where G is Green's function, H0 ' a zero-order Hankel function of the first kind and z0 the source depth. Note that $ is a function of the source depth (z0) and the receiver depth (z). The properties of these various functions are described in standard mathematical handbooks such as Abramowitz and Stegun (1964).

To obtain what is known as the normal-mode solution to the wave equation, the Green's function is expanded in terms of normalized mode functions (un). The eigenvalues, which are the resulting values of the separation constants, are represented by fn. These eigenvalues (or characteristic values) represent the discrete set of values for which solutions of the mode functions un exist. The infinite integral in Equation (4.13) is then evaluated by contour integration:

0 un(Z2 "nf0) Ho1) G r) • H + branch-cut integral (4.14)

The contour integral represents the trapped (or discrete) modes that propagate through the water column. The branch-cut integral is associated with the continuous mode spectrum, which represents those modes propagating through the ocean floor and which are strongly attenuated. The branch-cut integral describes the near-field conditions and corresponds in ray theory to those rays striking the bottom at angles greater than the critical angle. Thus, the contribution from the branch-cut integral is often neglected, particularly when horizontal distances greater than several water depths separate the source and receiver. For acoustic propagation problems using an impedance boundary condition at the bottom, the solution actually comprises three spectral intervals: the continuous, discrete and evanescent. The evanescent spectrum is associated with interface waves that decay exponentially away from the boundary (Jensen et al., 1994). Interface waves that propagate along a fluid-solid boundary (i.e. water-sediment interface) are called Scholte waves. In seismology, interface waves propagating along the boundaries between solid layers are called Stoneley waves.

By neglecting the branch-cut integral, evaluating the contour integral and replacing the Hankel function expression by its asymptotic expansion for large arguments:

(where f r » 1 is the far-field approximation), a simple solution for the potential function $ can be obtained:

0 = g(r,p)Y, Un(z)'/un(z0) exp[i(fnr - n/4)] exp(-Snr) (4.15)

V fn where g(r, p) is a general function of range (r) and water density (p). Each of the n-terms in Equation (4.15) corresponds to the contribution of a single normal mode of propagation. Each of these modal contributions is propagated independently of the others. Under idealized conditions, there is usually an upper limit on the number of modes to be calculated, and this number increases with increasing frequency.

The attenuation coefficient Sn can be written in the form (Miller and Wolf, 1980):

where s is the plane-wave attenuation coefficient in the sediment layer, sc, ss the compressional and shear plane-wave attenuation coefficients, respectively, of the basement, y„, y^, Yn} measures of nth mode interactions with sediment, and with basement compressional and shear wave mechanisms, Sn0),S^ the attenuation of the modal field due to the interaction of the nth mode with statistically rough boundaries at the pressure release boundary (sea surface) and the water-sediment boundary, respectively, and an the attenuation due to absorption by sea water.

One advantage of normal-mode solutions over ray-theoretical methods is that TL can easily be calculated for any given combination of frequency and source depth (zo) at all receiver depths (z) and ranges (r) [refer to Equation (4.14)]. Ray models, on the other hand, must be executed sequentially for each change in source or receiver depths (compare Figures 3.2 and 3.3).

A disadvantage associated with normal-mode solutions is the degree of information required concerning the structure of the sea floor, as indicated in Equation (4.16). In order to execute effectively, this type of model generally requires knowledge of the density as well as the shear and compressional sound speeds within the various sediment layers. Figure 4.10 shows an ideal three-layer physical model usable in normal-mode solutions. The fluid layer boundaries are defined by duct depths or by other notable features in the sound-speed profile. Accordingly, more layers can be added to the physical model as the complexity of the profile increases. The computational intensity increases in proportion to this complexity. In Figure 4.10, the sediment layer is treated as a second fluid layer (below the water column) and is characterized by a compressional sound speed only, thus ignoring shear effects. The effects of sediment attenuation can be included, as noted above in Equation (4.16).

