True Amplitude One-Way Wave Equations near Smooth Caustics: An Example

True amplitude one-way wave equations provide full waveform solutions that agree with leading order WKBJ solutions of the full, two-way wave equation away from caustics and other anomalies of the ray field. Both a proof of this fact and numerical examples are available in the literature. Near smooth caustics, ray-theoretic solutions fail for both the one-way and two-way wave equations; the WKBJ amplitude becomes infinite on the caustic although this is not true for the exact solution. For the full wave equation, the asymptotic solution in the neighborhood of smooth caustics is well understood A theory exists for generating asymptotic expansions in terms of Airy functions. These asymptotic solutions having the property that they remain finite on the caustic and accurately represent exact solutions in the neighborhood of the caustic. A corresponding theory for the one-way wave equation has proven to be elusive. Here, however, we present an example in which the one-way wave equation admits the same leading order asymptotic solution in the neighborhood of a smooth caustic as the two-way wave equation. This suggests that the derivation of a ray-theoretic asymptotic solution in the neighborhood of a smooth caustic should ultimately be achieved for the one-way wave equation, as well.