Transient axial solution for the reflection of a spherical wave from a paraboloidal mirror

Abstract

A method used previously [J. Acoust. Soc. Am. 93, 1256 (1993)] to derive a transient axial solution for a spherical wave reflected from an ellipsoidal mirror is applied to the case of a paraboloidal mirror. The incident spherical wave is radiated from the focus of the mirror. A solution for the impulse response of the reflected axial pressure is obtained in the form h(z,t)= δ(t−z/c0)−he(z)δ[t−te(z)]−(c0/zF)hw(z,t), where δ is the Dirac delta function, c0 is sound speed, z is axial distance from the base of the mirror, zF is distance to the focus, he is the relative amplitude of the edge wave, te its relative time of arrival, and hw is the wake. Simple expressions are obtained for he and hw. Beyond the focus, the geometrical acoustics result he∼(1+d/zF)−1 is recovered for the edge wave, where d is the mirror depth. In the far field, hw becomes a delta function, the impulse response reduces to h(z, t)∼(2z2F/c0z)ln(1+d/zF)δ’(t−z/c0), and the derivative of the source waveform is thus obtained. Calculations for various source waveforms are presented. Related measurements are discussed in the following presentation by Gelin etal. (Paper 1pPA5). [Work supported by ONR.]