## Mathematical models Part

5.1 Background

Mathematical models of underwater acoustic propagation include both numerical models and empirical models. Chapter 4 addressed the theoretical development of numerical models and summarized their availability. This chapter addresses the development of empirical models applicable to special propagation paths such as surface ducts, shallow water and Arctic half-channels. Where appropriate, comparisons are made with predictions generated by numerical models. Data support requirements for mathematical models of propagation are discussed and a select number of special applications are described in order to highlight promising areas for future research and development.

### 5.2 Surface duct models

Properties of the surface duct were discussed previously in Chapter 3. Both ray- and wave-theoretical solutions can be applied to propagation in the surface duct.

### 5.2.1 Ray-theory models

An expression for transmission loss (TL) in a surface duct may be obtained through simple ray-theoretical considerations (Urick, 1983). In Figure 5.1, let a nondirectional source of sound be located at P in a surface duct (or mixed layer). Also, let C0 denote the reference sound speed in the duct. Of all the rays leaving the source, only those within a certain limiting angle 20 remain in the duct. At a distance of 1 m, the power contained in this ray bundle is distributed over a portion of the spherical surface A\. At a long distance r, this same amount of power (in the absence of leakage and absorption) is distributed over a cylindrical surface A2. Because the power crossing areas A1 and A2 is conserved, the TL to range r, averaged over the duct thickness H, is

By geometry r9

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