Reports of the Academy of Sciences of the USSR
1966. Volume 166, No. 3

UDC 530.10+530.16

PHYSICS

Corresponding Member of the Academy of Sciences of the USSR D. I. BLOKHINTSEV

ON THE PROPAGATION OF HIGH-FREQUENCY SIGNALS IN A MEDIUM WITH RANDOM CHARACTERISTICS

We consider an equation for the propagation of a signal \(\Psi\) of the form

\[ A_{jk}\frac{\partial^2\Psi}{\partial x_j \partial x_k} + B_k\frac{\partial\Psi}{\partial x_k} + C\Psi = 0, \tag{1} \]

where the coefficients \(A_{jk}, B_k, C\) are random functions of the variables \(x_j\) \((j=1,2,3,4)\). It is assumed that, in the range of possible values of \(A_{jk}\), equation (1) remains hyperbolic. Next put

\[ A_{jk}=\overline{A}_{jk}+a_{jk},\qquad B_k=\overline{B}_k+b_k,\qquad C=\overline{C}+c, \tag{2} \]

where the bar denotes averaging over possible values of the random quantities \(A_{jk}, B_k, C\). This averaging has the meaning of functional integration over possible values of the random quantity \(a(x)\):

\[ \overline{\Phi} = \int \Phi\{a(x)\}\,dw\{a(x)\}, \tag{3} \]

where \(\Phi\) is a functional of \(a(x)\); \(dw\{a(x)\}\) is the probability that \(a=a(x)\). We shall assume that the random quantity \(a(x)\) can be represented in the form of a series

\[ a(x)=\sum_n a_n\varphi_n(x,\alpha_n), \tag{4} \]

where \(\varphi_n(x,\alpha_n)\) is some system of orthonormal functions, \(\alpha_n\) are random phases, and \(a_n\) are random amplitudes. In view of (4), \(dw\{a(x)\}\) may be regarded as the probability of one or another set of values of the quantities \(a_n,\alpha_n\); in particular, if \(a_n,\alpha_n\) are independent, then

\[ dw\{a(x)\}=\prod_n dw(a_n)\,d\Omega(\alpha_n). \tag{5} \]

We shall seek the solution \(\Psi\) in the form

\[ \Psi=Ae^{iS}, \tag{6} \]

where the frequency \(\omega\) considerably exceeds the frequencies characteristic of the spectrum of the random quantities \(A_{jk}, B_k, C\). In this case the amplitude \(A\) and the phase function \(S\) may be regarded as slowly varying functions of the variables \(x_j\) (the geometrical-optics approximation). Setting \(S=\overline{S}+\sigma\) and substituting (6) into (1), as \(\omega\to\infty\) we obtain:

\[ \overline{A}_{jk} \frac{\partial \overline{S}}{\partial x_j} \frac{\partial \overline{S}}{\partial x_k} = 0, \tag{7} \]

\[ \overline{A}_{jk} \frac{\partial \overline{S}}{\partial x_j} \frac{\partial \sigma}{\partial x_k} + \overline{A}_{jk} \frac{\partial \sigma}{\partial x_j} \frac{\partial \overline{S}}{\partial x_k} + a_{jk} \frac{\partial \overline{S}}{\partial x_j} \frac{\partial \overline{S}}{\partial x_k} = 0. \tag{8} \]

From the last equation one finds the random phase \(\sigma\), which will be a linear functional of the random quantities \(a_{jk}(x)\). Therefore \(\sigma(x)\) has the form

\[ \sigma(x)=\sum_n a_n\sigma_n(x,\alpha_n), \tag{9} \]

where \(\sigma_n(x,a_n)\) corresponds to the solution of the system (7), (8), if in (4) all \(a_m=0\) are set, except for \(a_n\).

