Reflection of Bounded Wave Beams and Pulses

L. M. Brekhovskikh

Contents

I. Reflection of bounded beams . . . . . . . . . . . . . . . . . . . . . . 539
    1. Representation of a bounded beam as a superposition of plane waves . . . . . . . . . . . . . . . . . . 540
    2. Field of the reflected beam. Displacement of the beam upon reflection . . . . . . . . . . . . . . . . . . 543
    3. Total internal reflection of a light beam . . . . . . . . . . . . . . 546
    4. Reflection of ultrasonic beams . . . . . . . . . . . . . . . . . . 548
    5. Distribution of energy in the cross section of the reflected beam . . . . . . . . . . . . . . . . . . 551
    6. On the energy flux under total internal reflection . . . . . . . . . . . . . . . . . . 553
    7. Reflection from an inhomogeneous medium . . . . . . . . . . . . . . . . . . 558

II. Reflection of pulses . . . . . . . . . . . . . . . . . . . . . . 564
    8. General relations. The law of conservation of the integral impulse . . . . . . . . . . . . . . . . . . 564
    9. Distortion of the pulse shape under total internal reflection . . . . . . . . . . . . . . . . . . 569

I. Reflection of Bounded Beams

The theory of reflection of plane waves from plane boundaries of separation, owing to its extreme idealization, proves in a number of cases to be insufficient for describing real physical phenomena. In real cases one usually has to deal not with an infinite plane wave, but with a bounded beam or with a spherical wave.

A separate work by the author was devoted to the study of the reflection and refraction of spherical waves[^1]. In the present article we consider the reflection of a bounded beam, as well as of a pulse, from a plane boundary. In a whole series of cases and, in particular, when a beam or pulse is incident on a boundary at an angle close to the angle of total internal reflection, or when beams are reflected from layers and plates, we encounter phenomena that are absent in the reflection of plane waves. In particular, the displacement of the beam along the boundary upon its reflection is of great interest, as is the change in the shape of the pulse. A number of practically interesting results are also obtained in considering the reflection of bounded beams from stratified-inhomogeneous media.

In considering these questions it is expedient to base oneself on the expansions of bounded beams and pulses in plane waves, which we have already used in the case of spherical waves.

1. Representation of a bounded beam as a superposition of plane waves

Let us suppose that a bounded beam is produced as a result of the passage of a plane wave through a slit \(cc\) in the screen \(AA\), shown in Fig. 1. Let us further suppose that the beam falls on the boundary of a stratified inhomogeneous medium, whose reflection is to be investigated.

Geometrical boundaries of the beam in Fig. 1 are marked by dashed lines. In reality, however, owing to diffraction, the beam spreads somewhat.

The angle of incidence of the plane wave we shall denote by \(\vartheta_0\). For simplicity we shall assume the problem to be two-dimensional, i.e. the slit in the screen is infinite, and the plane of incidence of the wave is perpendicular to the axial line of the slit. Under this condition, if the coordinate system is chosen as in Fig. 1, then the coordinate \(y\) drops out of the formulas.

Fig. 1. Beam formed when a plane wave is incident on an aperture in a screen.

We direct the \(z\)-axis perpendicular to the plane of the screen, the \(x\)-axis perpendicular to the axial line of the slit, and place the origin of coordinates on the axial line. We denote the width of the slit by \(2a\), and the distance from the slit to the reflecting plane by \(l\).

The field of the incident beam between the screen and the reflecting plane must satisfy the wave equation, and, in the plane of the screen, boundary conditions which, assuming the slit width to be large in comparison with the wavelength\(^*\), we shall prescribe in the approximate Kirchhoff form, namely, we shall assume that:

1. On the rear side of the screen the field is equal to zero.
2. In the plane of the slit the field is the same as in the absence of the screen.

By these boundary conditions and by the wave equation the field of the incident beam is determined completely.

We shall prescribe the field of the plane wave incident on the screen in the form

\[ \psi = e^{i(\alpha x+\gamma z)-i\omega t}, \tag{1} \]

\(^*\) In the exact theory of diffraction by a slit it is shown that, for Kirchhoff’s assumption to be valid, it is also necessary that the angle of incidence be small. However, for the question investigated here this is not of fundamental significance.

where

\[ \alpha=k\sin\vartheta_0,\qquad \gamma=k\cos\vartheta_0. \tag{2} \]

After passage through the slit at \(z=0\) we shall have, omitting everywhere for brevity the factor \(e^{-i\omega t}\),

\[ \left. \begin{aligned} \psi(x)&=e^{i\alpha x} &&\text{for } -a\ll x\ll a,\\ \psi(x)&=0 &&\text{for } |x|\gg a. \end{aligned} \right\} \tag{3} \]

It is expedient to generalize the problem somewhat and to consider the case of an arbitrary distribution of the wave amplitude over the cross section of the beam by the plane \(z=0\). Therefore in what follows we shall assume that the function \(\psi(x)\) has the form

\[ \psi(x)=F(x)e^{i\alpha x},\qquad -\infty<x<\infty, \tag{4} \]

where the function \(F(x)\) corresponds to the variable transparency of the screen, depending on the coordinate \(x\).

To represent the beam as a superposition of plane waves, it is expedient to represent the field in the plane \(z=0\) in the form of a Fourier integral:

\[ \psi(x)=\int_{-\infty}^{+\infty}\Phi(p)e^{ipx}\,dp. \tag{5} \]

The function \(\Phi(p)\) is then determined by the well-known formula

\[ \Phi(p)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\psi(x)e^{-ipx}\,dx =\frac{1}{2\pi}\int_{-\infty}^{+\infty}F(x)e^{i(\alpha-p)x}\,dx. \tag{6} \]

Thus, for example, in the case of a beam formed by the passage of a wave through a slit, we have:

\[ \left. \begin{aligned} F(x)&=1, && -a<x<a,\\ F(x)&=0, && |x|>a. \end{aligned} \right\} \tag{7} \]

Therefore for the given case we obtain:

\[ \Phi(p)=\frac{1}{2\pi}\int_{-a}^{+a} e^{i(\alpha-p)x}\,dx =\frac{\sin(\alpha-p)a}{\pi(\alpha-p)}. \tag{8} \]

It is useful to note that the function \(\Phi(p)\) has an appreciable magnitude (of order 1) only for small differences \(\alpha-p\), satisfying the condition \(a(\alpha-p)\ll1\). For \(a(\alpha-p)\gg1\) the value of the function \(\Phi(p)\) will be very small.

On the basis of the function \(\psi(x)\), specified in the form (5) and characterizing the field in the plane of the screen, we shall construct the following function of both

of the variables \(x\) and \(z\):

\[ \psi(x,z)=\int_{-\infty}^{+\infty}\Phi(p)e^{i(px+\mu z)}\,dp, \tag{9} \]

where

\[ \mu=\sqrt{k^2-p^2}. \]

This function will describe the field of the incident beam between the plane of the screen and the reflecting plane, since:

1. It satisfies the wave equation

\[ \frac{\partial^2\psi}{\partial x^2}+\frac{\partial^2\psi}{\partial z^2}+k^2\psi=0, \tag{10} \]

because this equation is satisfied by the expression under the integral.

1. For \(z=0\) it passes into the function \(\psi(x)\), specified by expression (5), i.e., it satisfies the boundary condition.

Owing to the uniqueness of the solution of the wave equation under the specified boundary conditions, this expression determines the beam field completely.

The exponential in the integrand of (9), for any definite \(p\), represents a plane wave propagating at the angle

\[ \vartheta=\arcsin\frac{p}{k} \tag{11} \]

with respect to the direction of the \(z\)-axis. Thus, each component of the expansion of the field in the plane \(z=0\) into a Fourier integral is continued in space in the form of a separate plane wave.

The sought expansion of the incident beam in plane waves will therefore be expression (9).

For \(p>k\) the angle \(\vartheta\), according to (11), will be complex, i.e., in the expansion there will also be inhomogeneous plane waves (see \(^{1}\)). Their amplitude will decrease with distance from the plane of the screen according to an exponential law. This is also seen from (9), since for \(p>k\) the exponential under the integral takes the form

\[ e^{ipx-\sqrt{p^2-k^2}z}. \]

The greatest role in (9) will be played by those plane waves whose direction is close to the direction of the plane wave incident on the target. Indeed, as was already indicated, the function \(\Phi(p)\) has an appreciable magnitude only for

\[ (\alpha-p)a\ll 1. \]

Since \(\alpha=k\sin\vartheta_0\) and \(p=k\sin\vartheta\), the last condition may be written in the form

\[ ak(\sin\vartheta_0-\sin\vartheta)\ll 1 \]

or, owing to the closeness of \(\vartheta_0\) and \(\vartheta\):

\[ \vartheta_0-\vartheta \ll \frac{1}{ak\cos\vartheta_0}. \tag{12} \]

The quantity \(a\cos\vartheta_0\) is equal to the width of the normal cross-section of the beam (whereas \(a\) is the width of the cross-section in the plane \(z=0\)). Since this width must, of course, be much greater than the wavelength, otherwise the concept of a beam loses its meaning, i.e. \(ak\cos\vartheta_0 \gg 1\), it follows that

\[ \vartheta_0-\vartheta \ll 1. \tag{13} \]

2. Field of the Reflected Beam. Displacement of the Beam upon Reflection

Let us denote the reflection coefficient of a plane wave incident on the reflecting boundary at an angle \(\vartheta\) by \(V(\vartheta)\). In the general case this will be a complex quantity, the square of whose modulus gives the energy reflection coefficient, while its argument gives the change in the phase of the wave upon reflection. We shall write \(V(\vartheta)\) in the form

\[ V(\vartheta)=\rho(p)e^{i\varphi(p)}, \tag{14} \]

where \(\rho(p)\) and \(\varphi(p)\) are functions of \(p\), and consequently also of the angle of incidence \(\vartheta\).

The field produced at the boundary by the incident beam we obtain from (9), putting there \(z=l\). As a result, at the boundary we shall have:

\[ \psi_{\mathrm{inc}}(x)=\int_{-\infty}^{+\infty}\Phi(p)e^{i(px+\mu l)}\,dp. \tag{15} \]

The field of the reflected beam at the boundary is obtained if the integrand in (15), representing a plane incident wave, is multiplied by the reflection coefficient \(V(\vartheta)\).

Consequently:

\[ \psi_{\mathrm{refl}}(x)=\int_{-\infty}^{+\infty}\Phi(p)\rho(p)e^{i\varphi(p)+i(px+\mu l)}\,dp. \tag{16} \]

Substituting into (16) the value of \(\Phi(p)\) from (6), one may write the fields of the incident and reflected beams in another form:

\[ \psi_{\mathrm{inc}}(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\int F(\xi)e^{i(\alpha-p)\xi+i(px+\mu l)}\,dp\,d\xi, \tag{17} \]

\[ \psi_{\mathrm{refl}}(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\int F(\xi)\rho(p)e^{i\varphi(p)}e^{i(\alpha-p)\xi+i(px+\mu l)}\,dp\,d\xi. \tag{18} \]

In what follows we shall consider the case in which the reflection coefficient, in its absolute value, changes insignificantly in the angular range of interest to us, or remains altogether constant (as, for example, in the case of total internal reflection).

