THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

618 21. Nonlinear Acousticsby∇ 2 φ = 1 c 2 0where φ is the velocity potential, t is the time, and c 0 is the small signal soundspeed listed in Tables A, B, and C of Appendix A. In linear acoustics a sound waveordinarily propagates through a medium without changing its shape, because eachpart on the wave travels with the same speed c 0 = dx/dt.In the case of the finite wave, the propagation speed varies from point to point.The variation occurs because a propagating wave engenders a longitudinal velocityfield u in the fluid medium through which it travels. The motion of the fluid addsto the propagation speed with respect to a fixed location:dx= c + u (21.1)dtwhere c represents the speed of sound with respect to the moving fluid (and it isnot the same as c 0 ). To understand this situation better, consider a fluid that is a gasfor which the speed varies as √ T [cf. Equation (2.2)]. The sound speed becomesa bit higher when the acoustic pressure p is positive (i.e., during the compressionphase that increases with the temperature) and a bit lower when p is negative (i.e.,during the expansion phase that lowers the temperature). Thenc = c 0 + γ − 1 u (21.2)2where γ is the ratio of specific heats of the gas. Combining Equations (21.1) and(21.2) results indx= c 0 + βu (21.3)dtwhere β, the coefficient of nonlinearity is given byβ = γ + 12From Equation (21.3) we observe that the propagation speed depends on the particlevelocity, as the result of the convective effect of the moving fluid and thenonlinearity of the traveling wave. Even in nonlinear acoustics, the particle velocityu is normally much smaller than c 0 . The impact of the varying propagationspeed is accumulative and leads to appreciable distortion that becomes even greaterwith increasingly stronger waves.Nonlinearity of the pressure–density relation is considerably more prominentin liquids and solids than for gases. In the case of liquids the coefficient of thefirst-order nonlinear term in the pressure–density relation is B/2A, where B/A istermed the parameter of nonlinearity. In this case, the analog of Equation (21.1)for liquids is given by∂ 2 φ∂t 2β = 1 + B2A