The mean value of the signal \(\Psi\) will be

\[ \overline{\Psi}=Ae^{i\omega\overline{S}}e^{i\overline{\omega\sigma}}. \tag{10} \]

If the distribution \(dw(a_n)\) is normal,

\[ dw(a_n)=\frac{1}{\sqrt{\pi}}\exp\left[-\frac{(a_n-\overline{a}_n)^2}{b_n^2}\right]\frac{da_n}{b_n}, \tag{11} \]

then, on the basis of (9), we obtain

\[ \overline{\Psi} = Ae^{i\omega\overline{S}} \prod_n \int_0^{2\pi} d\Omega(\alpha_n) \exp\left[ i\omega\overline{a}_n\sigma_n(x,\alpha_n) - \frac{b_n^2\omega^2}{5}\sigma_n^2(x,\alpha_n) \right]. \tag{12} \]

The final result depends on the form of \(\sigma(x,\alpha_n)\).

Let us consider several applications.

A. Scattering of sound in a turbulent flow. In the simplest case of a stationary, vortex-free flow, neglecting quantities of order \(u^2/c^2\) (\(u\) is the flow velocity, \(c\) is the speed of sound), the equation for the velocity potential of the sound wave \(\varphi\) is \({}^{(1)}\)

\[ \partial^2\varphi/\partial t^2+2u\,\partial^2\varphi/\partial x\,\partial t-c^2\partial^2\varphi/\partial x^2=0, \tag{13} \]

so that \(x_1=x,\ x_4=t\) and \(A_{44}=1,\ A_{41}=a_{41}=2u,\ A_{11}=-c^2\).

Equations (7) and (8) now have the form

\[ \left(\frac{\partial\overline{S}}{\partial t}\right)^2 - c^2\left(\frac{\partial\overline{S}}{\partial x}\right)^2 =0, \tag{7'} \]

\[ \frac{\partial\overline{S}}{\partial t}\frac{\partial\sigma}{\partial t} - 2c^2\frac{\partial\overline{S}}{\partial x}\frac{\partial\sigma}{\partial x} + 2u\frac{\partial\overline{S}}{\partial x}\frac{\partial\overline{S}}{\partial t} =0. \tag{8'} \]

From (7) for a plane wave we have \(\overline{S}=t\pm x/c\), so that

\[ \partial\sigma/\partial t \mp 2c\,\partial\sigma/\partial x + 2u/c=0. \tag{14} \]

For a stationary flow \(\partial u/\partial t=0\), and one may obtain \(\partial\sigma/\partial t=0\). Then we obtain

\[ \sigma(x)=\pm\int_0^x \frac{u(x')\,dx'}{c^2}. \tag{15} \]

Now putting

\[ u(x)=\sum_n u_n\cos(q_nx+\alpha_n), \tag{16} \]

for the normal distribution law of \(u_n\), with \(\overline{u}_n=0\), we obtain

\[ \overline{e^{i\omega\sigma}} = \prod_n \int \exp\left[ -\frac{\omega^2 b_n^2}{4q^2c^4}F_n^2(x,\alpha_n) \right] \frac{d\alpha_n}{2\pi}, \tag{17} \]

where

\[ F_n(x,\alpha_n)=\sin(q_nx+\alpha_n). \tag{18} \]

Since \(F^2(x,\alpha_n)>0\) and is bounded, this quantity in the exponent of (17) may be replaced by the effective mean \(F_n^2(x,\theta_n\alpha_n)\), \(0<\theta_n<1\). Then instead of (17) we obtain:

\[ \overline{e^{i\omega\sigma}}\exp\left[-\frac{\omega^2}{2}\Phi(x)\right], \tag{19} \]

where

\[ \Phi(x)=\sum_n \frac{b_n^2}{2q_n^2c^4}F_n^2(x,\theta_n\alpha_n). \tag{20} \]

Thus, the mean strength of the sound signal (10) drops sharply with increasing frequency \(\omega\) (according to a Gaussian curve).

