Ralph Goodman and Hamiltonian acoustics.

Abstract

Much of the underwater acoustic community that has worked with ray theory starts from the eikonal-ray equations. Since these equations, which are solved numerically, can be derived from a high-frequency asymptotic expansion, Fermat’s/Hamilton’s principle is typically bypassed. Historically, the introduction of the Hamiltonian formulation of underwater acoustics ray has come from scientists with a background in classical physics and/or from researchers whose goals were to study complicated environments, chaos, etc., using the well developed formalism of Hamilton’s equations and perturbation theory. Mostly, these latter applications go back to the 1980s. Among Ralph Goodman’s earliest work in underwater acoustics is a paper [R. Goodman and R. B. Duykers, J. Acoust. Soc. Am. 34, 960–962 (1961)] in which he derived the Hamiltonian for convergence zone rays in the harmonic oscillator approximation and then solved the equations by inspection. This paper must be one of the earliest applications of the Hamiltonian formulation to underwater ray acoustics and a part of the beginning of an illustrious career in the underwater acoustics community.