A Hamiltonian approach to horizontal ray tracing in adiabatic normal mode modeling

Abstract

Recently, Arnold and Felsen [J. Acoust. Soc. Am. Suppl. 1 78, S23 (1985)] and Kamel and Felsen [J. Acoust. Soc. Am. 73, 1120–1130 (1983)] have extended adiabatic mode theory to treat trapped‐to‐leaky mode transitions. These papers introduce a new spectral decomposition theory by identifying an adiabatic invariant for the discrete modes and then analytically continuing it to the whole complex horizontal wavenumber plane. The theory is formulated for waveguides composed of isovelocity layers with slowly varying thicknesses. Our work extends Felsen and Kamel's definition of the adiabatic invariant to arbitrary vertical sound‐speed profiles by applying Milne's amplitude and phase representation of Sturm‐Liouville solutions [Phys. Rev. 35, 865–867 (1930)]. This adiabatic invariant was employed to trace horizontal ray paths. In our approach, the horizontal ray paths are identified as the trajectories of a two‐dimensional dynamical system in which the Hamiltonian is the adiabatic invariant and the momentum is the horizontal wavenumber. By analytically continuing that Hamiltonian, ray paths can be defined for modes which pass through cutoff. A numerical model has been developed for horizontal ray tracing in waveguides with arbitrary vertical structure and slow but otherwise arbitrary horizontal variations. With this model the horizontal refraction of adiabatic modes for some waveguides of interest is examined.