Under this assumption the quantity \(\rho(p)\) may be taken outside the integral sign at the value \(p=\alpha\). In addition, in (17) and (18) we introduce the new quantity

\[ \Omega=p-\alpha, \tag{19} \]

and, since we have seen that under the integrals only the values of the functions corresponding to small \(\Omega\) will play an essential role, we expand the function \(\varphi(p)\) in a series in powers of \(\Omega\), i.e., we write:

\[ \varphi(p)=\varphi(\alpha)+\varphi'(\alpha)\Omega+\frac{1}{2}\varphi''(\alpha)\Omega^2+\cdots, \tag{20} \]

where

\[ \varphi'(\alpha)=\left(\frac{\partial\varphi}{\partial p}\right)_{p=\alpha}, \tag{21} \]

and for the present we confine ourselves only to the term of first degree in \(\Omega\). Then (17) and (18), taking also the value of \(\mu\) into account, may be written in the form:

\[ \psi_{\mathrm{inc}}(x)=e^{i\alpha x}\int\limits_{-\infty}^{+\infty}\int F(\xi)\, e^{i\Omega(x-\xi)+i\sqrt{k^2-(\Omega+\alpha)^2}\,l}\,d\xi\,d\Omega, \tag{22} \]

\[ \psi_{\mathrm{ref}}(x)=V(\vartheta_0)e^{i\alpha x}\int\limits_{-\infty}^{+\infty}\int F(\xi)\, e^{i\Omega[x+\varphi'(\alpha)-\xi]+i\sqrt{k^2-(\Omega+\alpha)^2}\,l}\,d\xi\,d\Omega. \tag{23} \]

Comparing (22) and (23), we find:

\[ \psi_{\mathrm{ref}}(x)=V(\vartheta_0)\psi_{\mathrm{inc}}\,[x+\varphi'(\alpha)]. \tag{24} \]

It follows from this that upon reflection the beam is displaced along the boundary by the amount

\[ \Delta=-\varphi'(\alpha)=-\left(\frac{\partial\varphi}{\partial p}\right)_{p=\alpha}. \tag{25} \]

The expression for \(\Delta\) in this form was apparently first obtained in work \(^{2}\), although by a considerably more complicated route than the derivation given above.

It should be noted that even Eichenwald \(^{3}\), on the basis of his very careful theoretical investigation of the phenomenon of total internal reflection, found that the point of entry of the lines of energy flux (the Umov–Poynting vector) into the reflecting, optically less dense, medium is separated by a certain distance from the point of their

output. This very significant result played a major role in clarifying the nature of the phenomenon of total internal reflection. However, one should not confuse, as is done in the review paper\(^{4}\), the above-found displacement \(\Delta\) with the displacement obtained by Eichenwald. The essential difference between these quantities is that \(\Delta\) is the displacement of a finite beam, whereas the Eichenwald displacement is the displacement of the line of energy flux in an unbounded plane wave. The magnitudes of these displacements likewise have nothing in common with one another.*)

One can indicate cases in which the calculation of \(\Delta\) by formula (25) gives an almost trivial result, easily obtained from ray considerations. We shall consider one such case in order that the reader may better understand the physical meaning of the quantity \(\Delta\).

Fig. 2. Case of an absolutely reflecting boundary.

Let a plane wave be incident on the boundary \(z=0\) (Fig. 2), whose reflection coefficient is equal to unity (the boundary of an absolutely reflecting body). Taking the plane \(xz\) to coincide with the plane of incidence, and choosing the direction of the \(z\)-axis as indicated in Fig. 2, we write the incident wave in the form \(\exp[i(\alpha x-\gamma z)-i\omega t]\) and the reflected wave in the form \(\exp[i(\alpha x+\gamma z)-i\omega t]\), where \(\alpha=k\sin\vartheta_0,\ \gamma=k\cos\vartheta_0\).

At \(z=0\) both waves have equal amplitudes and phases. In some plane \(z=h\), however, the incident and reflected waves will have the expressions \(\exp[i(\alpha x-\gamma h)-i\omega t]\) and \(\exp[i(\alpha x+\gamma h)-i\omega t]\). The ratio of the reflected wave to the incident wave, equal to \(V=e^{2i\gamma h}\), may be regarded as the reflection coefficient of the wave in the plane \(z=h\). The modulus of this coefficient is unity, while the phase

\[ \varphi=2\gamma h=2\sqrt{k^2-\alpha^2}\,h \]

corresponds to the phase lag of the wave during its passage from the plane \(z=h\) to the plane \(z=0\) and back.

In the general case of an arbitrary angle of incidence of the wave, the phase \(\varphi\) will be

\[ \varphi=2h\sqrt{k^2-p^2}, \]

where \(p\), as above, is the projection of the wave vector of the wave onto the \(x\)-axis \((p=k\sin\vartheta)\).

*) Here it is appropriate to note that it is hardly proper to attach a real meaning to the lines of the Umov–Poynting vector in a plane wave, since this vector is determined ambiguously and leads to unambiguous physical conclusions only when the energy flux is calculated through a closed surface. In the case of a plane wave, the pattern of energy fluxes can be traced only by selecting from the wave a beam by means of a diaphragm, large in comparison with the wavelength (otherwise the diaphragm will distort the field of the wave). In doing so we arrive at the problem considered here of the reflection of finite beams.

Expression (25) gives for the displacement \(\Delta\) in this case:

\[ \Delta=-\left(\frac{d\varphi}{dp}\right)_{p=\alpha}=2h\frac{\alpha}{\gamma}=2h\,\operatorname{tg}\vartheta_0. \]

The quantity \(\Delta\) is shown in Fig. 2. It is equal to the horizontal displacement of the ray as it passes from the plane \(z=h\) to the plane \(z=0\) and back.

Thus, in the case under consideration the displacement of the rays, calculated by formula (25), has a very simple and natural meaning. It is, of course, always computed without recourse to formula (25), but only on the basis of considering the path of the ray.

However, the main value of the theory set forth above, which led to formula (25), lies in cases where the ray concept is inapplicable. Below we shall consider several such cases.

Let us also note that investigation of the question of the displacement of a beam upon reflection and of the change in its shape (see below) by the method of resolving the beam into plane waves is completely analogous to the investigation of the propagation of a quasi-monochromatic pulse in a dispersive medium (see, for example, \(^{5}\)).

3. Total internal reflection of a light beam

According to formula (25), one should expect that the displacement of a beam upon reflection will be most significant in those cases where the phase of the reflection coefficient changes rapidly with angle. One such case is that of a beam incident on a boundary at an angle somewhat exceeding the limiting angle of total internal reflection. Indeed, according to the well-known Fresnel formulas, the reflection coefficient of an electromagnetic wave from the interface of two media can be written in the form

\[ \left. \begin{aligned} V_{\perp} &= \frac{\dfrac{\mu_1}{\mu}\cos\vartheta-\sqrt{\,n^2-\sin^2\vartheta\,}} {\dfrac{\mu_1}{\mu}\cos\vartheta+\sqrt{\,n^2-\sin^2\vartheta\,}}, \\[1.2em] \text{and}\qquad V_{\parallel} &= \frac{\dfrac{\mu}{\mu_1}n^2\cos\vartheta-\sqrt{\,n^2-\sin^2\vartheta\,}} {\dfrac{\mu}{\mu_1}n^2\cos\vartheta+\sqrt{\,n^2-\sin^2\vartheta\,}}, \end{aligned} \right\} \tag{26} \]

where \(\vartheta\) is the angle of incidence, \(n\) is the refractive index of the boundary, and \(\mu\) and \(\mu_1\) are the magnetic permeabilities of the media. The signs \(\perp\) and \(\parallel\) refer to the cases when the electric-field vector \(\mathbf{E}\) is directed respectively perpendicular and parallel to the plane of incidence.

For \(n<1\) and \(\sin\vartheta>n\) (the case of total internal reflection), the root entering expressions (26) must be written in the form \(i\sqrt{\sin^2\vartheta-n^2}\). In this case \(V_\perp\) and \(V_{\parallel}\) become complex quantities with moduli equal to unity.

Let us represent the reflection coefficients in the form

\[ V_\perp=\rho_\perp e^{i\varphi_\perp}\quad \text{and}\quad V_{\parallel}=\rho_{\parallel}e^{i\varphi_{\parallel}} . \tag{27} \]

Then, under total internal reflection, we obtain:

\[ \rho_\perp=\rho_{\parallel}=1 \]

and

\[ \left. \begin{aligned} \varphi_\perp&=-2\operatorname{arctg}\frac{\mu}{\mu_1}\, \frac{\sqrt{\sin^2\vartheta-n^2}}{\cos\vartheta},\\[4pt] \varphi_{\parallel}&=-2\operatorname{arctg}\frac{\mu_1}{\mu n^2}\, \frac{\sqrt{\sin^2\vartheta-n^2}}{\cos\vartheta}. \end{aligned} \right\} \tag{28} \]

It is not difficult to see that near the angle of total internal reflection \(\varphi_\perp\) and \(\varphi_{\parallel}\) are very rapidly varying functions of the angle \(\vartheta\). In particular, at \(\vartheta=\arcsin n\) the derivatives \(\dfrac{\partial\varphi_\perp}{\partial\vartheta}\) and \(\dfrac{\partial\varphi_{\parallel}}{\partial\vartheta}\) become infinite. Consequently, in this case one should expect an appreciable displacement of the beam.

Using, as above, the notation \(p=k\sin\vartheta\), we can write for \(\mu=\mu_1=1\):

\[ \varphi_\perp=-2\operatorname{arctg}\sqrt{\frac{p^2-n^2}{1-p^2}} . \tag{29} \]

According to (25), we obtain for the displacement of the beam, if its axial line makes an angle \(\vartheta_0\) with the normal to the boundary \((\sin\vartheta_0>n)\),

\[ \Delta_\perp=\frac{\lambda}{2\pi}\, \frac{\operatorname{tg}\vartheta_0}{\sqrt{\sin^2\vartheta_0-n^2}} . \tag{30} \]

Similarly, if the electric-field vector lies in the plane of incidence of the wave, then

\[ \Delta_{\parallel}=\frac{\lambda}{2\pi n^2}\, \frac{\operatorname{tg}\vartheta_0}{\sqrt{\sin^2\vartheta_0-n^2}} . \tag{31} \]

In these formulas \(\lambda\) is the wavelength in the medium from which the wave is incident.

