A modal Wentzel-Kramers-Brillouin approach to calculating the waveguide invariant for non-ideal waveguides

The frequency dependence of a waveguide's Green's function can be summarized by a single parameter known as the waveguide invariant, β. Although it has been shown analytically that β≈1 for ideal waveguides, numerical and experimental results have shown that β≈1 for many realistic shallow water waveguides as well. There is not much prior work explaining why the non-uniformities present in realistic sound speed profiles sometimes have such a small effect on the value of β. This paper presents a method for calculating β using a modal Wentzel-Kramers-Brillouin (WKB) description of the acoustic field, which reveals a straightforward relationship between the sound speed profile and β. That relationship is used to illustrate why non-uniformities in the sound speed profile sometimes have such a small effect on β and under what circumstances the non-uniformities will have a large effect on β. The method uses implicit differentiation and thus does not explicitly solve for the horizontal wavenumbers of the modes, making it applicable to waveguides with arbitrary sound speed profiles and fluid bottom halfspaces. Several examples are given, including an analytic estimate of β in a Pekeris waveguide.