### ASA 126th Meeting Denver 1993 October 4-8

## 3aSA2. Approximate traveling wave solution for impulsively perturbed
circular membrane.

**Alan Powell
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*Dept. of Mech. Eng., Univ. of Houston, Houston, TX 77204-4792
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It is instructive to transform the normal modes solution for a circular
membrane (radius R) suddenly displaced (xi)[sub 0] at the boundary into an
approximate purely periodic traveling waveform (xi)=(xi)[sub 0](Sigma) xA[inf
n][radical r/R[radical cos[(n -1/4)(pi)(r/R (plus or minus) ct/R) - (pi)/4],
where c=wave speed (although strictly good for n>>1, r/R<<1). The series
evidently has period T=8R/c. Individual mode periods T/3,T/7,T/11,... account
for the notable characteristic symmetry of the waveform about the center of the
periods T and asymmetry about the quarter points. The -(pi)/2 focal phase shift
(finite Hilbert transform) and the simple sign reversal ((plus or minus)(pi)
shift) at the boundary are readily deduced. The result also applies to ideal
supersonic jets, (xi)(implies) pressure perturbation, ct(implies) (axial
distance) [radical M[sup 2]-1[radical . For cell length (axial period) one may
take s[inf M] = 8R[radical M[sup 2]-1[radical for the series, or s[inf P]
=8/3R[radical M[sup 2]-1[radical for the first term. The two-dimensional case
is unambiguous s[sub M]=4(semiwidth) [radical M[sup 2]-1[radical . Of course,
the exact solution is not periodic at all, the major discrepancy being in first
term approximation (4/3 vs 1.307...), so a better approximation is to transform
just terms n(greater than or equal to)2: the characteristic periodic waveform
remains a dominant feature.