From formulas (30) and (31) it is seen, as might have been expected, that the displacement of the beam increases as its angle of incidence approaches the critical angle of total internal reflection.

For \(\sin\vartheta_0\to n\) we have from (30) and (31) \(\Delta_\perp\to\infty\) and \(\Delta_{\parallel}\to\infty\), i.e., our formulas lose their meaning. This occurs because

that for \(\sin \vartheta_0\) very close to \(n\), the phase \(\varphi\) of the reflection coefficient changes so rapidly with changes in the angle that, in the expansion (20), terms containing \(\Omega^2\), \(\Omega^3\), etc., can no longer be neglected (it is assumed here that \(\Omega=p-\alpha\) has an order of magnitude determined by relation (12)).

The experimental determination of the displacement of light beams under total internal reflection at the glass–air boundary is described in works \(^{6,7}\). Although at each reflection, according to formulas (30) and (31), the displacement may in magnitude reach many wavelengths, in order to make it clearly observable it was necessary to resort to multiple reflections. In this case the displacement was multiplied by a number equal to the number of reflections. The experiments fully confirmed the theoretical data.

4. Reflection of ultrasonic beams

The reflection coefficient of a plane sound wave from the interface of two liquids or gases has the form

\[ V=\frac{m\cos\vartheta-\sqrt{\,n^2-\sin^2\vartheta\,}} {m\cos\vartheta+\sqrt{\,n^2-\sin^2\vartheta\,}}, \tag{32} \]

where \(m=\dfrac{\rho_1}{\rho}\) is the ratio of the densities of the media, and \(n\), as above, is the refractive index of the boundary \((n=c/c_1)\).

Hence, for the phase of the reflection coefficient under total internal reflection, neglecting attenuation of the waves in the media (\(n\) real), we obtain:

\[ \varphi=-2\operatorname{arctg}\frac{\sqrt{\sin^2\vartheta-n^2}}{m\cos\vartheta}. \tag{33} \]

For the displacement of the beam, as in the case of electromagnetic waves, using formula (25) we obtain the expression

\[ \Delta=\frac{\lambda}{2\pi m}\, \frac{\operatorname{tg}\vartheta_0}{\sqrt{\sin^2\vartheta_0-n^2}}. \tag{34} \]

We again note that the displacement of the beam becomes especially noticeable near the angle of total internal reflection, when the difference \(\sin^2\vartheta_0-n^2\) is small.

When a sound wave is incident from a liquid onto the boundary of a solid, total internal reflection takes place for \(\sin\vartheta>c/b_1\), where \(c\) is the speed of sound waves in the liquid and \(b_1\) is the speed of shear waves in the solid. Near the limiting angle of total internal reflection there will occur, as in optics, a noticeable displacement of the beam, which can be calculated by formula (25), using the corresponding expression for the reflection coefficient. However, in addition to this

case, a noticeable displacement will occur at such an angle of incidence of the wave when its phase velocity along the boundary coincides with the velocity of the Rayleigh wave at the boundary of the solid. This angle will be somewhat greater than the critical angle of total internal reflection. Near it the reflection coefficient, as detailed analysis shows, is a very rapidly varying function. In particular, the phase of the reflection coefficient changes in the neighborhood of this angle by approximately \(2\pi\).

The calculation for the displacement gives in this case\(^8\)

\[ \Delta=\frac{2\lambda}{\pi}\,\frac{\rho_1}{\rho} \sqrt{ \frac{r(r-s)}{s(s-1)} \frac{1+6s^2(1-q)-2s(3-2q)}{s-q} }, \tag{35} \]

where \(\lambda\) is the wavelength in the liquid,

\[ s=\left(\frac{b_1}{v_R}\right)^2,\quad q=\left(\frac{b_1}{c_1}\right)^2,\quad r=\left(\frac{b_1}{c}\right)^2, \]

\(v_R\) is the velocity of the Rayleigh wave, \(b_1\) and \(c_1\) are the velocities of shear and longitudinal waves in the solid, and \(c\) is the speed of sound in the liquid.

The quantity \(s\) depends on Poisson’s ratio and varies for different solids within the limits from 1.1 to 1.3. For aluminum, for example, \(s=1.15\).

Substitution of numerical values into formula (35) gives for the boundaries:

a) water—aluminum \(\Delta=24.4\lambda\);

b) xylol—aluminum \(\Delta=33.4\lambda\).

The angle of incidence \(\vartheta_R\) at which this displacement occurs, determined from the relation

\[ \sin\vartheta_R=\frac{c}{v_R}, \]

is: for case a) \(\vartheta_R=31^\circ\), for case b) \(\vartheta_R=27^\circ\).

The displacement \(\Delta\) is large enough to be recorded in a simple manner. Figure 3 shows a shadow photograph

Fig. 3. Displacement of an ultrasonic beam (frequency \(1.6\cdot10^7\) cycles/sec) upon its reflection from the xylol–aluminum boundary at three angles of incidence differing slightly from one another.

obtained by the shadow method, of the total reflection of an ultrasonic beam (frequency \(1.6\cdot10^7\) cycles/sec) from the xylol—aluminum boundary for three angles of incidence (increasing from left to right\(^8\)). In the middle image

the angle of incidence is equal to $\vartheta_R$; the displacement of the reflected beam relative to the incident one is quite noticeable. This displacement is absent in the extreme images, where the angles of incidence differ somewhat from $\vartheta_R$. The total

Fig. 4. Displacement of the ultrasonic beam upon its reflection from the acid–aluminum boundary for a frequency of $5.6 \cdot 10^6$ cycles and for three different angles of incidence.

beam width is $12$ mm, the wavelength in the acid at this frequency is $0.08$ mm; the displacement $\Delta$, in agreement with the calculations, proves to be equal to $2.7$ mm.

Figure 4 shows a pattern obtained under the same conditions, but for a lower frequency, $5.6 \cdot 10^6$ cycles ($\lambda = 0.24$ mm). Since,

Fig. 5. Phase $\varphi$ of the reflection coefficient of a sound wave from an aluminum plate in air at various angles of incidence (calculation). Plate thickness $3.1$ mm, wave frequency $3.35 \cdot 10^6$ cycles.

according to theory, the displacement is proportional to the wavelength, here it proves to be still more noticeable; the beam is displaced by almost the entire

full width. In this case, owing to the increase in the wavelength, the boundaries of the beam become more blurred because of diffraction.

From the magnitude of the displacement the angle \(\vartheta_R\) is determined with such good accuracy that on this principle a sufficiently precise method for determining the Rayleigh-wave velocity can be based.

A very large displacement of the sound beam may arise when it is reflected from a plate, since in this case the phase of the reflection coefficient changes especially rapidly with the angle (see, for example, Fig. 5, where the phase of the reflection coefficient of a sound wave from an aluminum plate in air is shown as a function of the angle of incidence). In this case, the transmitted beam is also displaced with respect to the incident beam.

Two cases of reflection of an ultrasonic beam of frequency \(16.3\cdot 10^6\) cycles from an aluminum plate of thickness \(0.43\) mm in xylene are shown

Fig. 6. Reflection of an ultrasonic beam of frequency \(16.3\cdot 10^6\) cycles from an aluminum plate of thickness \(0.43\) mm in xylene.

in Fig. 6. In these cases the angles of incidence of the beam on the plate are somewhat different. In one case we have almost complete reflection of the wave, in the other—almost complete transmission through the plate. The displacements of the reflected and transmitted beams are, in magnitude, several plate thicknesses. The inhomogeneity of the reflected beam over its cross-section is also noticeable. We shall dwell on this in the next section.

5. Distribution of energy in the cross-section of the reflected beam

To investigate the “internal structure” of the reflected beam, i.e. the distribution of energy over its cross-section, it is necessary in formula (20), when expanding the phase of the reflection coefficient in a series in the angles, to take into account also the second-order terms*). In this case, instead of

*) The question considered here is formally analogous to the question of the change in the form of a wave packet during propagation in a dispersive medium (see\({}^{5}\), from which we also take the scheme of the calculation).

from formula (23) we obtain the following expression for the field of the reflected beam:

\[ \psi_{\mathrm{refl}}(x)=\frac{1}{2\pi}V(\vartheta_0)e^{i\alpha x} \int_{-\infty}^{+\infty}\int F(\xi)\exp\left\{ i\Omega\left[x+\varphi'(\alpha)-\xi\right]\right. \]
\[ \left. {}+i\sqrt{k^2-(\Omega+\alpha)^2}\,l+\frac{i\varphi''(\alpha)}{2}\Omega^2 \right\}\,d\xi\,d\Omega . \tag{36} \]

Here the factor \(\exp\left(i\sqrt{k^2-(\Omega+\alpha)^2}\,l\right)\) may be taken outside the integral sign at the value \(\Omega=0\), owing to the slowness of its variations when \(\Omega\) changes (in the case of not very large \(l\)). This factor does not depend on the change of the phase of the wave upon reflection and determines the spreading of the beam due to diffraction during its propagation from the target to the reflecting surface. This spreading has already been studied sufficiently well in diffraction theory. Here we shall not take it into account.

Introduce a new variable \(\eta\) according to the relation

\[ \varphi''\left(\Omega+\frac{x+\varphi'-\xi}{\varphi''}\right)^2=\pi\eta^2 . \]

Performing the integration with respect to \(\eta\) and taking into account that

\[ \int_{-\infty}^{+\infty} e^{\pm i\frac{\pi}{2}\eta^2}\,d\eta=1\pm i, \tag{37} \]

we obtain:

\[ \psi_{\mathrm{refl}}(x)=\frac{1+i}{2}V(\vartheta_0)\frac{1}{\sqrt{\pi\varphi''}} e^{i(\alpha x+\sqrt{k^2-\alpha^2}\,l)} \int_{-\infty}^{+\infty} e^{-i\frac{(x+\varphi'-\xi)^2}{2\varphi''}}F(\xi)\,d\xi . \]

We shall further denote

\[ \frac{x+\varphi'-\xi}{\sqrt{\pi\varphi''}}=u, \]

then we obtain:

\[ \psi_{\mathrm{refl}}(x)=\frac{1+i}{2}V(\vartheta_0)e^{i(\alpha x+\sqrt{k^2-\alpha^2}\,l)}\times \]
\[ \times\int_{-\infty}^{+\infty} e^{-i\frac{\pi}{2}u^2} F\left(x+\varphi'+\sqrt{\pi\varphi''}\,u\right)\,du . \tag{38} \]

For sufficiently small \(\varphi''\), taking (37) into account, we again obtain formula (24), which gives the displacement of the beam without changing its shape.

