## Mathematical models Part

4.1 Background

Chapter 1 described a framework within which all underwater acoustic models could be categorized. It was shown that propagation models formed the foundation for the category of models classified as basic acoustic models. In turn, basic acoustic models supported the more specialized category of sonar performance models. Propagation models are the most common (and thus the most numerous) type of underwater acoustic models in use. Their application is fundamental to the solution of all types of sonar performance problems.

Chapter 3 described the observations and physical (physics-based) models that are available to support the mathematical modeling of sound in the sea. Conceptually, it was convenient to separate propagation phenomena into the categories of boundary interactions, volumetric effects and propagation paths. A similar approach will be adopted here in the description of the mathematical models.

The mathematical models will first be distinguished on the basis of their theoretical treatment of volumetric propagation. Then, as appropriate, further distinctions will be made according to specification of boundary conditions and the treatment of secondary volumetric effects such as attenuation due to absorption, turbidity and bubbles. These secondary effects are generally accommodated by using the physical models described in Chapter 3. Special propagation paths such as surface ducts, shallow water and Arctic half-channels will be discussed in Chapter 5.

The various physical and mathematical models all have inherent limitations in their applicability. These limitations are usually manifested as restrictions in the frequency range or in specification of the problem geometry. Such limitations are collectively referred to as "domains of applicability," and vary from model to model. Most problems encountered in model usage involve some violation of these domains. In other words, the models are misapplied in practice. Therefore, considerable emphasis is placed on these restrictions and on the assumptions that ultimately give rise to them. Finally, model selection criteria are provided to guide potential users to those models most appropriate to their needs. Comprehensive summaries identify the available models and associated documentation. Brief descriptions have been provided for each model.

The emphasis in this chapter, as throughout the book, is placed on sonar (versus seismic) applications. Reviews of mathematical models of seismo-acoustic propagation in the ocean have been provided by Tango (1988) and by Schmidt (1991). Tango (1988) placed particular emphasis on the very-low-frequency (VLF) band.

### 4.2 Theoretical basis for propagation modeling

The theoretical basis underlying all mathematical models of acoustic propagation is the wave equation. The earliest attempts at modeling sound propagation in the sea were motivated by practical problems in predicting sonar performance in support of anti-submarine warfare (ASW) operations during the Second World War. These early models used ray-tracing techniques derived from the wave equation to map those rays defining the major propagation paths supported by the prevailing marine environment. These paths could then be used to predict the corresponding sonar detection zones. This approach was a forerunner to the family of techniques now referred to as ray-theoretical solutions.

An alternative approach, referred to as wave-theoretical solutions, was first reported by Pekeris (1948), who used the normal-mode solution of the wave equation to explain the propagation of explosively generated sound in shallow water.

As modeling technology matured over the intervening decades, the attendant sophistication has complicated the simple categorization of ray versus wave models. The terminology is still useful in distinguishing those models based principally on ray-tracing techniques from those using some form of numerical integration of the wave equation. Occasionally, a mixture of these two approaches is used to capitalize on the strengths and merits of each and to minimize weaknesses. Such combined techniques are referred to as hybrid approaches. Related developments in propagation modeling have been reviewed by Harrison (1989), McCammon (1991), Buckingham (1992), Porter (1993) and Dozier and Cavanagh (1993). Finite-element methods have also been used in underwater acoustics to treat problems requiring high accuracy [see Kalinowski (1979) for a good introduction to applications in underwater acoustics]. Developments in finite-element modeling will be discussed in appropriate sections throughout this book.

### 4.2.1 Wave equation

The wave equation is itself derived from the more fundamental equations of state, continuity and motion. Rigorous derivations have been carried out in numerous basic texts in physics. Kinsler et al. (1982: Chapter 5) presented a particularly lucid derivation. DeSanto (1979) derived a more general form of the wave equation that included gravitational and rotational effects. Accordingly, the derivation will not be repeated here. Rather, the mathematical developments described in this book will build directly upon the wave equation.

Formulations of acoustic propagation models generally begin with the three-dimensional, time-dependent wave equation. Depending upon the governing assumptions and intended applications, the exact form of the wave equation can vary considerably (DeSanto, 1979; Goodman and Farwell, 1979). For most applications, a simplified linear, hyperbolic, second-order, time-dependent partial differential equation is used:

where V2 is the Laplacian operator [= (d2/dx2) + (d 2/dy2) + (d 2/d z2)], * the potential function, c the speed of sound and t the time.

Subsequent simplifications incorporate a harmonic (single-frequency, continuous wave) solution in order to obtain the time-independent Helmholtz equation. Specifically, a harmonic solution is assumed for the potential function *:

where \$ is the time-independent potential function, m is the source frequency (2nf) and f the acoustic frequency. Then the wave equation (4.1) reduces to the Helmholtz equation:

where k = (m/c) = (2n/X) is the wavenumber and X the wavelength. In cylindrical coordinates, Equation (4.3a) becomes:

Equation (4.3a) is referred to as the time-independent (or frequency-domain) wave equation. Equation (4.3b), in cylindrical coordinates, is commonly referred to as the elliptic-reduced wave equation.

Various theoretical approaches are applicable to the Helmholtz equation. The approach used depends upon the specific geometrical assumptions made for the environment and the type of solution chosen for as will be discussed in the following sections. To describe the different approaches effectively, it is useful to first develop a classification scheme, with associated taxonomy, based on five canonical solutions to the wave equation: ray

theory, normal mode, multipath expansion, fast field and parabolic equation techniques.

Throughout the theoretical development of these five techniques, the potential function \$ normally represents the acoustic field pressure. When this is the case, the transmission loss (TL) can easily be calculated as:

This relationship necessarily follows from Equations (3.1) and (3.2). If phases are considered, the resulting TL is referred to as coherent. Otherwise, phase differences are ignored and the TL is termed incoherent.

### 4.2.2 Classification of modeling techniques

Although acoustic propagation models can be classified according to the theoretical approach employed, the cross-connections that exist among the various approaches complicate a strict classification, or taxonomic, scheme. Consequently, as the schemes become more detailed, more cross-connections will appear. A generalized classification scheme has been constructed using five categories corresponding to the five canonical solutions of the wave equation (also see Jensen and Krol, 1975; DiNapoli and Deavenport, 1979; Weston and Rowlands, 1979).

Within these five categories, a further subdivision can be made according to range-independent and range-dependent models. Range independence means that the model assumes a horizontally stratified ocean in which properties vary only as a function of depth. Range dependence indicates that some properties of the ocean medium are allowed to vary as a function of range (r) and azimuth (0) from the receiver, in addition to a depth (z) dependence. Such range-varying properties commonly include sound speed and bathymetry, although other parameters such as sea state, absorption and bottom composition may also vary. Range dependence can further be regarded as two dimensional (2D) for range and depth variations or three dimensional (3D) for range, depth and azimuthal variations.

In order to illustrate the relationships among the five approaches used to solve the wave equation, the rather elegant scheme developed by Jensen and Krol (1975) will be adopted with slight modifications (Figure 4.1). According to this classification scheme, there are three avenues that connect the five basic approaches applicable to underwater acoustic propagation modeling. These five categories of propagation models will be described in detail in the following sections, and Figure 4.1 will serve as a useful road map. For convenience, the general functions and equations represented in Figure 4.1 have been identified with the letters F and G. In the discussions that follow, different symbols will be substituted to facilitate identification with relevant physical properties or with other well-known mathematical functions.

> Ray theory

C2 dt2

Wave equation