B. Propagation of light in a medium with a turbulent metric.

The wave equation in this case is\(^2\)

\[ g^{\mu\nu}\partial^2\varphi/\partial x_\mu \partial x_\nu-\Gamma^\mu \partial\varphi/\partial x_\mu=0, \tag{21} \]

\[ \Gamma^\mu=-\frac{1}{\sqrt{-g}}\frac{\partial}{\partial x_\nu}\left(\sqrt{-g}\,g^{\mu\nu}\right), \tag{22} \]

where \(g\), as usual, is \(\det(g_{\mu\nu})\).

We shall regard \(g^{\mu\nu}\) as random functions of the variables \(x_1,x_2,x_3,x_4=t\). The physical reasons for such a possibility are considered in \((3)\).

Let us turn to the case when

\[ g^{\mu\nu}=g_0^{\mu\nu}+\varepsilon^{\mu\nu},\qquad \overline{\varepsilon^{\mu\nu}}=0. \tag{23} \]

From (21), (22), (7), and (8) we obtain the equation for \(\sigma\):

\[ \varepsilon^{\mu\nu}\frac{\partial \overline{S}}{\partial x_\mu}\frac{\partial \overline{S}}{\partial x_\nu} +g_0^{\mu\nu}\left( \frac{\partial \overline{S}}{\partial x_\mu}\frac{\partial \sigma}{\partial x_\nu} +\frac{\partial \overline{S}}{\partial x_\nu}\frac{\partial \sigma}{\partial x_\mu} \right)=0. \tag{24} \]

In particular, for a plane wave \(\overline{S}=t-x\) (we take the speed of light \(c\) to be equal to 1), equation (24) takes the form

\[ \frac{\partial \sigma}{\partial t}+\frac{\partial \sigma}{\partial x} =\frac{1}{2}\varepsilon(x,t),\qquad \varepsilon=\varepsilon^{44}+\varepsilon^{11}-2\varepsilon^{14}. \tag{25} \]

It has the solution

\[ \sigma(x,t)=\frac{1}{4}\int_{t_0}^{t}\varepsilon(t'-\xi,t')\,dt' +\frac{1}{4}\int_{x_0}^{t}\varepsilon(x',\xi+x')\,dx', \tag{26} \]

where \(\xi=t-x\). An analogous solution is obtained for the wave \(\overline{S}=t+x\). If \(\varepsilon(x,t)\) can be represented in the form

\[ \varepsilon(x,t)=\sum_{n,m}\varepsilon_{nm}\sin(q_nx+\alpha_n)\sin(\omega_m t+\beta_m), \tag{27} \]

where \(\varepsilon_{nm},\alpha_n,\beta_m\) are random quantities, then, for a normal distribution law of \(\varepsilon_{nm}\), \(\overline{\varepsilon_{nm}}=0\), \(\overline{\varepsilon_{nm}^{\,2}}=\frac{1}{2}b_{nm}^{\,2}\), and a uniform distribution of \(\alpha_n,\beta_m\), we obtain

\[ \sigma(x,t)=\sum_{n,m}\varepsilon_{nm}F_{nm}(x,t), \tag{28} \]

where the explicit form of \(F_{nm}(x,t)\) is not difficult to obtain from (26) and (27). Applying the same reasoning as in point A, we find that the light signal has the form (19), with \(\Phi(x)>0\) now equal to

\[ \Phi(x)=\frac{1}{2}\sum_{n,m} b_{nm}^2 F_{nm}^2(x,t,\theta_n\alpha_n,\theta_m\beta_m). \tag{29} \]

United Institute
for Nuclear Research

Received
20 X 1965

References

\(^1\) D. I. Blokhintsev, Acoustics of an Inhomogeneous and Moving Medium, 1946.
\(^2\) V. A. Fock, The Theory of Space, Time and Gravitation.
\(^3\) D. I. Blokhintsev, Nuovo Cim., 18, 193 (1960).