Introduction

There are several situations in which the methods of the previous chapters cannot be applied or fail outright. In particular, there are cases in which an unperturbed path cannot be identified. This occurs in shadow zone regions, in the finale of long-range deep–water transmissions, and in shallow-water propagation (Figure 1.9). Single-frequency transmissions in which several unperturbed paths add together coherently are a special case of this. There is also the case of long-range propagation for which ray chaos effects can become important and cumulative errors from the ray-tangent approximation become nonnegligible. Finally, there is the issue of attenuation, which is not treated in the previous methodologies and is critical for shallow-water applications.

Figure 8.1 shows the first observations that revealed the limitations of path integral theories for describing acoustic fluctuations. The data are from the 1989, 1000-km, 250-Hz SLICE89 experiment, where scattering into the deep shadow zone near the transmission finale is evident (also see Figures 1.6 and 1.7 showing ensonification of shadow zones for the early arriving time fronts). The SLICE89 observations also revealed that internal-wave-induced scattering leads to a loss of the coherent time front branch pattern in the finale. This branch pattern is needed to identify unperturbed ray paths.

This leads to the method of modal transport equations that have been known for some time outside of the ocean acoustics community. The development of this body of work appears in the physics and chemistry literature (see Van Kampen, 1981, and references therein). Transport equations have been used to describe diverse stochastic phenomena, such as waves in plasmas, Anderson localization, random walks and oscillations, many body problems, and spectral line broadening and narrowing. The key connection to the acoustics problem comes from the fact that the one-way coupled mode equation in Chapter 2 (Eq. 2.128) is in the same form as the Schrödinger equation in the “interaction” representation.

Introduction of the transport theory method to the ocean acoustics community is due to Dozier and Tappert (1978a), whose “Master Equations” for mode energy were far ahead of their time. After a long period of neglect, development of the mode transport equation approach in the last decade has seen remarkable success. Comparisons to Monte Carlo simulations in both deep and shallow-water environments (Dozier and Tappert, 1978b; Colosi and Morozov, 2009; Colosi et al., 2013a, 2011) show that the method is extremely accurate.