Abstract
Asymptotic methods are used to find approximate solutions of the acoustic wave equation in a medium in which the velocity is a continuously variable function of one coordinate. It is shown that, when the velocity function has a minimum, undamped normal mode solutions exist and are closely analogous to the internally reflected waves in the case of a medium made up of discrete layers. By converting the sum of the high order normal modes into an equivalent integral, it is shown that superposition of these modes leads to geometrical ray theory modified by diffraction in a manner that may be computed from the incomplete fresnel and airy integrals.
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