Frequency- and time-domain asymptotic fields near the critical cone in fluid-fluid configuration

Abstract

Near the critical cone of a point source placed at the interface between two half-space fluid media, investigation is made of the asymptotic fields in the frequency domain and their synthetic wave forms in the time domain. While the leading-order (and the head-wave) components give a good description of the true fields well off the critical cone, the uniform asymptotic (UA) analysis has to be made for the approximation near the critical cone. The UA analysis splits into two cases, depending on the medium densities and wave speeds. For Case 1, the UA1 approximation is employed that takes into account the proximity of the stationary-phase point to the branch point. In Case 2, the UA2 approximation is employed in which consideration is also given to the proximity of the stationary point to the pole and to the combined effect of the stationary point, the branch point, and the pole. The validities of the asymptotic fields are checked in the time domain by comparing the asymptotic field wave forms against the wave-number-synthetic wave forms. The UA fields show good accuracy and causal behaviors, with the causality of the UA2 fields previously unreported in the literature.

Asymptotic Fields in Frequency and Time Domains Generated by a Point Source at the Horizontal Interface Between Vertically Uniaxial Media

“…Especially, the acoustic fluid-fluid problem [16] is in close analogy to the electromagnetic problem in this paper. …”

“…From the asymptotic analyses in [17] and [18] the RHSs are given by [16] (61) where the path runs left to right, parallel to the real axis with a small positive imaginary part. Then the factors in (58)-(60) are evaluated (fully or partially) as [16] (70) …”

A semi-numerical model for near-critical angle scattering

“…Whether a critical zone overlaps a penumbra or not the distance dependence of the scattered field u aðscÞ T has been shown 1,12 to be roughly a À3=4 T , so that in critical zones the difference between the respective correction coefficients [Eqs. (19) and (14)] has to be decreasing as a can be modeled using a relatively fast semi-analytical schedule where for clarity of presentation, the arguments of the diffraction and scattering coefficients have been omitted.…”

“…No high-frequency asymptotics are available when the critical transition zones overlap penumbras, 12,13 that is when all three critical points coalesce, but when they do not the critical zones can be described using asymptotics involving the Weber functions or their variant, the parabolic cylinder functions. 1,11,12,14 In some situations of practical interest none of these asymptotics prove satisfactory. 15 The need to resolve this difficulty has been the main rationale for the present study.…”