Dept. of Mech. Eng., Univ. of Houston, Houston, TX 77204-4792
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.