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Figure 5.5 Sound-speed profile and bottom properties for a shallow-water area located in the eastern North Atlantic Ocean: c = sound speed, p = density, @ = attenuation, S1 = rms bottom roughness. Subscripts "C" and "S" refer to compressional and shear waves, respectively. Source (S) and receiver (R) depths are indicated on the sound-speed profile (Jensen and Kuperman, 1979).

Range (km)

Range (km)

Figure 5.6 Transmission loss contours (in 2-dB intervals) for the environment presented in Figure 5.5: (a) experimental measurements and (b) predictions generated by the SNAP propagation model (Jensen and Kuperman, 1979).

Range (km)

Figure 5.6 Transmission loss contours (in 2-dB intervals) for the environment presented in Figure 5.5: (a) experimental measurements and (b) predictions generated by the SNAP propagation model (Jensen and Kuperman, 1979).

optimum frequency predicted by the model is about 200 Hz, as evidenced by the elongated axis of low-loss values. These predicted results compared favorably with experimental results (Figure 5.6(a)).

Shallow-water modeling techniques include both numerical models and empirical models. Eller (1984b, 1986) summarized important development in shallow-water acoustic modeling. Katsnelson and Petnikov (2002) reviewed experimental results in shallow-water acoustics together with approximate approaches for modeling such phenomena.

5.3.3 Numerical models

Both ray- and wave-theoretical approaches have been used to numerically model sound propagation in shallow water. Since shallow water environments are best approximated by range-dependent geometries (e.g. sloping bottom and high spatial variability of water-column and sediment properties), attention will be focused on the range-dependent modeling approaches. Moreover, the high-angle boundary interactions encountered in shallow water have traditionally limited consideration to two basic approaches: ray theory and normal-mode solutions (with either adiabatic approximations or mode coupling). More recently, appropriately modified PE models have also been utilized successfully in shallow-water environments (Jensen, 1984; Jensen and Schmidt, 1984). Approximately 18 percent of the numerical modeling inventory is specifically tailored for shallow-water applications (Etter, 2001c).

Much emphasis has been placed on modeling sound propagation over a sloping bottom. This geometry is commonly referred to as a "wedge problem" and involves both upslope and downslope propagation. The direction of propagation (i.e. upslope or downslope) considerably alters the observed propagation characteristics. Consequently, this problem is of great practical interest to sonar operations in shallow water. This geometry has also been used as a benchmark problem in model evaluation (see Chapter 11).

The basic mechanisms involved in acoustic propagation in a horizontally stratified (i.e. range-invariable) waveguide are spreading loss, attenuation due to bottom-interaction effects, and intermode phasing effects. In a rangevariable waveguide, an additional mechanism must be considered. This mechanism is related to changes in the acoustic energy density that occur with bathymetric changes and is often referred to a renormalization loss, or megaphone effect. The term renormalization is used because the so-called megaphone effect is manifested as a change in the normalization of the normal mode depth function due to changes in the waveguide depth (Koch et al., 1983). The megaphone effect produces a gain in upslope propagation and a loss in downslope propagation.

The processes involved in upslope propagation can be better understood by using the ray-mode analogy. This analogy is a heuristic concept which states that any given mode trapped in the water column can be associated with upgoing and downing rays corresponding to specific grazing angles at the bottom (e.g. Urick, 1983: 174-6; Boyles, 1984: 197-204). As sound propagates upslope, the horizontal wavenumber associated with each mode decreases. In the ray analogy, the grazing angle at the bottom increases. For each mode, then, a point on the slope will be reached at which the grazing angles of the analogous rays will approach the critical angle at the bottom. At this point, the energy essentially leaves the water column and enters the bottom. In the ray analogy, the bottom-reflection losses associated with those rays become very large. In the wave analogy, the modes transition from the trapped (waterborne) to the continuous (bottom propagating) spectrum. This point is called the "cutoff depth" for the equivalent modes. Upslope propagation is then said to exhibit a transition from a trapped to a radiative state (e.g. Arnold and Felsen, 1983; Jensen, 1984).

In the case of downslope propagation, the transition is from a radiative to a trapped state.

Conventional modal formulations fail to adequately explain circumstances in which a mode suddenly disappears from the water column with its energy being radiated into and dissipated within the sea floor (Pierce, 1982). Thus, the particular case of mode coupling where discrete modes (trapped in the water column) couple into continuous modes (which propagate in the bottom) has been further explored as a matter of practical interest (e.g. Miller et al., 1986). Evans and Gilbert (1985) developed a stepwise-coupled-mode method that overcame the failure of previous coupled-mode techniques to properly conserve energy over sloping bottoms. Their method of stepwise-coupled modes avoided problems associated with sloping bottoms by using only horizontal and vertical interfaces. The full solution thus included both forward and backscattered energy (Jensen and Ferla, 1988). Also see the discussion of wedge modes in Chapter 4, Section 4.4.7.

