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Wave Propagation in a Random Medium
By Lev A. Chernov, Richard A. Silverman Dover Publications, Inc.
Copyright © 2013 Lev A. Chernov
All rights reserved.
ISBN: 978-0-486-82147-4
CHAPTER 1
STATISTICAL CHARACTERISTICS OF THE MEDIUM
In a medium with random inhomogeneities the index of refraction is a random function of the coordinates and the time. The changes in refractive index in the sea and in the atmosphere are usually caused by temperature fluctuations. In addition, salinity fluctuations in the sea and humidity fluctuations in the atmosphere can play a role. Apparently, these fluctuations are small in the majority of cases. Liebermann has made an experimental study of temperature inhomogeneities in the ocean using a fast-acting thermometer mounted on a submarine. The curve of temperature fluctuations obtained by him is shown in Fig. 1.
The choice of a submarine instead of a surface vessel was dictated by the need to avoid pitching, which introduces additional errors into the measurements. The submarine moved at depths of 30 to 60 meters. At these depths the mean temperature fluctuations amounted to 0.04 °C, with a ~60 cm as the mean size of the inhomogeneities. Such small temperature fluctuations correspond to small fractional changes in the index of refraction; the mean square fluctuation of the acoustic index of refraction is equal to [MATHEMATICAL EXPRESSION OMITTED]. Even if the temperature fluctuations amounted to a few degrees, the fractional changes in refractive index would still not exceed 0.01. Apparently, the influence of salinity fluctuations is even less significant.
1. The Correlation Function. We shall assume that the fluctuations in refractive index represent a random process in space and time, described by the random function of coordinates and time µ(x, y, z, t). Regarding this random process as stationary in time, we shall characterize it by the correlation function
N12 = µ(x1, y1, z1, t)µ(x2, y2, z2, t), (1)
where the overbar designates averaging with respect to the time t, or, because of the ergodic hypothesis, averaging over the ensemble of realizations corresponding to the different possible states of the medium. Understanding the overbar to mean time averaging, we write
[MATHEMATICAL EXPRESSION OMITTED] (2)
For a spatially homogeneous process the correlation function depends only on the coordinate differences x = x2 - x1, y = y2 - y1, z = z2 - z12, i.e.
N12 = N12(x, y, z). (3)
For x = y = z = 0 the function N12 achieves its maximum N11, equal to the mean square fluctuation of refractive index [MATHEMATICAL EXPRESSION OMITTED]. The correlation coefficient N is defined as the ratio of the correlation function N12 to the mean square fluctuation [MATHEMATICAL EXPRESSION OMITTED], i.e.
[MATHEMATICAL EXPRESSION OMITTED], (4)
so that
[MATHEMATICAL EXPRESSION OMITTED]. (5)
As the distance between the points is increased, the correlation coefficient decreases from its maximum value of unity and becomes small compared to unity at a distance called the correlation distance, i.e., the statistical dependence between the fluctuations disappears. If the properties of the medium are not spatially homogeneous, then the correlation function will depend not only on the coordinate differences but also on the coordinates themselves. However, in what follows we shall consider only the statistically homogeneous case.
In addition to the three dimensional spatial correlation function (l), we introduce the four dimensional space-time correlation function
[MATHEMATICAL EXPRESSION OMITTED] (6)
Setting t2 = t1 + t and taking the overbar to mean time averaging, we write
[MATHEMATICAL EXPRESSION OMITTED] (7)
For a stationary and spatially homogeneous random process, the four dimensional correlation function depends only on the coordinate differences and the time difference t = t2 - t1, i.e.
N12= N12 (x, y, z, t). (8)
The corresponding correlation coefficient is defined by Eq. (4). As t grows the correlation coefficient decreases and becomes small compared to unity for a time t ~ T called the correlation time. The three dimensional correlation function is the special case of the four dimensional one for t1 = t2.
In a brief note Fine considers the problem of averaging not only in time but also in space when defining the correlation coefficient for refractive index fluctuations. However, the equivalence of these two averaging operations for a stationary and spatially homogeneous process cannot be doubted.
