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

21.3 Progressive Waves in Fluids 619u(x,t)t = 0 t = t 1t = t 2xc 0 t 1 c 0 (t 2 -t 1 )u(x,t)(a)x = 0 x = x 1 x = x 2 x = x 3t(b)Figure 21.1. Cumulative distortion as seen from the viewpoints of (a) initial-value problem,u as a function of x with increasing time t, and (b) signaling problem, u as a functionof x with increasing x, from the signaling location x = 0.Nonlinear acoustics covers a considerable variety of topics. Problems arisingfrom nonlinearity fall into two categories: (1) source problems, where the timevariation of an acoustic field variable (e.g., pressure or particle velocity) is specifiedat a single location and (2) initial value problems where the spatial variationof the field is specified for a specific time. Figure 21.1 illustrates the cumulativedistortion by linear convection as seen from the viewpoints of (a) initialvalue problems, with u as a function of x at increasing times t, and (b) signalingproblems, where u is a function of t at increasing ranges x, from the signalinglocation.21.3 Progressive Waves in FluidsWhen a single wave moves through a fluid, we have a case of a progressive wavefield. The governing second-order wave equation∂ 2 u∂x = 1 ∂ 2 u2 c02 (21.4)∂t 2