# Opening rate of the transverse cusp diffraction catastrophe in scattering from oblate penetrable spheroids

*Copyright 2009 Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America.*

#### Abstract

Sound scattered by an oblate penetrable spheroid should produce a transverse cusp caustic in the region associated with the rainbow in optics (for relative speed of sound *c*scatt/*c* < 1). The principal curvatures of the generic local wave front that produces the far‐field transverse cusp are examined. This wave front is shown to generate a caustic curve (U − Uc)3 = d∝ V3" role="presentation" style="display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">(U − Uc)3 = d∝ V3(U − Uc)3 = d∝ V3, where *U* and *V* are horizontal and vertical scattering angles, and *U*c is the cusp point direction. The far‐field opening rate *d*∝ is calculated for the transverse cusp. It is shown that *d*∝ has a simple dependence on the parameters of the generic wave front. Define the aspect ratio *q* = *D/H*, where *H* is the height and *D* is the equatorial width of the penetrable spheroid. Generalized ray tracing is used to relate *q* to principal curvatures and shape parameters of the outgoing wave front and hence to *d*∝. Measurements of *d*∝ in the optically analogous problem appear to support the calculation. As *q* goes to *q*14 ≈ 1.31, the critical value for the generation of a hyperbolic umbilic focal section, the predicted *d*∝ goes to infinity. The nature of the divergence was numerically investigated as was the rate at which *d*∝ vanishes as *q* approaches other critical values. The analysis suggests benchmarks for testing numerical scattering codes.

*This paper has been withdrawn.*