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,\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}
\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).