The range-dependent 2-D models assume azimuthally-symmetric environments and we violate this assumption in applying the model with different profiles for each bearing. In practice, this approximation is generally adequate and indeed is normally implicit in a 2-D run of a range-dependent model. That is, we normally use environmental information on a bearing-line between source and receiver, not intending the slice to define an azimuthally-symmetric environment.
In some cases, effects of horizontal refraction must be included. In principle, this is easily done. The derivation follows the pattern developed in the previous section for adiabatic modes in two-dimensions. We start with the Helmholtz equation in three-dimensions:
or, written out in full,
We next seek a solution of the form
where are the local modes. Substituting in the Helmholtz equation and applying the operator
The adiabatic approximation can then be obtained by neglecting the contributions of the coupling matrices A, B and C. This yields the horizontal refraction equation:
Using the local normal modes, we have eliminated the z-dimension from the problem and obtained a new Helmholtz equation, but now in the lateral coordinates x and y. The effective index of refraction is given by the horizontal wavenumber so that every mode generates a corresponding Helmholtz equation. Such 2-D Helmholtz equations can be solved using normal modes, PE, ray or spectral integral methods. Weinberg and Burridge present results using ray theory to solve the horizontal refraction equations.
Note that the horizontal refraction equation leads to the usual results for the cases of stratified range-independent or range-dependent but cylindrically symmetric profiles. For instance, if we rewrite the horizontal refraction equation in cylindrical coordinates and assume a cylindrically symmetric sound speed profile, we obtain,
The WKB approximation to the solution is then,
Using this result in Eq. () yields the usual adiabatic mode result of Eq. ().
In KRAKEN the horizontal refraction equations are solved using Gaussian beams which refract in the horizontal plane. As we have just seen, the ``refractive medium'' for each set of Gaussian beams is constructed from the phase velocity field corresponding to that particular mode. The field due to an individual beam involved a Gaussian decay in the horizontal plane and a depth dependence determined by the local mode.
The Gaussian beam solution to the horizontal refraction equation is straightforward. As discussed in references [59,60,61] the method begins by tracing a fan of beams originating from the source. The beams are defined by a central ray (which obeys the usual ray equations), the beam radius, and the curvature, , which are functions of arclength along the ray. The beams are Gaussian in that at any point on the central ray of an individual beam the intensity falls-off in a Gaussian form as a function of normal distance from the central ray. The final step is to sum up the contributions of all of the beams to compute the solution.
In more detail, the process is as follows. We begin by solving for the central rays of the beams which satisfy ray equations:
where, s denotes arclength along the ray and is the phase speed of the jth mode. A beam is then constructed about each ray using ray-centered coordinates, where n denotes the normal distance from the ray. The result is,
where the complex quantities and are obtained by integrating an auxiliary set of differential equations,
where the subcript n in indicates the derivative of the sound speed in a direction normal to the central ray.
The term in Eq. () is the phase delay or travel time which satisfies,
It is convenient to think of the p-q functions as defining the evolution of the beam in terms of its radius and curvature. Consider,
Clearly, characterizes the beam radius in terms of the normal distance at which the beam decays by . Furthermore, is a representation of the number of phase fronts crossed as one travels normal to the central ray and is a measure of curvature of the beam.
To summarize, the beam is defined in terms of , , , and representing respectively the central ray of the beam, the tangent to the central ray, the beam width and curvature, and the travel time. All of these functions are obtained by integrating a set of ordinary differential equations. To complete the definition we must provide initial conditions. For the central ray these are simply,
where is the source coordinate and is the take-off angle of the central ray.
The optimal choice of initial conditions for the p-q equations is a matter of current research. The two extremes of infinitely wide and infinitely narrow beams may be favored in cases with strong variation of wave speeds or when head waves are generated. Fortunately, the horizontal refraction equations are especially benign. In contrast to the usual r-z ocean acoustic problem, there are no boundaries to worry about. Secondly, the effective sound speed is so gradually varying that in may cases there is no clear need to even try to account for it. We have obtained good results using the analytic result for an isovelocity medium and selecting the initial beam width to minimize the beam width at the receiver. The beam curvature is chosen to be zero at the origin.
The final step is to synthesize the field by summing up the Gaussian beams. Thus, we seek a solution,
where is the lth beam. Following the standard procedure, we construct the expansion by the method of canonical problems, i.e., we require that the beam expansion be reasonable for a homogeneous medium. In a homogeneous medium, the equations which define the beam may be solved analytically. In addition analytical representations for the receiver coordinate in terms of the ray-centered coordinates may also be obtained without fanfare. Finally, one applies the saddle point method to compute the high-frequency asymptotic expansion of the integral. The result is,
On the other hand, the exact solution for a single-mode propagating in a stratified medium is
Evidently the forms match if the weighting for each beam, is . Putting all these things together, we obtain
The procedure then is to begin a loop over each mode. For each mode one has a Helmholtz equation with that mode's phase speed acting like the usual ocean sound speed. Each such horizontal refraction equation is solved by looping over an azimuthal take-off angle and each azimuthal take-off angle leads to a beam which propagates out from the source. As each beam is traced out its contribution to the field is summed up. This procedure is repeated for each mode to obtain the complete field representation.