### ASA 128th Meeting - Austin, Texas - 1994 Nov 28 .. Dec 02

## 1pPA4. Transient axial solution for the reflection of a spherical wave
from a paraboloidal mirror.

**Mark F. Hamilton
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*Dept. of Mech. Eng., Univ. of Texas at Austin, Austin, TX 78712-1063
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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)=(delta)(t-z/c[sub 0])-h[sub e](z)(delta)[t-t[sub e](z)]-(c[sub 0]/z[sub
F])h[sub w](z,t), where (delta) is the Dirac delta function, c[sub 0] is sound
speed, z is axial distance from the base of the mirror, z[sub F] is distance to
the focus, h[sub e] is the relative amplitude of the edge wave, t[sub e] its
relative time of arrival, and h[sub w] is the wake. Simple expressions are
obtained for h[sub e] and h[sub w]. Beyond the focus, the geometrical acoustics
result h[sub e]~(1+d/z[sub F])[sup -1] is recovered for the edge wave, where d
is the mirror depth. In the far field, h[sub w] becomes a delta function, the
impulse response reduces to h(z,t)~(2z[sub F][sup 2]/c[sub 0]z)ln(1+d/z[sub
F])(delta)'(t-z/c[sub 0]), 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 et al. (Paper
1pPA5). [Work supported by ONR.]