2. Determination of the Form of the Correlation Function. It seems that only one case can be given where the correlation function can be determined theoretically. This is the case of homogeneous isotropic turbulence. The field of temperature fluctuations caused by turbulence was investigated by Obukhov, who found for the mean square temperature difference at two points a law similar to the "two-thirds law". However, even if (under familiar conditions) the state of the atmosphere can "be described in a satisfactory way by the theory of homogeneous isotropic turbulence, the question of whether this theory is applicable to the ocean has yet to be settled. In hydroacoustical investigations it is appropriate to start with correlation functions found empirically. Moreover, in atmospheric acoustics (as well as in optics and radiophysics) it is also appropriate to begin this way if the conditions for the applicability of the theory of homogeneous isotropic turbulence are violated for one reason or another.
The correlation coefficient for temperature fluctuations in the ocean was determined experimentally by Liebermann, who used a correlator which first multiplied together two temperature curves T(x1) and T(x1 + x) and then averaged them over the entire record. In Fig. 2 the points indicate the results found by Liebermann for different distances x. The dependence of the correlation coefficient on the distance is satisfactorily approximated by the function
N(x) = e-|x|/a, (9)
if we set the correlation distance a = 60 cm. For a statistically isotropic medium we can write
N(r) = e-r/a (10)
instead of {9), where r = [square root of x2 + y2 + x2].
A peculiarity of the correlation function e-|x|/a should be pointed out, namely, its derivative at x = 0 differs from zero. This is possible only in the case where the refractive index fluctuation µ(x) is a discontinuous function. Actually, we assume the opposite, i.e., that the refractive index fluctuation µ(x) is a continuous function of x. Differentiating the product µ(x1)µ(x1 + x) with respect to x, we obtain
[MATHEMATICAL EXPRESSION OMITTED]. (11)
Since
d/dx µ(x1 + x) = d/dx1 µ(x1 + x),
in the right hand side of (11) we can differentiate with respect to x1 instead of with respect to x:
d/dx [µ(x1)µ(x1 + x)] = µ(x1) d/dx1 µ(x1 + x). (12)
Averaging (12) with respect to the coordinate x1 we obtain
[MATHEMATICAL EXPRESSION OMITTED] (13)
where
[MATHEMATICAL EXPRESSION OMITTED]
We now find the value of the derivative at x = 0:
[MATHEMATICAL EXPRESSION OMITTED].
Since µ(x) is everywhere bounded, the limit of the last expression is zero, i.e.
[dN12(x)/dx]x=0 = 0. (14)
The correlation coefficient e-|x|/a does not satisfy this condition. Consequently, the refractive index fluctuation µ(x) is a discontinuous function for this case. Actually, the temperature discontinuities, are smoothed out as a result of heat conductivity, so that there is not a full correspondence between the correlation coefficient e-|x|/a and the actual conditions. To achieve such a correspondence the function e-|x|/a must be modified in such a way that the new function differs from the function e-|x|/a only in the immediate neighborhood of zero and has a vanishing derivative at x = 0. For example, the function
[MATHEMATICAL EXPRESSION OMITTED]
satisfies these conditions. The smallest value of a corresponding to Liebermam's experimental data is 0.5 cm-1.
The experimental data are also described in a satisfactory way by a function of the form
[MATHEMATICAL EXPRESSION OMITTED]
which is especially convenient for theoretical investigations, and has a vanishing derivative at x = 0. The corresponding correlation coefficient for a statistically isotropic medium has
[MATHEMATICAL EXPRESSION OMITTED] (15)
In the following we shall use the correlation coefficients (15) and (10) in studying special cases. In this regard it should not he forgotten that the correlation coefficient (10) corresponds to discontinuous changes in the refractive index fluctuations.
CHAPTER 2
RAY STATISTICS
We turn now to a consideration of ray propagation in a medium with random inhomogeneities, assuming that a, the scale of the inhomogeneities, is large compared to the wavelength λ. In hydroacoustics this condition is often satisfied for ultrasonic -waves; in the atmosphere the condition is met for light waves, since the inner dimension of the turbulent fluctuations in the atmosphere is of the order of 1 cm. It should be noted that the condition that the -wavelength be small compared with the scale of the inhomogeneities is only a necessary condition for the geometrical approximation to be suitable, but not a sufficient condition. If this condition is met, then the ray theory can be used in regions of linear dimension L, where L satisfies the condition [square root of λL] << a. This condition has a simple physical meaning: the size of the first Fresnel zone for the distance in question must be small compared to the scale of the inhomogeneities. At larger distances which do not satisfy this condition the ray approximation cannot be used, and in this case diffraction theory is necessary.
In this chapter we shall restrict ourselves to the ray model and we shall assume both of the conditions λ << a and [square root of λL] << a. The necessity of the second condition will be rigorously justified later (Section 21). Moreover, we shall assume that the transit time of the ray is small compared to the characteristic scale of changes of the inhomogeneities in time.
