Ray theory for wide classes of sound-speed profiles with two-dimensional variation

Abstract

This paper presents closed-form ray-path and travel-time equations using solutions of the eikonal equation for wide classes of sound-speed profiles which vary with range as well as depth. The geometric intensity equation is also given. In the past, ray theory for two-dimensional sound-speed variation has been treated mainly by numerical methods. Closed-form solutions have been available for only a few profiles of a simple form. In this paper the sound-speed profile is expressed in terms of x and y of a transformed coordinate system in which x and y are functions of depth and range. This transformation and sound-speed function are such that the transformed eikonal equation reduces to a partial differential equation separable in x and y. The closed-form ray equations are calculated in terms of x and y, then the results are converted to and presented in depth and range coordinates. The general characteristics of large classes of allowable transformations are presented. Five specific examples of allowable transformation are presented. Three different profile forms are treated in detail for the polar coordinate transformation. Numerical results, including ray diagrams, are presented for these three forms. These three numerical examples provide controls suitable for test cases testing approximate numerical ray-trace methods.