Zhong-Yue Jiang
James F. Greenleaf
Biodynam. Res. Unit, Dept. of Physiol. and Biophys., Mayo Clinic and Foundation, Rochester, MN 55905
Earlier studies [P. J. Westervelt, J. Acoust. Soc. Am. 29, 199--203, 934--935 (1957)] of the mutual nonlinear interaction of two plane waves of sound with each other are extended to include the viscous effect. The viscous effect is considered both from the equations of motion and the equation of state of the medium. An analytical solution to the lowest order scattering process is obtained if the viscous effect of second order and higher can be neglected. In fact, it is shown that the scattered density (rho)[inf s] of two interacting plane waves having the frequencies (omega)[inf 1] and (omega)[inf 2],respectively, satisfies the following equation: (open square)[inf v][sup 2](rho)[inf s]=(open square)[inf v][sup 2]{c[inf 0][sup -2]E[inf 12]+[(2cos(theta) +B/A)/2(omega)[inf 1](omega)[inf 2] (1-cos(theta))](del)[sup 2]W[inf 12] +[DB(c[inf 0][sup 2](omega)[inf 1](omega)[inf 2])[sup -1] / 4Ax(1-cos(theta))[sup 2]](del)[sup 2]((cursive beta)W[inf 12]/(cursive beta)t)+[DB(3+cos(theta))/2Ac[inf 0][sup 4](1-cos(theta))]((cursive beta)V[inf 12]/(cursive beta)t)}, where E[inf 12], V[inf 12], and W[inf 12] are, respectively, the total, potential, and special defined energy densities, D is the sound diffusivity, B and A are nonlinearity parameters, (theta) is the intersecting angle, and (open square)[inf v][sup 2] is a modified d'Alembertian operator. [Work supported by CA43920 NIH.]