absorption coefficient:

\[ 2\gamma_x=\frac{P}{2Q}\, \frac{2}{C u_2\left(\dfrac{1}{4KT^3}+\sqrt{\dfrac{2}{C_T\chi\omega}}\right)}. \tag{17.17} \]

Here \(P\) is the perimeter of the cylinder, \(Q\) is the cross-sectional area\(^*\). For a circular cylinder, therefore, we have

\[ \gamma_x=\frac{1}{r C u_2}\, \frac{1}{\dfrac{1}{4KT^3}+\sqrt{\dfrac{2}{C_T\chi\omega}}}. \tag{17.18} \]

Formula (17.18) determines the absorption coefficient of second sound at the walls of a cylindrical vessel, associated with the thermal conductivity of the walls and the temperature discontinuity at the boundary between liquid helium II and the solid body. The absorption coefficient of second sound at the walls of a cylindrical vessel was calculated by Peshkov\(^{38}\). It is composed of three parts, due to viscous losses, heat-conduction losses, and, finally, edge effects. That part of the absorption coefficient which is due to viscous losses, according to\(^{38}\), is equal to

\[ \gamma_\eta=\frac{1}{r u_2 \rho}\,\frac{\rho_s}{\rho_n}\sqrt{\frac{\rho_n\eta\omega}{2}}. \tag{17.19} \]

As for the heat losses, in the cited work the temperature discontinuity at the boundary between the solid body and liquid helium, which substantially changes the magnitude of \(\gamma_\eta\), was not taken into account. Finally, for the absorption coefficient of second sound at the walls of a cylinder, at

\(^*\) In the case of sound propagation along the axis of the tube it is necessary to introduce an additional factor 2 in the denominator of (17.17). This is due to the fact that upon reflection of sound, as is seen from formula (17.2), the temperature \(T\) is practically doubled in comparison with the temperature in the incident (traveling) wave.

with the aid of (17,18) and (17,19) we obtain the following expression:

\[ \gamma = \frac{1}{r u_{3}\rho}\frac{\rho_s}{\rho_n} \sqrt{\frac{\rho_n \eta \omega}{2}} + \frac{1}{r C u_2} \frac{1}{\dfrac{1}{4KT^3}+\sqrt{\dfrac{2}{C_T\varkappa\omega}}} +\gamma_{\mathrm{b}} . \tag{17,20} \]

The term \(\gamma_{\mathrm{b}}\) is due to boundary losses.

The absorption of second sound at the walls is substantial only at small sound frequencies. At higher frequencies, the volume absorption, which depends on the frequency according to the law \(\omega^2\), noticeably exceeds the surface effects.

Dingle\(^{18}\), in considering the phenomena of absorption of second sound at the walls of a cylindrical tube, made a crude error. He denoted by the same letter the coordinates along the axis and normal to the wall. Failing to notice this, he then made no distinction between differentiating the velocity \(v_n\) in different directions. As a result, in the coefficient of sound absorption there was obtained a term having no physical meaning, in which the effects of the viscosity of helium II and the thermal conductivity of the solid wall are confused. This effect proved to be so large that, in Dingle’s opinion, it alone determines the absorption of second sound at the walls of the tube.

Surface effects in the absorption of second sound were observed by K. Zinov’eva\(^{16}\). In these experiments the attenuation of second sound in a cylindrical tube was determined. In the region of low frequencies the absorption varied according to the law \(\sqrt{\omega}\). There were no effects depending on thermal conductivity; on passing from glass tubes to copper ones no noticeable change in absorption was observed. This is explained by the fact that, owing to the large thermal resistance of the solid-body—liquid-helium-II boundary, second sound is practically completely reflected from the solid body, and temperature oscillations do not penetrate into the solid body. This follows directly from formula (17,20), in which, because of the extremely large value of \(1/KT^3\), characterizing the thermal resistance of the boundary, the second term proves to be negligible.

The experimental values of the absorption coefficient of second sound obtained by K. Zinov’eva agree well with the calculation by formula (17,20).

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2. *) The difference of the values of some quantity on the two sides of the surface is denoted by square brackets. ↩