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Figure 4.11 Comparison of measured and calculated amplitude functions (using arbitrary scales) for the first and second modes: (a) at 400 Hz with a downward-refracting (negative-gradient) profile; (b) at 750 Hz with and upward-refracting (positive-gradient) profile; and (c) at 750 Hz with a nearly constant sound-speed profile (Ferris, 1972).

by "mode coupling" or by "adiabatic approximation." Mode coupling considers the energy scattered from a given mode into other modes. Adia-batic approximation assumes that all energy in a given mode transfers to the corresponding mode in the new environment, provided that environmental variations in range are gradual.

Three-dimensional propagation modeling using normal-mode theory has been attempted using two different approaches. The first approach employs the N x 2D technique in which the 3D problem is solved using N horizontal radials (or sectors) in conjunction with range-dependent (2D) adiabatic mode (or coupled mode) theory along each radial. The resulting quasi-3D propagation fields can then be contoured on polar plots, for example. The second approach directly includes the effects of horizontal refraction through use of the lateral wave equation. One implementation by Kuper-man et al. (1988) and Porter (1991) employed Gaussian beam tracing (Porter and Bucker, 1987) to solve the lateral wave equation. In essence, the horizontal field of each modal wavenumber is translated into a horizontal sound-speed field that defines the Gaussian beam environment for each mode. In related work, a new development called wide-area rapid acoustic prediction (WRAP) precomputed local acoustic eigenvalues and normal modes for complex 3D ocean environments comprising a number of distinct local environments (Perkins et al., 1990). For mild horizontal variability, the full 3D acoustic field was constructed by adiabatic mode computations. Another 3D model called CMM3D included horizontal refraction and radial mode coupling (Chiu and Ehret, 1990, 1994). Ainslie et al. (1998a) demonstrated the importance of leaky modes in range-dependent environments with variable water depth. In this particular investigation, the bottom-interacting field was computed by mode summation. Gabrielson (1982) investigated the application of normal-mode models to leaky ducts.

4.4.6 High-frequency adaptations

Normal-mode approaches tend to be limited to acoustic frequencies below about 500 Hz due to computational considerations (and not due to any limitations in the underlying physics). Specifically, the number of modes required to generate a reliable prediction of TL increases in proportion to the acoustic frequency. However, by invoking some simplifying assumptions regarding the complexity of the ocean environment, upper frequency limits in the multi-kilohertz range can be achieved (Ferla et al., 1982).

4.4.7 Wedge modes

Primack and Gilbert (1991) investigated so-called "wedge modes," which are the intrinsic normal modes in a wedge (i.e. sloping-bottom) coordinate system. Wedge modes are identical to the usual normal modes of a rangeinvariant waveguide except that the mode functions are referenced to the arc of a circle rather than to a vertical line. This difference derives from use of a polar coordinate system, with its origin at the apex of the wedge, rather than the usual range-depth coordinate system in a wedge domain (Fawcett et al., 1995). For shallow-water acoustic propagation, the acoustic wavelength is commensurate with the water depth, but short compared to the horizontal extent of the problem. Under these conditions, a sloping bottom causes the development of normal modes having wavefronts that are curved in the vertical direction. Using simple slopes, for example, wedge modes were found to propagate with cylindrical wavefronts (Mignerey, 1995).

Fawcett et al. (1995) developed an efficient coupled-mode method based on the concept of wedge modes. Leaky modes were also included because of their importance in range-dependent waveguide geometries. In related work, Tindle and Zhang (1997) developed an adiabatic normal-mode solution for a well-known benchmark wedge problem (discussed in Chapter 11) that included both fluid and solid attenuating bottom boundaries. The continuous-mode contribution was treated as a sum of leaky modes, and each trapped mode gradually transitioned into a leaky mode as the water depth decreased.

4.5 Multipath expansion models

Multipath expansion techniques expand the acoustic field integral representation of the wave equation [Equation (4.13)] in terms of an infinite set of integrals, each of which is associated with a particular ray-path family. This method is sometimes referred to as the "WKB method" since a generalized WKB (Wentzel, Kramers and Brillouin) approximation is used to solve the depth-dependent equation derived from the normal-mode solution [Equation (4.11)]. Each normal mode can then be associated with corresponding rays (see the discussion in the latter part of Section 4.4.4). Multipath expansion models do not currently accommodate environmental range dependence.

The WKB approximation (sometimes also referred to as the WKBJ or Liouville-Green approximation) facilitates an asymptotic solution of the normal mode equation by assuming that the speed of sound varies gradually as a function of depth. Advanced versions of the WKB method provide connection formulae to carry the approximation through "turning points" (i.e. depths where an equivalent ray becomes horizontal). Unlike ray-theoretical solutions, however, the WKB method normally accounts for first-order diffraction effects and caustics.

The specific implementation of this approach is accomplished by directly evaluating the infinite integral of Equation (4.13) over a limited interval of the real H-axis. Thus, only certain modes are considered. By using a restricted number of modes, an angle-limited source can be simulated. The resulting acoustic pressure field $ in Equation (4.13) is then expressed as a sum of finite integrals, where each integral is associated with a particular ray family. As implemented, this approach is particularly applicable to the modeling of acoustic propagation in deep water at intermediate and high frequencies. Multipath expansion models thus have certain characteristics in common with ray models. Moreover, the pressure field is properly evaluated in caustics and shadow zones. Weinberg (1975) provided a brief summary of the historical development of this technique. This approach is explored in more detail in Section 4.8 where the RAYMODE model (which is based in part on the multipath expansion approach) is described.

