Figure 4.9 A typical ray path with beam displacement included (Tindle and Bold, 1981).

at the water-sediment interface is:

where kh is the horizontal wavenumber in the water layer

A typical ray path for this simple model is presented in Figure 4.9. The source and receiver depths are zo and z, respectively. A ray leaving the source at an angle 0 relative to the horizontal travels in a straight (unrefracted) path in the water layer. There is no beam displacement at the sea surface and the ray is reflected in the conventional manner. At the bottom, the beam is displaced horizontally by an amount A before traveling upward again at angle 0.

Recent efforts to improve ray-theory treatments of bottom attenuation and beam displacement have been conducted by Siegmann et al. (1987) and Westwood and Tindle (1987), among others. Modified ray theory with beam and time displacements has advantages over wave theory in that the interaction of the acoustic energy with the bottom can be intuitively visualized. Moreover, Jensen and Schmidt (1987) computed complete wave-theory solutions for a narrow Gaussian beam incident on a water-sediment interface near the critical grazing angle. They observed that the fundamental reflectivity characteristics of narrow beams could be explained entirely within the framework of linear acoustics.

Normal-mode solutions are derived from an integral representation of the wave equation. In order to obtain practical solutions, however, cylindrical symmetry is assumed in a stratified medium (i.e. the environment changes as a function of depth only). Then, the solution for the potential function $ in Equation (4.3b) can be written in cylindrical coordinates as the product of a depth function F(z) and a range function S(r):

Next, a separation of variables is performed using f2 as the separation constant. The two resulting equations are:

dr2 r dr

Equation (4.11) is the depth equation, better known as the normal mode equation, which describes the standing wave portion of the solution. Equation (4.12) is the range equation, which describes the traveling wave portion of the solution. Thus, each normal mode can be viewed as a traveling wave in the horizontal (r) direction and as a standing wave in the depth (z) direction.

The normal-mode equation (4.11) poses an eigenvalue problem. Its solution is known as the Green's function. The range Equation (4.12) is the zero-order Bessel equation. Its solution can be written in terms of a zero-

order Hankel function (H0 0. The full solution for $ can then be expressed by an infinite integral, assuming a monochromatic (single-frequency) point source:

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