Opening rate of the transverse cusp diffraction catastrophe in scattering from oblate penetrable spheroids
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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 cscatt/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 Uc 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 q14 ≈ 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.