4.6 Fast-field models

In underwater acoustics, fast-field theory is also referred to as "wavenum-ber integration." In seismology, this approach is commonly referred to as the "reflectivity method" or "discrete-wavenumber method." In fast-field theory, the wave-equation parameters are first separated according to the normal-mode approach. Then, the Hankel function expression in Equation (4.13) is replaced by the first term in the asymptotic expansion (DiNapoli and Deavenport, 1979):

The infinite integral is then evaluated by means of the fast Fourier transform (FFT), which provides values of the potential function $ at n discrete points for a given source-receiver geometry. Evaluation of the Green's function can be simplified by approximating the sound-speed profile by exponential functions. Such an approximation facilitates the matrizant solution, but complicates specification of the sound-speed profiles.

Historically, models based on fast-field theory did not allow for environmental range dependence. However, two early developments introduced the possibility of range-dependent calculations of TL. First, Gilbert and Evans (1986) derived a generalized Green's function method for solving the oneway wave equation exactly in an ocean environment that varied discretely with range. They obtained an explicit marching solution in which the source distribution at any given range step was represented by the acoustic field at the end of the previous step. Gilbert and Evans (1986) further noted that their method, which they called the range-dependent fast-field program (RDFFP) model, was computationally intensive. Second, Seong (1990) used for f r ยป 1

Equation (4.13) can now be written as

a hybrid combination of wavenumber integration and Galerkin boundary element methods (BEM), referred to as the SAFRAN model, to extend the fast-field theory technique to range-dependent ocean environments (see also Schmidt, 1991). The experimental nature of these early methods imposed uncertain restrictions on their application to system performance modeling in range-dependent ocean environments. As described below, however, further research proved these methods useful in modeling range-dependent wave propagation.

One approach to range-dependent modeling partitioned the ocean environment into a series of range-independent sectors called "super elements" (Schmidt et al., 1995). Goh and Schmidt (1996) extended the spectral super-element approach for acoustic modeling in fluid waveguides to include fluid-elastic stratifications. Their method used a hybridization of finite elements, boundary integrals and wavenumber integration to solve the Helmholtz equation in a range-dependent ocean environment. It provided accurate, two-way solutions to the wave equation using either a global multiple scattering solution or a single-scatter marching solution.

Grilli et al. (1998) combined BEM and eigenfunction expansions to solve acoustic wave propagation problems in range-dependent, shallow-water regions. Their hybrid BEM technique (HBEM) was validated by comparing outputs to analytical solutions generated for problems with simple boundary geometries including rectangular, step and sloped domains. Hybrid BEM was then used to investigate the transmission of acoustic energy over bottom bumps while emphasizing evanescent modes and associated "tunneling" effects. Related developments in BEM in shallow water were reported by Santiago and Wrobel (2000).

The FFP approach has been modified to accommodate acoustic pulse propagation in the ocean by directly marching the formulation in the time domain (Porter, 1990). Applications included specification of arbitrary source time series instead of the more conventional time-harmonic sources used in frequency-domain solutions of the wave equation.

4.7 Parabolic equation models

Use of the parabolic approximation in wave propagation problems can be traced back to the mid-1940s when it was first applied to long-range tro-pospheric radio wave propagation (Keller and Papadakis, 1977: 282-4). Subsequently, the parabolic approximation method was successfully applied to microwave waveguides, laser beam propagating, plasma physics and seismic wave propagation. Hardin and Tappert (1973) reported the first application to problems in underwater acoustic propagation (also see Spofford, 1973b: 14-16). Lee and Pierce (1995) and Lee et al. (2000) carefully traced the historical development of the parabolic equation (PE) method in underwater acoustics.

4.7.1 Basic theory

The PE (or parabolic approximation) approach replaces the elliptic reduced equation [Equation (4.3b)] with a PE. The PE is derived by assuming that energy propagates at speeds close to a reference speed - either the shear speed or the compressional speed, as appropriate (Collins, 1991).

The PE method factors an operator to obtain an outgoing wave equation that can be solved efficiently as an initial-value problem in range. This factorization is exact when the environment is range independent. Range-dependent media can be approximated as a sequence of range-independent regions from which backscattered energy is neglected. Transmitted fields can then be generated using energy-conservation and single-scattering corrections. The following derivation is adapted from that presented by Jensen and Krol (1975).

The basic equation for acoustic propagation, Equation (4.3a), can be rewritten as:

where ko is the reference wavenumber (m/co), m(=2nf) the source frequency, co the reference sound speed, c{r, 0, z) the sound speed in range (r), azimuthal angle (0) and depth (z), n the refraction index {co/c), 0 the velocity potential and V2 the Laplacian operator.

Equation (4.18) can be rewritten in cylindrical coordinates as:

where azimuthal coupling has been neglected, but the index of refraction retains a dependence on azimuth. Further, assume a solution of the form:

and obtain:

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