Consider a sound wave propagating through a static, inhomogeneous fluid with no gravity. Explain why the unperturbed fluid has velocity v = 0 and pressure P_o = constant, but can have variable density and sound speed, ρ_o(x) and C(x, t). By repeating the analysis in Eqs. (16.47)–(16.50), show that the wave equation is ∂²δP/∂t² = C²ρ_o ∇· (ρ_o⁻¹∇δP), which can be rewritten as

where W = (C²ρ_o)⁻¹.

Equation (16.54) is an example of the prototypical wave equation (7.17) that we used in Sec. 7.3.1 to illustrate the geometric-optics formalism. The functional form of W and the placement of W and C² (inside versus outside the derivatives) have no influence on the wave’s dispersion relation or its rays or phase in the geometric-optics limit, but they do influence the propagation of the wave’s amplitude. See Sec. 7.3.1.

Equations

W asP at - V. (WCV8P) = 0, (16.54)