Further analysis of expression (38) without neglecting the term containing \(\sqrt{\pi\varphi''}\) is possible only when a definite distribution of the wave amplitude over the cross section of the incident beam is specified, i.e., when the form of the function \(F\) is specified. We shall assume here that the function \(F\)

is specified by relations (7), i.e., the incident beam has width \(2a\) and an intensity constant over its entire cross section. Then the integral (38) will in fact be evaluated over the limits \((u_1,u_2)\), determined from the relation

\[ x+\varphi' + \sqrt{\pi\varphi''}\,u_{1,2}=\mp a, \tag{39} \]

or:

\[ u_1=-\frac{1}{\sqrt{\pi\varphi''}}\,(a+x+\varphi'); \]

\[ u_2=\frac{1}{\sqrt{\pi\varphi''}}\,(a-x-\varphi'), \tag{40} \]

since outside these limits the function \(F\) becomes zero. Taking into account that within these limits \(F=1\), we obtain from (38)

\[ \psi_{\mathrm{refl}}(x)=\frac{1+i}{2}\,V(\vartheta_0)\, e^{i(ax+\sqrt{k^2-a^2}\,l)} \int_{u_1}^{u_2} e^{-i\frac{\pi}{2}u^2}\,du. \tag{41} \]

The same can be written in a somewhat different form:

\[ \psi_{\mathrm{refl}}(x)=\frac{1+i}{2}\,V(\vartheta_0)\, e^{i(ax+\sqrt{k^2-a^2}\,l)} \left[f(u_2)-f(u_1)\right], \tag{42} \]

where

\[ f(u_1)=\int_{0}^{u_1} e^{-i\frac{\pi}{2}u^2}\,du = C(u_1)-iS(u_1) \tag{43} \]

and similarly for \(f(u_2)\). Here \(C\) and \(S\) are Fresnel integrals, whose values can be found in tables.

Since the functions \(C\) and \(S\) are oscillatory functions of their arguments, the intensity in the reflected beam will vary nonmonotonically over the cross section. Qualitatively, this is confirmed by experiment (see, for example, Fig. 6). A quantitative comparison of the theory with experiment is not yet possible, owing to the absence of quantitative measurements of the intensity distribution over the cross section of the reflected beam.

6. On the energy flux in total internal reflection

The displacement of a beam upon its reflection can also be obtained by considering the energy flux in the incident and reflected beams \(^{10,11}\).

As has already been indicated, the lines of the energy flux (the Umov–Poynting vector) in total internal reflection of an unbounded plane wave were first investigated by Eichenwald \(^{3}\). However,

these results is difficult, since in order to measure the energy-flux density at each point of space it is necessary to isolate, by a diaphragm, a bounded beam.

From the very beginning we shall assume that a bounded beam is incident on the boundary. For definiteness let us consider the acoustic case (using here the results of work \(^{12}\)), although the case of an electromagnetic wave in principle differs in no way from it.

The energy-flux density in any direction is given by the Umov vector

\[ \mathbf I=p\mathbf v, \tag{44} \]

where \(p\) is the sound pressure, \(\mathbf v\) is the velocity of the particles in the sound wave. If \(p\) and \(\mathbf v\) are expressed in terms of the sound potential \(\varphi\), then

\[ \mathbf I=-\rho\,\frac{\partial \varphi}{\partial t}\operatorname{grad}\varphi . \tag{45} \]

If one uses the complex form of writing wave processes, as we do everywhere, then here \(\varphi\) must be understood as the real part of the potential. Therefore we write \(\varphi\) in the form

\[ \varphi=\operatorname{Re}\left[\psi(x,y,z)e^{-i\omega t}\right], \tag{46} \]

where the symbol \(\operatorname{Re}\) denotes the real part. But

\[ \operatorname{Re}(\psi e^{-i\omega t})=\frac{1}{2}\left(\psi e^{-i\omega t}+\psi^{*}e^{i\omega t}\right) \tag{47} \]

and, evidently,

\[ \operatorname{grad}\varphi=\frac{1}{2}\left(\operatorname{grad}\psi e^{-i\omega t}+\operatorname{grad}\psi^{*}e^{i\omega t}\right); \]

\[ \frac{\partial\varphi}{\partial t}=\frac{i\omega}{2}\left(\psi^{*}e^{i\omega t}-\psi e^{-i\omega t}\right). \tag{48} \]

Let us find the time average of the energy flux \(\mathbf I\). Multiplying (47) and (48) and averaging over time (where the time average of terms containing the factors \(e^{i\omega t}\) and \(e^{-i\omega t}\) becomes zero), we obtain:

\[ \overline{\mathbf I}=\frac{i\omega\rho}{4}\left(\psi\,\operatorname{grad}\psi^{*}-\psi^{*}\operatorname{grad}\psi\right). \tag{49} \]

If the last formula is applied to compute the energy flux in a plane wave, then, taking into account that \(\psi(x,y,z)=\psi_{0}e^{i\mathbf k\mathbf r}\), we obtain \(\operatorname{grad}\psi=i\mathbf k\psi\), \(\operatorname{grad}\psi^{*}=-i\mathbf k\psi^{*}\), and formula (49) gives:

\[ \overline{\mathbf I}=\frac{\omega\rho}{2}\psi_{0}^{2}\mathbf k . \tag{50} \]

In the case of a bounded beam, in order to determine the energy flux it is necessary to use the general formula (49).

Let us consider the density of the mean energy flux at the boundary in the direction perpendicular to the boundary. From (49) we have:

\[ \overline{I}_z=\frac{i\omega\rho}{2}\left(\psi\,\frac{\partial\psi^*}{\partial z}-\psi^*\,\frac{\partial\psi}{\partial z}\right). \tag{51} \]

According to (14)—(16), the total field of the incident and reflected beams at the boundary can be written in the form

\[ \psi(x)=\psi_{\mathrm{inc}}(x)+\psi_{\mathrm{refl}}(x) =\int_{-\infty}^{+\infty} A(p)[1+V(p)]e^{ipx}\,dp, \tag{52} \]

where

\[ A(p)=\Phi(p)e^{i\sqrt{k^2-p^2}\,l}, \tag{53} \]

and \(V(p)\) is the reflection coefficient of a plane wave whose inclination with respect to the boundary is characterized by the parameter \(p\). The value of \(\dfrac{\partial\psi}{\partial z}\) is obtained by multiplying the integrand for the incident beam (see (9)) by \(i\mu\), and the integrand for the reflected beam by \(-i\mu\), where \(\mu=\sqrt{k^2-p^2}\). As a result, analogously to (52), we shall have:

\[ \frac{\partial\psi}{\partial z} =i\int_{-\infty}^{+\infty}\mu A(p)[1-V(p)]e^{ipx}\,dp. \tag{54} \]

Substituting now (52) and (54) into formula (51), we obtain:

\[ \begin{aligned} \overline{I}_z =&-\frac{i\omega\rho}{4}\Bigg\{ \int\!\!\int_{-\infty}^{+\infty} \mu\big[A(p')A^*(p)e^{ix(p'-p)} + A^*(p')A(p)e^{-ix(p'-p)}\big]\,dp'\,dp \\ &-\int\!\!\int_{-\infty}^{+\infty} \mu\big[A(p')A^*(p)V(p')V^*(p)e^{ix(p'-p)} \\ &\qquad\qquad\qquad + A^*(p')A(p)V^*(p')V(p)e^{-ix(p'-p)}\big]\,dp'\,dp \\ &+\int\!\!\int_{-\infty}^{+\infty} \mu\big[A(p')A^*(p)\big(V(p')-V^*(p)\big)e^{ix(p'-p)} \\ &\qquad\qquad\qquad + A^*(p')A(p)\big(V^*(p')-V(p)\big)e^{-ix(p'-p)}\big]\,dp'\,dp \Bigg\}. \tag{55} \end{aligned} \]

The expression for \(\overline{I_z}\) is thereby split into three terms \(I_{\mathrm I}+I_{\mathrm{II}}+I_{\mathrm{III}}\). Of these, \(I_{\mathrm I}\) contains only quantities pertaining to the incident beam and thus corresponds to the energy flux only in the incident beam. Similarly, \(I_{\mathrm{II}}\) gives the energy flux in the reflected beam and therefore has the sign opposite to that of \(I_{\mathrm I}\). Finally, in \(I_{\mathrm{III}}\) the quantities for the incident and reflected waves are mixed. This term characterizes the nonadditivity of the energy fluxes in the incident and reflected waves.

In the case of total reflection considered by us, \(\rho(p)\) in expression (14) for the reflection coefficient is equal to unity; the function \(\varphi(p)\) we shall again represent in the form of the expansion (20) in powers of \(\Omega\), retaining the first power of \(\Omega\). Recall that the legitimacy of this expansion is due to the smallness of the angular aperture of the beam under investigation. As a result, after simple transformations we obtain

\[ \overline{I}_{\mathrm I}(x)=\frac{\omega p}{2}\,k\cos\vartheta_0 \int_{-\infty}^{+\infty}\!\!\int_{-\infty}^{+\infty} A(p')A^*(p)e^{ix(p'-p)}\,dp'\,dp; \]

\[ \overline{I}_{\mathrm{II}}(x)=-\frac{\omega p}{2}\,k\cos\vartheta_0 \int_{-\infty}^{+\infty}\!\!\int_{-\infty}^{+\infty} A(p')A^*(p)e^{i(x-\Delta)(p'-p)}\,dp'\,dp = \]

\[ =-I_{\mathrm I}(x-\Delta), \]

\[ I_{\mathrm{III}}(x)=\frac{i\omega\rho}{2}\sin\varphi_0 \int_{-\infty}^{+\infty}\!\!\int_{-\infty}^{+\infty} (\mu'-\mu)A(p')A^*(p) e^{i\left(x-\frac{\Delta}{2}\right)(p'-p)}\,dp'\,dp, \]

where \(\varphi_0\) denotes the value of the phase of the reflection coefficient \(\varphi(p)\) at \(p=a=k\sin\vartheta_0\), i.e., at the angle of inclination corresponding to the inclination of the axial line of the beam; \(\Delta\) has the meaning indicated above in (25).

In transforming the expressions for \(I_{\mathrm I}\) and \(I_{\mathrm{II}}\), the function \(\mu\), as slowly varying, has been taken outside the integral sign at the value \(\mu=k\cos\vartheta_0\). This, however, cannot be done in \(I_{\mathrm{III}}\), since then \(I_{\mathrm{III}}\) would vanish identically. Therefore the difference
\[ \mu'-\mu=\sqrt{k^2-p'^2}-\sqrt{k^2-p^2} \]
will be expanded in a series in the neighborhood of the point \(p'=p=a\). In this case we shall have:

\[ \mu'-\mu= \left(\frac{\partial\sqrt{k^2-p'^2}}{\partial p'}\right)_{p'=a}(p'-a) - \]

\[ -\left(\frac{\partial\sqrt{k^2-p^2}}{\partial p}\right)_{p=a}(p-a) = -\operatorname{tg}\vartheta_0\,(p'-p). \]

After this we obtain:

\[ I_{\mathrm{III}}=-\frac{\sin\varphi_{0}\operatorname{tg}\vartheta_{0}}{k\cos\vartheta_{0}}\, \frac{\partial}{\partial x} I_{\mathrm{I}}\left(x-\frac{\Delta}{2}\right). \tag{56} \]

We see that \(I_{\mathrm{II}}\) represents the same energy flux as the flux \(I_{\mathrm{I}}\) in the incident beam, but only with the opposite sign and with its displacement along the boundary by the amount \(\Delta\), which also means a displacement of the reflected beam with respect to the incident one.

