contours (for both positive and negative arrival angles), which represent bottom-bounce paths, are referred to as order contours. Rays with the same number of bottom reflections belong to the same order. Thus, the first set of contours represents one bottom bounce and is of order 1. The contour associated with the positive angle within this first order has traveled from the source to the receiver over a RBR path and arrives at the receiver from below the horizontal axis. The corresponding contour with the negative angle has traveled from source to receiver over a refracted-bottom-reflected-surface-reflected (RBRSR) path and arrives at the receiver from above the horizontal axis.

These order contours can be used to identify all ray paths connecting the source and receiver that encounter the bottom at any range r. This information is very useful in the calculation of bottom reverberation. For example, only those rays encountering the bottom will contribute to bottom reverberation. The concept of order contours also applies to convergence zone (CZ) paths.

The 0-r contours can also be used to identify caustic formations. An instructive example was provided by Franchi et al. (1984). Figure 4.7 illustrates order contours obtained for a typical deep-ocean sound-speed profile. The two rays that just graze the bottom at ranges Xb and Xb< determine points B and B' on the contour. The two rays that just graze the surface at ranges Xa and Xa< occur at shallower angles and determine points A and A' on the contour. Thus, rays with angles in the interval (0b<, 0a<) and (0b, 0a) hit the surface but not the bottom. Caustics are associated with points C and C' on these contours.

Propagation models based on ray-tracing techniques generally treat bottom reflection as specular and reduce the intensity through application of a bottom reflection loss. However, acoustic energy can be transmitted into the bottom where it is subsequently refracted, attenuated and even transmitted back into the water column at some distance down range (Figure 4.8). This spatial offset is referred to as "beam displacement." Time displacements would also be associated with this process while the ray is absent from the water column.

In their initial study of the effects of beam displacement in ray calculations, Tindle and Bold (1981) considered a simple two-fluid Pekeris model as a good first approximation to many shallow-water environments. The Pekeris model (Pekeris, 1948) consists of a fluid (water) layer of depth H, density p\ and sound speed c\ overlying a semi-infinite fluid (sediment) layer of density P2 and sound speed c2, where c2 > c1. Attenuation was neglected and only rays totally reflected at the interface were considered to propagate to a range (r) greater than the water depth (H). The vertical wavenumbers

Initial angle 6

Initial angle 6

Range x

Figure 4.7 Smoothed order contours showing caustic behavior (Franchi etal., 1984).

Range x

Figure 4.7 Smoothed order contours showing caustic behavior (Franchi etal., 1984).

(Y1, y2) in the two layers were defined as:

c where 0 is the grazing angle and & the angular frequency of the wave field. The lateral displacement (A) of a beam of finite width undergoing reflection

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