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where a is the absorption coefficient (dB km 1 ), aL is the leakage coefficient (dBkm-1 ) that expresses the rate at which acoustic energy leaks out of the duct and r is the range (m). This expression is applicable to all ducts.

For a duct with a constant sound-speed gradient (g), in which the rays are arcs of circles with radius of curvature R = co/g, the following relationships result from geometry when R ^ H and when sin Go ^ 1 at source depth d (refer to Figure 5.1):

R = â€” = radius of curvature of rays g x = V8RH = skip distance of limiting ray i 2h do = -râ€”- = maximum angle of limiting ray (rad) R

0 = J---= angle of limiting ray at source depth (rad)

Kinsler etal. (1982: 402-6) also presented an insightful development of these relationships.

Marsh and Schulkin (1955) developed empirical surface duct equations based on an extensive set of data collected during Project AMOS in 1953-54. These equations have been incorporated into many ray-theoretical models to handle the special case of surface duct propagation. Graphical summaries of the AMOS data were presented in Chapter 3. Other empirically derived formulae have been reviewed by Urick (1982: chapter 6).

### 5.2.2 Wave-theory models

Pedersen and Gordon (1965) adapted Marsh's (1950) normal-mode approach to short ranges for a bilinear gradient (positive gradient overlying a negative gradient) model of a surface duct (Figure 5.2). This now classic bilinear surface duct model has been incorporated into some ray-theoretical models to augment their capabilities.

Assuming that leakage due to surface roughness can be neglected, TL is given as a function of range by

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