2pPA1. Second-order solutions for arbitrary piston radiators.

Session: Tuesday Afternoon, June 17

Author: Peter J. Westervelt
Location: Dept. of Phys., Brown Univ., Box 1843, Providence, RI 02912


Eckart's equation in the presence of the volume source density q is (open square)[sup 2][(pi)+2((Psi)[sup 2]),[inf 00]]=(Gamma)V,[inf 00]-(rho)[inf 0]c[inf 0]q,[inf 0]. The interaction terms [((rho)[inf 0]-(rho))c[inf 0]q,[inf 0]-qc[inf 0](rho),[inf 0]] are absent, a consequence of (open square)[sup 2](rho)[inf s]=-(rho)[inf 0]c[inf 0]q,[inf 0]. If the approximate first-order solution for a radiator with directivity D[inf 1]((omega),(theta),(phi)), p[inf 1]=P[inf 0]r[inf 0]|r[sup ']|[sup -1]D[inf 1][cos((omega)t-kr+(Psi))]exp[-r(alpha)((omega))] in which r[sup ']=r[inf 0]+ir, r[inf 0]=Rayleigh distance, (Psi)=arctan(rr[inf 0][sup -1]), P[inf 0]=(rho)[inf o]c[inf 0]u[inf 0] with piston velocity u[inf 0], is introduced into Eckart's equation the second-order pressure is obtained: p[inf 2]=(Gamma)(4(rho)[inf 0]c[inf 0][sup 2])[sup -1]P[inf 0][sup 2]D[inf 1][sup 2]kr[inf 0][sup 2]|r[sup ']|[sup -1][(ln|r[sup ']|r[inf 0][sup -1])[sup 2]+(Psi)[sup 2]][sup 1/2][cos(2(omega)t-2kr+(Psi)-(chi))]exp[-2r(alpha)((omega))] in which (chi)=arctan[(Psi)(ln|r[sup ']|r[inf 0][sup -1])[sup -1]]. Matching solutions at r=r[inf 0] yields the term generating fingers: p[inf f]=P[sup *]r[inf 0]r[sup -1]D[inf 2][cos(2(omega)t-2kr+1/4(pi)-(delta))]exp[(r[inf 0]-r)(alpha)(2(omega))-2r[inf 0](alpha)((omega))] in which (delta)=arctan[1/4(pi)(ln[radical 2])[sup -1]], D[inf 2]=D(2(omega),(theta),(phi)), and P[sup *]=(Gamma)(4(rho)[inf 0]c[inf 0][sup 2])[sup -1]P[inf 0][sup 2]kr[inf 0]2[sup -1/2][(ln[radical 2])[sup 2]+(1/4(pi))[sup 2]][sup 1/2]. For rectangular radiators the zero's of D[inf 1] coincide with the zero's of D[inf 2] therefore fingers do not occur in the case. The total pressure p[inf t]=p[inf 2]+p[inf f]. Fingers arise from colinear interaction and have nothing to do with noncolinear scattering of sound by sound which does not exist.

ASA 133rd meeting - Penn State, June 1997