3. The Ray Equation. The ray equation in the form most convenient for the considerations to follow can be obtained from Fermat's principle
[MATHEMATICAL EXPRESSION OMITTED] (16)
Introducing the refractive index n = co/C, Eq. (l6) can be written in the form
[MATHEMATICAL EXPRESSION OMITTED] (17)
We shall assume that the ray trajectories belong to the family of curves expressed by the equations x = x(u), y = y(u), z = z(u) and passing through given points A and B. The parameter u is supposed to be chosen so that it takes fixed values u1 and u2 at the points A and B. For example, any of the three Cartesian coordinates satisfies this condition. The arc length σ of the curve does not satisfy this condition, since it changes in going from one curve to another. Since dσ = [square root of x'2 + y'2 + z'2] du (where the prime denotes differentiation with respect to u), the Fermat principle can be written as follows:
[MATHEMATICAL EXPRESSION OMITTED] (18)
By introducing the parameter u the problem is reduced to an ordinary variational problem. Designating
F(x, y, z, x', y', z') [equivalent to] n(x, y, z) [square root of x'2 + y'2 + z'2, (19)
we write the Euler equations of the variational problem as
[MATHEMATICAL EXPRESSION OMITTED]
Because of (19) they take the form
[MATHEMATICAL EXPRESSION OMITTED] (20)
Introducing the unit vector [??] tangent to the ray, with components
[MATHEMATICAL EXPRESSION OMITTED]
and returning to the variable σ, we rewrite Eq. (20) as
[MATHEMATICAL EXPRESSION OMITTED] (21)
Instead of the three ray equations (21) we can write one vector equation
d(n[??]/dσ - [for all]n = 0. (22)
If the refractive index is given as a function of the coordinates, then Eq. (22), together ¦with the equation [??] = d[??]/dσ, allows us to find the equation of the trajectories [??] = [??](σ) with given initial conditions.
4. The Ray Diffusion Coefficient. We shall assume that the index of refraction deviates only slightly from a mean value equal to unity, i.e.
n(x, y, z) = 1 + µ(x, y, z), |µ| << 1. (23)
Then Eq. (22) can be rewritten as
d(n[??])/dσ - [for all]µ = 0. (24)
We now calculate the mean square deviation [MATHEMATICAL EXPRESSION OMITTED] of the ray from its initial direction after going a distance Δσ. We choose the path Δσ so that it is large compared to the correlation distance of the refractive index, but so that the deviation of the ray along the path is still small. Integrating (2l) along the path Δσ, we obtain
[MATHEMATICAL EXPRESSION OMITTED] (25)
It is physically clear that the angle of deviation of the ray from its initial direction is determined by the change of the refractive index along the whole path Δσ, and that the longer the path Δσ compared to the correlation distance, the less the deviation depends on the values n and n' of the refractive index at the ends of the path. Therefore setting the random values of the refractive index at the ends of the path Δσ equal to the mean value (unity) we obtain
[MATHEMATICAL EXPRESSION OMITTED] (26)
Squaring both sides of (26) and bearing in mind that
[MATHEMATICAL EXPRESSION OMITTED]
we find
[MATHEMATICAL EXPRESSION OMITTED] (27)
Averaging the correlation coefficient for the refractive index fluctuations over the ensemble of realizations of the medium,, we obtain
[MATHEMATICAL EXPRESSION OMITTED] (28)
Since the curvature of the ray is small, the integration along the ray can be replaced by integration along a straight line:
[MATHEMATICAL EXPRESSION OMITTED] (29)
Introducing relative coordinates
x = x1 - x2, y = y1 - y2, z = z1 - z2, r = r1 - r2
and center-of-mass coordinates
[MATHEMATICAL EXPRESSION OMITTED]
we have
[MATHEMATICAL EXPRESSION OMITTED]
Since Δσ >> a, we can integrate with respect to r from the limits - ∞ to + ∞. Eq. (29) takes the form
[MATHEMATICAL EXPRESSION OMITTED] (30)
Since the correlation coefficient N is an even function, we finally obtain
[MATHEMATICAL EXPRESSION OMITTED] (31)
(Continues...)
Excerpted from Wave Propagation in a Random Medium by Lev A. Chernov, Richard A. Silverman. Copyright © 2013 Lev A. Chernov. Excerpted by permission of Dover Publications, Inc..
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