Three-dimensional ray acoustics: new expressions for the amplitude, eikonal, and phase functions

New three-dimensional ray-acoustics expressions for the amplitude, eikonal, and phase along a ray path are derived. These expressions clearly indicate the numerical calculations that must be performed in order to evaluate these functions. The ocean medium is characterized by a three-dimensional random index of refraction which is decomposed into deterministic random components. >

is the constant, reference speed of sound at the source position ro = (XO, YO, ZO). Note that the wave number amplitude, eikonal, and phase along a ray path are derived. phase functions. k ( r ) = 2n f / c ( r ) I. INTRODUCTION HE main purpose of this paper is to derive new T expressions for the amplitude, eikonal, and phase along a ray path based on three-dimensional ray acoustics. The new expressions clearly indicate the numerical calculations that must be performed in order to evaluate these functions. The Ocean medium is characterized by a three-dimensional random index of refraction which is decomposed into deterministic and random components.

ANALYSIS
The propagation of small-amplitude acoustic signals in the ocean can be described by the following linear, homogeneous wave equation: where p ( t , r ) is the velocity potential in square meters per second at time t and position r = (x, y , z ) , and c(r) is the speed of sound in the ocean in meters per second. If we assume a time-harmonic dependence for the velocity potential, that is, if can be expressed as and that k(r0) = ko since n(ro) = 1 .

The index of refraction is commonly written as El], [2]
n(r)=nD(r) + nR(r) or n ( r ) = nD(r) + U(r)nNR(r) (10) where nD(r) is the deterministic component and is sometimes referred to as the deterministic or background sound channel,

n~( r )
is the random, zero-mean component, a(r) is the standard deviation of nR(r), and is the normalized random component with zero mean and variance equal to unity. Note that the expected (average) value of n (r) is equal to nD(r).
An approximate solution of the Helmholtz wave equation given by (3), based on the method of three-dimensional ray acoustics, is given by is the constant, reference wave number in radians per meter, is the random, three-dimensional dimensionless index of refraction, and Equations (13) through (15) are the most common threedimensional ray acoustics' expressions for the amplitude, eikonal, and phase, respectively. The disadvantage of these common expressions is that there is no additional information given on how to actually solve for the cross-sectional areas SO and S in ( 1 3 ) and how to evaluate the integrals appearing in (14) and (15). However, new expressions for these quantities can be derived that clearly indicate the numerical calculations that must be performed.

A . The Amplitude Function
and, noting that the real index of refraction is always positive An alternate, new expression for the amplitude a@) along a and that n (ro) = 1, we obtain the following expression for the ray path can be obtained as follows. We begin with the amplitude along a ray path: transport equation as given by [lo]

ds W r )
The solution of (16) can be expressed as or, upon substituting (5) into (27),

V 2 W ( r )
Next, in order to make (27) and (

ds ds ds ds
which is the desired result. Therefore, either (27) or (28), in conjunction with (34), represent new expressions for the amplitude along a ray path. By comparing (13) and (27), it can evaluating the integral of the divergence of the unit vector along a ray path; that is, be seen that the task of computing So and S is equivalent to Equation ( In order to determine the amplitude along a ray path according to (28) and (34), and in order to determine the phase along a ray path according to (45) through (48), the direction cosines u(r), U@), and w(r) must first be determined by solving the ray equations [ 171, [ 181. This is not necessarily a shortcoming, since in order to draw ray diagrams for a threedimensional sound-speed profile, the ray equations must be solved anyway. And, finally, note that if the index of substituting (39) through (41) into the right-hand side of (38) az and refraction is random, the direction cosines will be random as well. However, for problems dealing with wave propagation in a random, inhomogeneous medium, amplitude and phase calculations are performed by carrying out integrations along deterministic unperturbed ray paths [ 191, that is, using the deterministic components of the direction cosines in (34), (43), and (45) through (48).

ACKNOWLEDGMENT
Discussions held with Prof. S. M. Flank of the University of California at Santa Cruz and several of his graduate students, and with Prof. J. H. Miller of the Naval Postgraduate School regarding the content of an earlier version of this paper, were very helpful and very much appreciated.