For clarity, Fig. 7 schematically shows the dependence of the constituent parts of the energy flux \(I_{\mathrm{I}}, I_{\mathrm{II}}, I_{\mathrm{III}}\) on the coordinate \(x\). In the inner part of the beam cross-section the resulting energy flux

\[ I_z=I_{\mathrm{I}}+I_{\mathrm{II}}+I_{\mathrm{III}} \]

is equal to zero owing to the mutual compensation of the fluxes of the incident and reflected beams. In the edge regions of the beam, however, energy flows on one side into the reflecting medium, and on the other returns back (Fig. 8).

Fig. 7. Distribution of the constituent parts of the component of the energy flux \(I_z\) normal to the boundary.

The total amount of energy participating in this “exchange,” i.e. the amount of energy flowing into the reflecting medium and flowing out of it per unit time, is obtained by integrating \(\overline{I_z}\) from \(x=-\infty\) to \(x=x_m\), where \(x_m\) corresponds to the middle of the beam. In this way we obtain:

\[ \int_{-\infty}^{x_m}\overline{I_z}\,dx = \int_{+\infty}^{+\infty}(I_{\mathrm{I}}+I_{\mathrm{II}}+I_{\mathrm{III}})\,dx = I_{\mathrm{I}}(x_m)\left(\Delta-\frac{\sin\varphi_{0}\operatorname{tg}\vartheta_{0}}{k\cos\vartheta_{0}}\right). \tag{57} \]

Fig. 8. Total internal reflection of a beam. The arrows indicate the direction of the energy flux in the edge regions of the beam and in the refracted wave.

Here, since the width of the beam is assumed to be sufficiently large in comparison with the wavelength, \(I_{\mathrm{I}}(x_m)\) is practically equal to the projection onto the \(z\)-axis of the energy-flux density in a plane wave with the same amplitude as the amplitude at the center of the beam.

It is interesting to note that, as the angle of incidence approaches the limiting angle of total internal reflection \((\sin\vartheta_{0}\to n)\), the magnitude of the total flux rapidly increases, since in this case, according to formula (34), the beam displacement \(\Delta\) increases. At the same time, as can be shown, the depth of penetration of the wave into the medium from which reflection is being investigated also increases.

7. Reflection from an Inhomogeneous Medium

Of very great interest is the investigation of the reflection of a bounded beam from a medium with continuously varying properties. Here the application of formula (25) for the displacement leads to interesting results that supplement the usual ray representations of wave refraction[^13].

Let us consider a concrete example, when the refractive index of the medium is the following function of the coordinate \(z\):

\[ \begin{array}{ll} n=1 & \text{for } z \le 0,\\[4pt] n=\sqrt{1-az} & \text{for } z>0. \end{array} \tag{58} \]

It will be seen below that the results obtained in this case have a general significance.

Suppose that from a homogeneous medium \((z<0)\) a bounded beam is incident on the boundary \(z=0\). To find the form and position of the reflected beam, it is necessary, as we have seen, first of all to find the reflection coefficient of a plane wave from this boundary \(V(\vartheta)\) as a function of the angle of incidence.

For the field of the incident and reflected plane waves in the homogeneous medium we have:

\[ \psi=\left[e^{ik_{0}z\cos\vartheta}+V(\vartheta)e^{-ik_{0}z\cos\vartheta}\right]e^{ik_{0}x\sin\vartheta}, \tag{59} \]

where the first term represents the incident wave, and the second the reflected wave; \(k_{0}\) is the wave number in the homogeneous medium.

The field in the inhomogeneous medium is a solution of the wave equation:

\[ \frac{\partial^{2}\psi_{1}}{\partial x^{2}}+ \frac{\partial^{2}\psi_{1}}{\partial z^{2}}+ (k_{0}n)^{2}\psi_{1}=0 \tag{60} \]

under the boundary conditions

\[ \frac{\partial\psi}{\partial z}=\frac{\partial\psi_{1}}{\partial z},\quad \psi=\psi_{1}\quad \text{for } z=0 \tag{61} \]

and under the condition

\[ \psi_{1}\to 0,\quad z\to\infty . \tag{62} \]

The boundary conditions (61) mean, in acoustics, the continuity of the pressure and of the normal component of the particle velocity, and in the electromagnetic case the continuity of the tangential components of the electromagnetic field.

We seek the solution of equation (60) in the form

\[ \psi_{1}=\eta(z)e^{ik(z)x\sin\vartheta(z)},\quad k(z)=k_{0}n(z). \tag{63} \]

The function \(\vartheta(z)\) is related to the angle of incidence of the wave \(\vartheta\) by the relation \(n(z)\sin \vartheta(z)=\sin \vartheta\), so that \(\vartheta(0)=\vartheta\). For \(\eta(z)\) we then obtain the equation

\[ \frac{d^2\eta}{dz^2}+k_0^2\left[n^2(z)-\sin^2\vartheta\right]\eta=0. \tag{64} \]

For the form of the function \(n(z)\) chosen by us (see (58)), introducing the new variable \(\zeta=\cos^2\vartheta-az\), the last equation can be written in the form

\[ \frac{d^2\eta}{d\zeta^2}+\left(\frac{k_0}{a}\right)^2\zeta\eta=0. \]

This equation has the solution:

\[ \eta=Aw^{1/3}\left[H_{1/3}^{(2)}(w)+e^{i\frac{\pi}{3}}H_{1/3}^{(1)}(w)\right],\qquad \zeta>0, \tag{65} \]

\[ \eta=-iAe^{i\frac{\pi}{3}}w_1^{1/3}H_{1/3}^{(1)}(iw_1),\qquad \zeta<0, \tag{65'} \]

where \(H\) is the Hankel function, \(A\) is a constant quantity,

\[ w=2\frac{k_0}{3a}\zeta^{3/2}, \]

\[ w_1=\frac{2}{3}\frac{k_0}{a}(-\zeta)^{3/2}. \]

The expressions (65) and (65′) are connected with each other by the circuit relation\({}^{13}\) and satisfy the conditions of continuity of \(\eta\) and \(d\eta/dz\) at \(\zeta=0\). The solution (65) also satisfies condition (62).

From (59), (61), (63), and (65) we find the reflection coefficient:

\[ V= \frac{ J_{2/3}(w_0)-J_{-2/3}(w_0)-i\left[J_{1/3}(w_0)+J_{-1/3}(w_0)\right] }{ J_{2/3}(w_0)-J_{-2/3}(w_0)+i\left[J_{1/3}(w_0)+J_{-1/3}(w_0)\right] }; \tag{66} \]

\[ w_0\equiv w_{z=0}=\frac{k_0}{3a}\cos^3\vartheta. \]

Here it is expedient to express the Bessel functions in terms of Airy functions according to the relations\({}^{14}\):

\[ \left. \begin{aligned} v(-t)&=\frac{1}{3}\sqrt{\pi t}\left[J_{-1/3}(w_0)+J_{1/3}(w_0)\right];\\ v'(-t)&=-\frac{1}{3}\sqrt{\pi}\,t\left[J_{-2/3}(w_0)-J_{2/3}(w_0)\right];\\ t&=\left(\frac{3}{2}w_0\right)^{2/3}=\left(\frac{k_0}{a}\right)^{2/3}\cos^2\vartheta. \end{aligned} \right\} \tag{67} \]

As a result, if we write

\[ V(\vartheta)=e^{i\varphi(\vartheta)}, \tag{68} \]

then we obtain:

\[ \varphi(\vartheta)=-2\operatorname{arctg}\left[\sqrt{t}\,\frac{v(-t)}{v'(-t)}\right]-\pi. \tag{69} \]

Before proceeding to the analysis of this expression in the general case, it is useful to investigate the case \(|t| \gg 1\). The latter condition, as is clear from (67), is satisfied if:

a) \(\dfrac{k_0}{a} \gg 1\), i.e., the change of the refractive index over a wavelength is small;

b) \(\cos \vartheta\) is not too small (the incidence of the wave is not too grazing).

For \(|t| \gg 1\) we have the asymptotic expressions\({}^{14}\)

\[ v(-t)=(t)^{-\frac14}\sin\left(w_0+\frac{\pi}{4}\right); \]

\[ v'(-t)=-t^{\frac14}\cos\left(w_0+\frac{\pi}{4}\right). \tag{70} \]

Substituting them into (69), we obtain:

\[ \varphi=\frac{4k_0}{3a}\cos^3\vartheta-\frac{\pi}{4}. \tag{71} \]

It is interesting to note that this expression for the phase, apart from the term \(\dfrac{\pi}{2}\) (which, under the assumptions we have made, is small in comparison with the first term), can be obtained with the aid of the concepts of geometrical optics. Indeed, according to these concepts, the wave penetrates into the inhomogeneous medium to a distance \(z_m\), determined from the condition

\[ n(z_m)=\sin\vartheta . \tag{72} \]

Fig. 9. Ray picture of total reflection from an inhomogeneous medium.
(Labels in the figure: homogeneous medium; inhomogeneous medium; \(x\); \(z\); \(\Delta\); \(z_m\); \(\vartheta\); \(\vartheta(z)\).)

In the plane \(z=z_m\) the ray incident on the boundary at an angle \(\vartheta\) and having at each point a direction determined by the angle \(\vartheta(z)\), satisfying the condition

\[ n(z)\sin\vartheta(z)=\sin\vartheta, \tag{73} \]

assumes a horizontal direction (Fig. 9).

The phase advance during the passage of the wave from the boundary \(z=0\) to the plane \(z=z_m\) and back will evidently be:

\[ \varphi=2\int_0^{z_m} k_z\,dz =2k_0\int_0^{z_m} n(z)\cos\vartheta(z)\,dz . \]

This will be the phase of the reflection coefficient.