Collins (1990a) suggested using a rotated parabolic equation (PE) in those wedge geometries involving complicated bottom-boundary conditions. Specifically, by rotating the coordinate system, the PE could be marched parallel to the sea floor. The sea surface was then a sloping boundary with simplistic boundary conditions (pressure-release surface) that could be approximated by a sequence of range-independent regions in which the surface was specified as a series of stair steps.

Eigenray formulations can be useful in determining significant propagation paths and propagation mechanisms in a wedge geometry. One particular phenomenon of interest in the upslope problem is that of backscattered eigenrays. These rays have paths that travel up the slope, past the receiver and then back down the slope before arriving at the receiver (Westwood, 1990). The method of images has also been used to construct ray-path solutions in shallow-water environments with a sloping bottom (Macpherson and Daintith, 1967).

The Shallow Water Acoustic Modeling (SWAM) Workshop, held in Monterey, California, in September 1999, provided a forum for the comparison of single-frequency (CW) and broadband (pulse) propagation models in synthetic (i.e. virtual) environments. Test cases included up-sloping, down-sloping, flat and 3D bathymetries. Additional cases considered the effects of internal waves and a shelf break. The goal was to determine which shallow-water environmental factors challenged existing propagation models and what details were important for constructing accurate, yet efficient, solutions. The results of this workshop, designated SWAM '99, were published in a series of papers in the Journal of Computational Acoustics (see Tolstoy et al, 2001).

Jensen (1984) examined both upslope and downslope propagation using an appropriately modified PE model. These results will be discussed below. Other researchers have investigated the wedge problem utilizing normal mode solutions (Evans, 1983; Tindle and Zhang, 1997) and ray theory (Arnold and Felsen, 1983; Westwood, 1989c).

5.3.3.1 Upslope propagation

For the wedge problem involving upslope propagation, Jensen (1984) considered the particular environment illustrated in Figure 5.7. [Note that the material presented by Jensen (1984) is also available in a report by Jensen and Schmidt (1984), which contains the papers presented by these two researchers at the same conference.] The water-bottom interface is indicated on the contour plot by the heavy line starting at 350 m depth and then inclining towards the sea surface beyond a range of 10 km. The bottom slope is 0.85°. The frequency is 25 Hz and the source is located at a depth of 150 m. The sound speed is constant at 1,500 m s-1. The bottom has a sound speed of 1,600 ms-1, a density of 1.5 g cm-3 and an attenuation coefficient of 0.2 dB per wavelength. Shear waves were not considered.

In Figure 5.7, Jensen displayed transmission loss contours between 70 and 100 dB in 2-dB intervals. Thus, high-intensity regions (where the loss is less than 70 dB) appear as black areas while low-intensity regions (where the loss is greater than 100 dB) appear as white areas. The PE solution was started by a Gaussian initial field, and there are four propagating modes. The high intensity within the bottom at ranges less than 10 km corresponds to the radiation of continuous modes into the bottom. As sound propagates upslope, four well-defined beams (numbered 1-4) are seen, one corresponding to each of the four modes. This phenomenon of energy leaking out of the water column as discrete beams has been confirmed experimentally by Coppens and Sanders (1980). These points correspond to the cutoff depths associated with each mode.

5.3.3.2 Downslope propagation

For the wedge problem involving downslope propagation, Jensen (1984) considered the environment illustrated in Figure 5.8. The initial water depth is 50 m and the bottom slope is 5°. The sound speed is constant at 1,500 ms-1. The sound speed in the bottom is 1,600 ms-1 and the attenuation coefficient is 0.5 dB per wavelength. The density ratio between the bottom and the water is 1.5. The contour plot is for a source frequency of 25 Hz. The initial field for the PE calculation was supplied by a normal mode model, and only the first mode was propagated downslope. Shear waves were not considered.

Figure 5.8 shows that some energy propagates straight into the bottom at short ranges. That is, it couples into the continuous spectrum. Beyond the near field, however, propagation within the wedge is adiabatic, with the one mode apparently adapting well to the changing water depth. At a range of 20 km, the energy is entirely contained in the local first mode, even though as many as 21 modes could exist in a water depth of 1,800 m.

Range (km)

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