The nonlinear interaction of two plane waves in a viscous medium

Abstract

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 ρs of two interacting plane waves having the frequencies ω1 and ω2, respectively, satisfies the following equation: ⧠2vρs=⧠2v{c0−2E12+[(2cosθ +B/A)/2ω1ω2 (1−cosθ)]∇2W12 +[DB(c20ω1ω2)−1 / 4A ×(1−cosθ)2]∇2(∂W12/∂t)+[DB(3+cosθ)/2Ac40 (1−cosθ)](∂V12/∂t)}, where E12, V12, and W12 are, respectively, the total, potential, and special defined energy densities, D is the sound diffusivity, B and A are nonlinearity parameters, θ is the intersecting angle, and ⧠2v is a modified d’Alembertian operator. [Work supported by CA43920 NIH.]