Taking relation (73) into account, we obtain:

\[ \varphi=2k_0\int_0^{z_m}\sqrt{n^2(z)-\sin^2\vartheta}\,dz . \tag{74} \]

For the particular case under consideration, when the function \(n(z)\) for \(z>0\) is specified by the second of relations (58), we find:

\[ \varphi=2k_0\int_0^z \sqrt{\cos^2\vartheta-az}\,dz \tag{75} \]

or, after the substitution \(\zeta=\cos^2\vartheta-az\),

\[ \varphi=\frac{2k_0}{a}\int_0^{\cos^2\vartheta}\zeta^{1/2}\,d\zeta =\frac{4k_0}{3a}\cos^3\vartheta, \tag{76} \]

which indeed agrees with (71) up to the term \(\pi/2\).

It is also of interest to calculate the displacement of the beam upon its reflection in the approximation of geometrical optics. Expressing \(\vartheta\) in terms of \(p\) according to the relation

\[ p=k_0\sin\vartheta, \tag{77} \]

we obtain:

\[ \varphi=\frac{4k_0}{3a}\left(1-\frac{p^2}{k_0^2}\right)^{3/2}. \tag{78} \]

After this, from (25), also taking into account that \(\alpha=k_0\sin\vartheta_0\), where \(\vartheta_0\) is the angle of incidence of the beam, we find:

\[ \Delta=-\frac{2\sin 2\vartheta_0}{a}. \tag{79} \]

We obtain the same expression for the displacement by calculating the path of the ray in Fig. 9 and taking into account relations (73) and (58). We leave it to the reader to verify this.

Let us now proceed to consider the general case. According to (67) and (77) we have:

\[ t=\left(\frac{k_0}{a}\right)^{2/3}\left(1-\frac{p^2}{k_0^2}\right). \tag{80} \]

Taking into account, moreover, that \(\alpha=k_0\sin\vartheta_0\), we obtain:

\[ \left(\frac{d\varphi}{dp}\right)_{p=\alpha} = \frac{4k_0}{a}\cos^2\vartheta_0\sin\vartheta_0 - 2\left(\frac{k_0}{a}\right)^{1/3}\sin\vartheta_0 \frac{vv'}{v'^2+tv^2}, \tag{81} \]

where

\[ v \equiv v(-t), \qquad v' \equiv \frac{\partial}{\partial t}v(-t). \]

In the differentiation, the differential equation for the Airy function has also been taken into account:

\[ v''(t)=t v(t). \tag{82} \]

The displacement \(\Delta\) of the beam upon reflection is determined by relation (25). The expression thereby obtained for the displacement is of interest only for the case of angles of incidence \(\vartheta_0\) close to \(\frac{\pi}{2}\), since otherwise it is expedient to use the simpler formula (79), obtained in the geometrical approximation. Introducing, instead of the angle of incidence \(\vartheta_0\), the glancing angle \(\alpha_0=\frac{\pi}{2}-\vartheta_0\), and assuming that \(\alpha_0 \ll 1\), we obtain

\[ \left(\frac{a^2}{4\tilde{\lambda}_0}\right)^{1/3}\Delta = 2^{4/3}\sqrt{-t}\left[ 1+\frac{1}{2t}\frac{v(t)v'(t)}{v'^2(t)-t v^2(t)} \right], \tag{83} \]

where now

\[ t=-\frac{\alpha_0}{(a\tilde{\lambda}_0)^{2/3}}, \qquad \tilde{\lambda}=\frac{1}{k_0}. \]

In Fig. 10 the quantity \(\left(\frac{a^2}{4\tilde{\lambda}_0}\right)^{1/3}\Delta\) is plotted graphically as a function of \(t\) (solid line). This graph may be used for practical calculations.

Fig. 10. Displacement of the beam upon reflection as a function of the parameter \(t\).

It is of interest to compare the obtained value of the displacement \(\Delta\) with the approximate value (79), obtained with the aid of ray representa-

tions. In this case it is expedient to write expression (79) in the form

\[ \left(\frac{a^2}{4\lambda}\right)^{1/3}\Delta = 2^{1/3}(-t)^{1/2}. \tag{84} \]

The graph of this function in Fig. 10 is shown by the dashed line.

From a comparison of the solid and dashed curves we see that ray theory gives an approximately correct result under the condition

\[ |t| \gg 1 \quad \text{or} \quad \alpha_0 \gg (a\lambda_0)^{1/3}. \tag{85} \]

If this condition is not satisfied, substantial discrepancies arise between the exact and the ray theories.

Let us note that expression (83) for the displacement in the essential region for \(|t| < 1\) is valid not only for the case when the refractive index \(n(z)\) is given by the second of formulas (58), but also in all those cases in which the expansion of the function \(n(z)\) in powers of \(z\) has a linear term, i.e., when for sufficiently small \(z\) one can write

\[ n(z)=1-\frac{a}{2}z. \tag{86} \]

Such broad applicability of the result obtained is due to the fact that, at small grazing angles, the process of wave reflection, for any law of variation of \(n(z)\), takes place in the layer where \(az \ll 1\), and consequently the expansion (86) is valid. We shall show that at large grazing angles, when the wave upon reflection enters deeper layers of the inhomogeneous medium, ray representations may be used. In this case \(\Delta\) is computed by an elementary method for any law \(n(z)\), and the need for the theory developed above disappears.

As the smallest grazing angle lying on the boundary of the region of applicability of ray theory, one may, according to (85), take the angle

\[ \alpha_{\mathrm{gr}} \simeq (a\lambda_0)^{1/3} \ll 1. \tag{87} \]

A ray incident on the boundary of homogeneous and inhomogeneous media at this grazing angle penetrates into the inhomogeneous medium to a depth \(z_m\), determined from the relation (see (72))

\[ n(z_m)=\cos \alpha_{\mathrm{gr}} \simeq 1-\frac{1}{2}\alpha_{\mathrm{gr}}^{2}. \]

Using the expansion (86) of the function \(n(z)\) in powers of \(z\), we obtain from the last relation:

\[ \frac{1}{2}az_m = \frac{1}{2}\alpha_{\mathrm{gr}}^{2} \ll 1. \]

At smaller grazing angles, where ray theory is no longer applicable, the depth of penetration of the wave into the inhomogeneous medium will be

in any case, not large*), so that the condition \(az \ll 1\) may indeed be regarded as fulfilled, which is what had to be proved.

The problem of the complete reflection of waves from the boundary of homogeneous and inhomogeneous media was also treated in the work\(^ {15}\) (see also\(^5\), § 19). The author of that work calculated the path of a ray near the turning point in an inhomogeneous medium. In doing so he found that the ray forms a cusp at the very turning point. It seems to us that his result has no physical meaning, since it is impossible to trace the path of a ray where ray theory is not applicable.

II. REFLECTION OF PULSES

8. General relations. Conservation law for integral momentum

Let us consider the reflection of a pulse with a plane front from a plane boundary. To do this we shall use the expansion of the pulse into plane monochromatic waves incident on the boundary at the same angle of incidence as the pulse.

For definiteness, let us again assume that the plane of incidence of the pulse coincides with the plane \(xz\) (Fig. 11). Then the general expression for an arbitrary pulse with a plane front, incident on the plane \(z=0\) at an angle \(\vartheta\), may be written in the form:

\[ F(x,z,t)=F\left(\frac{x\sin\vartheta+z\cos\vartheta}{c}-t\right). \tag{88} \]

Here by \(F\) we shall mean the sound pressure, if an acoustic case is being considered. In the electromagnetic case, by \(F(\xi)\) one must understand the magnitude of the electric field \(E_y\), if \(\mathbf E\) is perpendicular to the plane of incidence, and the magnitude of the magnetic field \(H_y\), if \(\mathbf H\) is perpendicular to this plane. Introduce the notation

\[ \xi=\frac{x\sin\vartheta+z\cos\vartheta}{c}-t. \tag{89} \]

Fig. 11. Incidence of a pulse with a plane front on a plane boundary.

Then the function \(F(\xi)\) can be represented in the form of a Fourier integral

\[ F(\xi)=\frac{1}{2}\int_{0}^{\infty}\Phi(\omega)e^{i\omega\xi}d\omega+\text{c.c.}, \tag{90} \]

*) This, generally speaking, requires proof carried out on the basis of wave theory. The reader may do this for \(n(z)\) given by formula (58), using expressions (65) and (65′) for the field in an inhomogeneous medium.

where

\[ \Phi(\omega)=\frac{1}{\pi}\int_{-\infty}^{+\infty} F(\xi)e^{-i\omega \xi}\,d\xi . \tag{91} \]

By the symbol C. C., for brevity, we denote the term complex-conjugate to the term written before it.

The exponential \(e^{i\omega \xi}\) under the integral in (90) is, as is easy to see from (89), a harmonic plane wave of frequency \(\omega\). Consequently, expression (90) represents the expansion of the pulse in plane harmonic waves.

Denoting, as before, the reflection coefficient of a plane harmonic wave by \(V\), we shall write the reflected pulse in the form

\[ \left. \begin{aligned} F_{\mathrm{refl}}(\xi^-) &=\frac{1}{2}\int_{0}^{\infty}V\Phi(\omega)e^{i\omega \xi^-}\,d\omega+\text{C. C.},\\[4pt] \xi^-&=\frac{x\sin\vartheta-z\cos\vartheta}{c}-t. \end{aligned} \right\} \tag{92} \]

In the general case \(V\) is a complex quantity depending on the frequency \(\omega\) (for example, in the case of reflection of a wave from a layer). Therefore the reflected pulse may have a form quite different from the form of the incident pulse. The form of the pulse is unchanged only in the case when \(V\) is a real quantity independent of \(\omega\). Indeed, in this case \(V^{*}=V\), and \(V\), as a constant quantity, is taken outside the integral sign, and from (92) we obtain an expression analogous to (90), multiplied only by the reflection coefficient \(V\). However, if we have total internal reflection from the boundary between two homogeneous media, where \(V\), though it does not depend on frequency, is a complex quantity, the function \(F_{\mathrm{refl}}(\xi^-)\) will be essentially different from the function \(F(\xi)\), i.e. the form of the pulse will change. We shall examine this case in greater detail below.

By analogy with (92), the transmitted pulse is written as

\[ F_{\mathrm{tr}}(\xi_1) = \frac{1}{2}\int_{0}^{\infty}D\Phi(\omega)e^{i\omega \xi_1}\,d\omega+\text{C. C.}, \tag{93} \]

\[ \xi_1=\frac{x\sin\vartheta_1+z\cos\vartheta_1}{c_1}-t, \]

where \(c_1\) is the propagation velocity in the medium into which the wave penetrates, \(\vartheta_1\) is the angle of refraction, related to the angle of incidence by the relation \(\sin\vartheta_1=\dfrac{c_1}{c}\sin\vartheta\), and \(D\) is the transmission coefficient.

Let us dwell on the proof of the following very interesting theorem: the total (integral) impulse at any point of the upper medium is equal to the total impulse at any point of the lower medium. Here by the upper and lower media we mean, respectively, the medium from which the wave is incident, and the medium from which reflection is being investigated.

