Corrections to complex ray tracing for modeling ultrasonic beam interaction with single interfaces when the reflection/transmission coefficient vanishes

A high-frequency high-order asymptotic solution is presented to model the interaction of time-harmonic ultrasonic beams with single planar interfaces. The zeroth-order term of the asymptotic expansion yields the well-known ray acoustics solution often used as the basis, for instance, of fast ultrasonic modeling and simulation codes. Whenever the interface reflection or transmission coefficient vanishes in the neighborhood of the specular angle (selected by the stationary-phase argument), the zeroth-order term becomes inaccurate as it underestimates the beam amplitude and yields the wrong phase. The higher-order terms of the asymptotic solution provide a correction to this shortcoming while retaining the computational efficiency of the asymptotic algorithm. This is demonstrated with pulsed beams incident on a solid-solid interface. The asymptotic solution is derived, in a generic form for general applicability, by assuming a plane-wave integral representation of the exact solution to the reflection or transmission problem. A numerical integration of the latter provides reference data to assess the domain of validity of the asymptotic solution. The ultrasonic (quasi-Gaussian) beam generation and propagation are modeled via the complex-source-point technique but other beam schemes can also be used so long as they allow for the asymptotic treatment (i.e., explicit expression for the beam phase and amplitude).