Some normal-mode models (Stickler, 1975; Bartberger, 1978b) partition the sound-speed profile into N layers such that the square of the index of refraction in each layer can be approximated by a straight line and the density can be assumed constant. Under these conditions, it is then possible to represent the depth-dependent portion of the pressure field in terms of Airy functions and thus improve computational efficiency. The model by Stickler (1975) is noteworthy in that it includes the continuous modes as well as the trapped modes. Fast finite difference methods have been used to accurately determine the real (versus imaginary) eigenvalues (Porter and Reiss, 1984). Errors in these eigenvalues would otherwise appear as phase shifts in the range dependence of the acoustic field.

Unlike ray-theoretical solutions, wave-theoretical solutions inherently treat dispersion effects. Dispersion is the condition in which the phase velocity is

Sound speed

Sound speed

Figure 4.10 Simple physical model depicting an infinite half-space consisting of: two fluid layers with depths H1, H2; densities p1, p2; and sound speeds c1, c2. A semi-infinite third layer (representing the bottom) has density p3, and compressional (c3c) and shear (c3s) sound speeds (adapted from Miller and Ingenito, 1975).

Figure 4.10 Simple physical model depicting an infinite half-space consisting of: two fluid layers with depths H1, H2; densities p1, p2; and sound speeds c1, c2. A semi-infinite third layer (representing the bottom) has density p3, and compressional (c3c) and shear (c3s) sound speeds (adapted from Miller and Ingenito, 1975).

a function of the acoustic frequency. If present, dispersion effects are most noticeable at low frequencies. In oceanic waveguides, dispersion depends on the characteristics and geometry of the waveguide and is referred to as geometrical dispersion. This is distinguished from intrinsic dispersion as might result from sound propagation through bubbly water layers near the sea surface (Clay and Medwin, 1977: 311-12).

Ferris (1972) and Ingenito et al. (1978) summarized results from field measurements at a site near Panama City, Florida. The experiments were conducted over tracks having a range-independent water depth and soundspeed profile. The sea floor was composed of hard-packed sand. The modal field distribution was calculated on the basis of Equation (4.14). If all acoustic and environmental parameters except for the receiver depth are held constant, then the dependence of the pressure amplitude ($n) of the nth-order mode on receiver depth (z) is directly proportional to the corresponding mode function (un):

If range and receiver depths are held constant, then the dependence of the pressure amplitude ($n) of the nth-order mode on source depth (z0) is

Comparisons of measured and calculated pressure amplitude distributions at the receiver (using an arbitrary amplitude scale) are presented in Figure 4.11 for the first and second modes. Two frequencies (400 and 750 Hz) were considered, as were three different sound-speed profiles (positive gradient, negative gradient and isospeed). The influence of the sound-speed gradient on the low-order modes is evident. Specifically, the first mode is nearly symmetrically distributed about mid-depth in the isospeed case (Figure 4.11(c)). The sound-speed gradients alter this symmetry and concentrate the energy in the low-speed portion of the water column (Figures 4.11(a), (b)).

Boyles (1984: chapter 6) provided a comprehensive discussion demonstrating that certain groupings of normal modes could be associated with particular ray families propagating in specific oceanic waveguides. This was accomplished by explicitly calculating the TL associated with selected subsets of modes using Equation (4.14). These calculations were then compared to the full TL curve. One example involved a typical North Atlantic Ocean sound-speed profile during the winter season. This profile was characterized by a surface duct, a sound channel and sufficient depth excess to support CZ propagation (refer back to Figure 2.7). The source and receiver were both placed within the surface duct, and a frequency of 30 Hz was selected. Only 27 modes were trapped in the water column at this frequency. Of these, only one mode (mode 11) contributed to the TL in the surface duct. Modes that uniquely contribute to surface duct propagation are referred to as "virtual modes" (Labianca, 1973). Thus, mode 11 is also referred to as virtual mode 1. Modes 13-27 accounted for the TL associated with the CZ paths. The remaining modes contributed very little to the overall TL. The association of certain normal modes with particular ray families will be explored further in discussions of the RAYMODE model later in this chapter.

4.4.5 Range dependence

As noted previously in Figure 4.1, normal-mode models assume range independence. Extensions to range dependence can be accommodated either

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