This theorem may be called the law of conservation of total impulse. Mathematically this theorem corresponds to the identity

\[ \int_{-\infty}^{+\infty} (F + F_{\text{ref}})\,dt = \int_{-\infty}^{+\infty} F_{\text{tr}}\,dt, \tag{94} \]

which remains valid independently of the form of the incident impulse and of the points in space for which the integrals on the right- and left-hand sides of the equality are taken*). Thus, for example, under total internal reflection of an impulse, the maximum value of \(F_{\text{tr}}\) in the impulse, as we shall see below, will decrease with increasing distance into the medium from which reflection takes place. However, in this case the impulse must be stretched out in time in such a way that its area, given by the integral on the right-hand side of (94), remains constant for arbitrarily large distances from the boundary.

To verify the validity of (94), let us first consider the expression for the integral quantity of the incident impulse. Taking into account that, according to (91), we have:

\[ \Phi^{*}(\omega)=\Phi(-\omega), \tag{95} \]

the expression (90) for the incident impulse may be written in the form

\[ F(\xi)=\frac{1}{2}\int_{0}^{\infty}\Phi(\omega)e^{i\omega \xi}\,d\omega + \frac{1}{2}\int_{0}^{\infty}\Phi(-\omega)e^{-i\omega \xi}\,d\omega \]

or, replacing \(-\omega\) by \(\omega\) in the second integral, we obtain:

\[ F(\xi)=\frac{1}{2}\int_{-\infty}^{+\infty}\Phi(\omega)e^{i\omega \xi}\,d\omega. \tag{96} \]

The integral with respect to \(t\) over infinite limits is equivalent to the integral with respect to \(\xi\). Therefore,

\[ \int_{-\infty}^{+\infty} F(\xi)\,dt = - \int_{-\infty}^{+\infty} F(\xi)\,d\xi = -\frac{1}{2} \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \Phi(\omega)e^{i\omega \xi}\,d\xi\,d\omega . \tag{97} \]

*) As far as we know, up to the present time the literature has contained an indication of the validity of this law only for the special case of an impulse reflected from a plane boundary of two homogeneous media\(^{16}\).

But, as is known, the relation\(^ {17}\)

\[ \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{i\omega \xi}\,d\xi=\delta(\omega), \tag{98} \]

holds, where \(\delta(\omega)\) is Dirac’s function. Taking into account the principal property of this function:

\[ \int_{-\infty}^{+\infty} f(\omega)\delta(\omega)=f(0), \tag{99} \]

where \(f(\omega)\) is an arbitrary continuous function, we find:

\[ \int_{-\infty}^{+\infty} F(\xi)\,d\xi = -\pi\int_{-\infty}^{+\infty}\Phi(\omega)\delta(\omega)\,d\omega = -\pi\Phi(0). \tag{100} \]

This result could have been anticipated in advance, since it is known that the area under a curve is given by the constant component (corresponding to \(\omega=0\)) of the expansion of this curve in a Fourier series or integral.

Each of the incident plane waves gives, according to (92), a reflected wave

\[ \frac{1}{2}V(\omega)\Phi(\omega)e^{i\omega\xi^{-}}+\text{c.c.}, \]

which, if we denote

\[ V(\omega)\Phi(\omega)=A(\omega)+iB(\omega), \tag{101} \]

is written as

\[ \frac{1}{2}A(\omega)\left(e^{i\omega\xi^{-}}+e^{-i\omega\xi^{-}}\right) + B(\omega)\sin\omega\xi^{-}. \]

Upon integration with respect to \(\xi^{-}\), the last term becomes zero by virtue of its oddness with respect to \(\xi^{-}\), while the first, as in the case of the incident wave, gives

\[ -\pi A(0),\quad \text{i.e. } -\pi \operatorname{Re}[V(0)\Phi(0)]. \]

An analogous expression, with \(V\) replaced by \(D\), is also obtained for the integral of the refracted pulse at an arbitrary point of space.

Thus, in order to prove (94), it is sufficient to verify the validity of the equality

\[ \Phi(0)+\operatorname{Re}[V(0)\Phi(0)]=\operatorname{Re}[D(0)\Phi(0)]. \]

However, according to (95) we have \(\Phi^*(0)=\Phi(0)\), i.e. \(\Phi(0)\) is real. Therefore the last equality reduces to the relation

\[ 1+\operatorname{Re} V(0)=\operatorname{Re} D(0). \tag{102} \]

It is not difficult to see that a more general equality holds,

\[ 1+V(0)=D(0), \tag{103} \]

by taking the real part of which we obtain (102).

In the case of reflection from the interface between two homogeneous media, the coefficients \(V\) and \(D\) do not depend on the frequency, and the equality \(1+V=D\) follows simply from the continuity conditions at the boundary for the sound pressure or the tangential components of the electromagnetic field. It can be shown that in more complicated cases, when reflection occurs from a layer or from an aggregate of layers, for the frequency \(\omega\to 0\) (i.e. for an infinitely large wavelength) this entire aggregate will constitute a concentrated system which will in no way affect the reflection process, and the latter will occur as if the media separated by this aggregate were in direct contact, i.e. equality (103*) will be satisfied.

Another proof of law (94) is also of interest; we shall give it for the case of an acoustic impulse.**

First of all, it is clear without proof that at two points \(A\) and \(A'\) (Fig. 12), lying in a plane parallel to the reflecting boundary (in the present case in the plane \(z=\mathrm{const}\)), the integral impulse will be the same, since these two points are situated in exactly the same way with respect to the reflecting boundary. Therefore it is sufficient to compare the integral impulses only for points \(A\) and \(B\), lying on one and the same normal to the boundary, irrespective of whether they are on the same side of the reflecting boundary or on different sides. Consider two such points \(A\) and \(B\). Imagine an elementary cylinder with an infinitely small base area, on which the points \(A\) and \(B\) are located. This cylinder was at rest before the passage of the wave and will remain at rest after the passage of the wave. This means that the total impulse received by it is equal to zero, i.e. the integral of the pressure acting on the surface of the cylinder over its entire surface

Fig. 12. Toward the proof of the conservation law for the integral impulse.

* This “dropping out” of the influence of the layer at \(\omega=0\) follows from the general formulas for the reflection coefficient from a layer of arbitrary form [18].

** The possibility of this proof was pointed out to the author by M. A. Isaakovich.

and over time from \(-\infty\) to \(+\infty\), is equal to zero. Since the action of the forces on the lateral surface (the arrows in Fig. 12) is mutually compensated, the integral impulses acting on the bases of the cylinder must be equal, i.e., the expressions:

\[ dS \int_{-\infty}^{+\infty} p_A(t)\,dt \quad \text{and} \quad dS \int_{-\infty}^{+\infty} p_B(t)\,dt, \]

where \(dS\) is the area of the bases of the cylinder, and \(p_A(t)\) and \(p_B(t)\) are the sound pressures at points \(A\) and \(B\) as functions of time. Thus, the equality holds

\[ \int_{-\infty}^{+\infty} p_A(t)\,dt = \int_{-\infty}^{+\infty} p_B(t)\,dt, \]

which is what we had to prove.

Among the general laws that are fulfilled in the reflection of a pulse of arbitrary form there also belongs, of course, the law of conservation of energy, which is written in the form:

\[ S_z + S_z^{\mathrm{refl}} = S_z^{\mathrm{pr}}, \tag{104} \]

where \(S_z\), \(S_z^{\mathrm{refl}}\), and \(S_z^{\mathrm{pr}}\) denote the components of the energy flux along the \(z\)-axis respectively in the incident, reflected, and refracted waves.

Thus, in the case of a sound pulse we have:

\[ S_z = \int_{-\infty}^{+\infty} p v_z\,dt, \tag{105} \]

where \(p\) and \(v_z\) are the sound pressure and the component of velocity along the \(z\)-axis in the incident wave. \(S_z^{\mathrm{refl}}\) and \(S_z^{\mathrm{pr}}\) are written analogously.

It is not difficult to verify the validity of law (104) for a pulse, since it is fulfilled for each of the harmonic waves into which the pulse can be decomposed. In total internal reflection \(S_z^{\mathrm{pr}}=0\), since the refracted pulse propagates along the boundary of separation. For the proof of (104) in this case, see \({}^{19}\).

9. Distortion of the Pulse Shape under Total Internal Reflection

Let us consider this question for several special types of pulse. Let us first take a pulse whose shape is specified by the function

\[ F(\xi)=\frac{1}{\pi}\frac{\varepsilon}{\varepsilon^2+\xi^2}, \tag{106} \]

where \(\xi\) is given by expression (89), and \(\varepsilon\) is a parameter characterizing the width of the pulse. The form of this pulse is shown in Fig. 13. Its expansion in a Fourier integral, as is easy to verify, has the form:

\[ F(\xi)=\frac{1}{2\pi}\int_0^\infty e^{-\varepsilon\omega+i\omega\xi}\,d\omega+\text{c.c.} \tag{107} \]

or

\[ F(\xi)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{-\varepsilon|\omega|+i\omega\xi}\,d\omega . \tag{108} \]

Thus, if (108) is represented in the general form (96), then one must put

\[ \Phi(\omega)=\frac{1}{\pi}e^{-\varepsilon|\omega|}. \tag{109} \]

Let us note that as \(\varepsilon\to 0\) the pulse under consideration passes into a pulse characterized by the \(\delta\)-function, which is immediately evident from comparing (98) and (108).

Fig. 13. Pulse shape adopted for the calculation.

The incident pulse (106) can also be written in the form:

\[ F(x,z,t)= \frac{1}{\pi}\, \frac{\varepsilon} {\varepsilon^2+\left(\dfrac{x\sin\vartheta+z\cos\vartheta}{c}-t\right)^2}. \tag{110} \]

If ordinary, and not total internal, reflection takes place, then, as we have seen, the form of the reflected and refracted pulses will be the same, i.e.

\[ F_{\text{refl}}(x,z,t)= \frac{V}{\pi}\, \frac{\varepsilon} {\varepsilon^2+\left(\dfrac{x\sin\vartheta-z\cos\vartheta}{c}-t\right)^2}; \tag{111} \]

\[ F_{\text{tr}}(x,z,t)= \frac{D}{\pi}\, \frac{\varepsilon} {\varepsilon^2+\left(\dfrac{x\sin\vartheta_1+z\cos\vartheta_1}{c_1}-t\right)^2}, \tag{112} \]

where \(V\) and \(D\) are the coefficients of reflection and transparency.

In the case of total internal reflection, \(V\) and \(D\) are complex quantities:

\[ V=\frac{m\cos\vartheta-is}{m\cos\vartheta+is}, \qquad D=1+V, \qquad s=\sqrt{\sin^2\vartheta-n^2}. \tag{113} \]

where in acoustics \(m=\dfrac{\rho_1}{\rho}\) (see (32)); in electrodynamics \(m=\dfrac{\mu_1}{\mu}\) or \(\dfrac{\mu}{\mu_1}n^2\), depending on the polarization (see (26)).

Put

\[ V=A+iB, \]

where

\[ A=\frac{m^2\cos^2\vartheta-s^2}{m^2\cos^2\vartheta+s^2},\qquad B=-\frac{2Sm\cos\vartheta}{m^2\cos^2\vartheta+S^2}. \tag{114} \]

Then the reflected pulse, taking (101) and (109) into account, is written in the form

\[ F_{\mathrm{refl}}(\xi^-)=\frac{A}{2}\int_{-\infty}^{+\infty}e^{-\varepsilon|\omega|+i\omega\xi^-}\,d\omega -2B\int_{0}^{\infty}e^{-\varepsilon\omega}\sin\omega\xi^-\,d\omega. \tag{115} \]

Both integrals in the last expression are evaluated without difficulty, and as a result, taking into account the value of \(\xi^-\), we obtain:

\[ F_{\mathrm{refl}}= \frac{A}{2\pi}\, \frac{\varepsilon}{ \varepsilon^2+\left(\dfrac{x\sin\vartheta-z\cos\vartheta}{c}-t\right)^2} - \frac{B}{\pi}\, \frac{\dfrac{x\sin\vartheta-z\cos\vartheta}{c}-t}{ \varepsilon^2+\left(\dfrac{x\sin\vartheta-z\cos\vartheta}{c}-t\right)^2} \tag{116} \]

or, for \(\varepsilon\to 0\):

\[ F_{\mathrm{refl}}= \frac{A}{2}\delta\!\left(\frac{x\sin\vartheta-z\cos\vartheta}{c}-t\right) -\frac{B}{\pi}\, \frac{1}{ \dfrac{x\sin\vartheta-z\cos\vartheta}{c}-t}. \tag{117} \]

Thus, the reflected pulse consists of two parts, one of which corresponds to a pulse of the same form as the incident one, and the other to a pulse of modified form.

Let us now consider the pulse penetrating into the lower medium. In expressions (93) for it we have:

\[ \xi_1=\frac{x\sin\vartheta_1}{c_1}-t+\frac{is}{c}z. \tag{118} \]

As a result, also taking (109) into account, we obtain:

\[ F_{\mathrm{tr}}=\frac{D}{2}\int_{0}^{\infty}e^{-\varepsilon\omega-\frac{\omega s}{c}z} e^{i\omega\left(\frac{x\sin\vartheta_1}{c_1}-t\right)}\,d\omega+\text{c.c.} \]

Or, denoting for brevity

\[ \frac{x\sin\vartheta_1}{c_1}-t=g;\qquad \varepsilon+\frac{s}{c}z=h \tag{119} \]

and substituting the value of the integral

\[ \int_0^\infty e^{-h\omega-ig\omega}\,d\omega=\frac{1}{h+ig}, \tag{120} \]

we shall have:

\[ F_{\mathrm{pr}}=-\frac{1}{2}\left(\frac{D}{h+ig}+\frac{D^*}{h-ig}\right). \tag{121} \]

Taking into account the equality \(D=1+V\) and equality (114), the last expression can be transformed to the form

\[ F_{\mathrm{pr}}=\frac{(1+A)h+Bg}{h^2+g^2}, \tag{122} \]

or, expressing \(h\) and \(g\) in terms of the coordinates and time:

\[ F_{\mathrm{pr}}= \frac{(1+A)\left(\varepsilon+\frac{s}{c}z\right)+ B\left(\frac{x\sin\vartheta_1}{c_1}-t\right)} {\left(\varepsilon+\frac{s}{c}z\right)^2+ \left(\frac{x\sin\vartheta_1}{c_1}-t\right)^2}. \tag{123} \]

In the case \(\varepsilon=0\), when the incident impulse is described by a \(\delta\)-function, from (123) we have:

\[ F_{\mathrm{pr}}= \frac{ B\left(\frac{x\sin\vartheta_1}{c_1}-t\right)+A\frac{s}{c}z } {\left(\frac{s}{c}z\right)^2+ \left(\frac{x\sin\vartheta_1}{c_1}-t\right)^2}. \tag{124} \]

We see that the refracted impulse, with respect to its form, essentially has nothing in common with the incident impulse. Indeed, in formulas (123) and (124) the time \(t\) enters only in the combination

\[ \frac{x\sin\vartheta_1}{c_1}-t. \]

It follows from this that in the lower medium the impulse propagates along the boundary with the velocity

\[ \frac{c_1}{\sin\vartheta_1}, \]

which, according to the law of refraction, is equal to

\[ \frac{c}{\sin\vartheta} \]

—the velocity of propagation of the impulse in the upper medium along the interface.

It is interesting to note that on the straight line

\[ B\left(\frac{x\sin\vartheta_1}{c_1}-t\right)+A\frac{s}{c}z=0 \tag{125} \]

the field of the refracted impulse is equal to zero (and has different signs on different sides of this line). In Fig. 14 there is schematically shown the pattern of the incident, reflected, and refracted impulses at the instant of time \(t=0\). In this figure \(AA\) is the interface of the media, \(OB\) is the front of the incident impulse, given by the equation \((x\sin\vartheta+z\cos\vartheta)/c-t=0\); \(OD\) is the front of that part of the reflected

of the pulse, which corresponds to the first term in (117). Hatching with solid lines on one side of \(OD\) and with dashed lines on the other schematically represents the second term in (117). Dashed hatching corresponds to a negative field, hatching with solid lines to a positive field. The field decreases with distance from the line \(OD\), which corresponds to a decrease in the density of the hatching.

The field of the refracted pulse everywhere, except at the origin of coordinates \(O\), has a finite magnitude. On the line \(OE\) it vanishes and has different signs on the two sides of it. The arrows indicate the directions of propagation of the incident, reflected, and refracted pulses.

Let us note that the dependence of the field of the reflected pulse on the coordinate \(z\) for

\[ x=\frac{c_1 t}{\sin \vartheta_1}, \]

according to (124), will be

\[ F_{\mathrm{pr}}=\frac{Ac}{sz}. \]

Thus, with distance from the boundary the field decreases not according to an exponential law, as in the case of a plane wave, but according to a law of inverse proportionality to the magnitude of the distance.

Fig. 14. Schematic representation of the pattern of reflection and refraction of a \(\delta\)-shaped pulse.

Fig. 15. Sections of the field, schematically shown in Fig. 14, by planes \(x=\mathrm{const}\) (\(\mathrm{const}>0\)).

In Figs. 15 and 16 are shown\({}^{16}\) sections of the field, schematically represented in Fig. 14, by planes \(x=\mathrm{const}\) for the incidence of a \(\delta\)-shaped pulse on a boundary characterized by the parameters: \(n=\dfrac{c}{c_1}=0.26\); \(m=0.07\), which corresponds, for example, in acoustics to the boundary between air \((c=343\ \mathrm{m/sec},\ \rho=1.2\cdot 10^{-3}\ \mathrm{g/cm^3})\) and hydrogen \((c_1=1305\ \mathrm{m/sec},\)

\(\rho_1 = 8.4 \cdot 10^{-5}\ \text{g}/\text{cm}^3\), taken at a temperature of \(20^\circ\text{C}\) and normal pressure. The angle of total internal reflection for this case is \(15^\circ 25'\). In the case shown in Figs. 15 and 16, the angle of incidence is \(\vartheta = 60^\circ\).

Along the ordinate axis in the graphs is plotted the sound pressure or the electric-field intensity in relative units.

Fig. 16. Sections of the field schematically shown in Fig. 14 by the planes \(x=\mathrm{const}\) \((\mathrm{const}<0)\).

Since the functions \(F_{\mathrm{otr}}\) and \(F_{\mathrm{pr}}\), describing the reflected and refracted pulses, according to (117) and (124), at \(t=0\) are homogeneous functions of \(x\) and \(z\), i.e., a change of scale in \(x\) and \(z\) changes nothing except the scale along the ordinate axis, the quantities \(x\) and \(z\) can also be expressed in relative units, as has been done

on Figs. 15 and 16. The incident pulse and that part of the field of the reflected pulse which is described by the \(\delta\)-function are not plotted on the graphs. The values of the coordinate \(z\) at which the front of these pulses intersects the corresponding planes \(x=\mathrm{const}\) are marked by arrows on the abscissa axis.

In the literature one can find studies of the change in pulse shape under total internal reflection also for other forms of the incident pulse. Thus, in paper \(^{20}\) the reflection and refraction of a “column-shaped” pulse was investigated, i.e., a pulse whose field has a constant value on some time interval \((t_1,t_2)\) and is equal to zero outside this interval. In doing this, the author of that paper did not resort to expanding the pulse in a Fourier series.

In paper \(^{19}\) the reflection of an exponential pulse specified by the equation

\[ F(\xi)= \begin{cases} 0, & \text{for } \xi<0,\\ F_0 e^{-\lambda \xi}, & \text{for } \xi \geq 0, \end{cases} \tag{126} \]

is considered. This function gives a fair description of the pulse shape in the shock wave from an underwater explosion. The authors compared their theoretical results with experimental results obtained by recording an explosive pulse in a layer of water bounded above by the free surface of the water and below by the bottom (compacted sea sand, \(\rho_1 \simeq 2.7\), \(c_1 \simeq 1{,}800\ \mathrm{m/sec}\), angle of total internal reflection at the boundary with water \(58^\circ\)).

At the top of Fig. 17 is shown the theoretically calculated form of the curve of sound pressure as a function of time; at the bottom—the experimental record.

Fig. 17. Theoretical curve characterizing the change in shape of an exponential pulse under successive reflections from the surface of the water and the bottom (top), and an experimental curve obtained from an explosion in water (bottom). Experimental conditions: charge weight \(0.225\ \mathrm{kg}\), distance \(143\ \mathrm{m}\), receiver depth \(7.4\ \mathrm{m}\), explosion depth \(12\ \mathrm{m}\), water depth \(25\ \mathrm{m}\).

\(M\) denotes the number of reflections from the water surface, \(N\)—from the bottom, \(\theta\)—the angle of incidence of the pulse on the bottom, \(\varphi\)—the phase jump of a plane harmonic wave upon reflection for this angle of incidence.

We see that the incident pulse is followed by a pulse reflected from the water surface \((M=1,\ N=0)\), which has the same form as the incident one but the opposite sign. It is followed in the receiver by a pulse reflected from the bottom \((M=0,\ N=1)\), whose form is completely different from the form of the incident pulse. Then pulses are recorded that have one reflection each from the water surface and from the bottom (and that differ in the order of these reflections). The experimental record corresponds on the whole fairly well to the theoretical curve. Agreement of these curves in detail could not have been expected, in view of a certain idealization of the form of the incident pulse.

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