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

2.8 Derivation of the Acoustic Equations 27The above expression can be differentiated with respect to time:1 ∂p= γ ∂ρp 0 ∂t ρ 0 ∂tCombining Equations (2.21) and (2.23),(2.23)∂p= γ p 0 ∂ρ ∂u= γ p 0∂t ρ 0 ∂t ∂xand then differentiating with respect to time t we obtain∂ 2 p∂t = γ p ∂ 2 u2 0∂t∂xDifferentiating Equation (2.22) with respect to x results in∂ 2 p∂x = ρ ∂ 2 u2 0∂x∂tEquating the above two cross-differential terms to each other, as we consider themto be equivalent regardless of their order of differentiation, we obtain the result∂ 2 p∂x = ρ 0 ∂ 2 p2 γ p 0 ∂t = 1 ∂ 2 p(2.24)2 c 2 ∂t 2wherec 2 = γ p 0= γ RTρ 0Here c, R, and T are respectively the propagation velocity of sound, the gasconstant, and absolute temperature of the (ideal) gas. In three-dimensional formthe wave equation (2.17) appears as∇ 2 p = 1 ∂ 2 p(2.25)c 2 ∂t 2We also could have eliminated p in favor of u by reversing the differentiationprocedure between Equations (2.22) and (2.23), in which situation we would get∂ 2 u∂x = 1 ∂ 2 u(2.26)2 c 2 ∂t 2for the one-dimensional situation, and∇ 2 V = 1 ∂ 2 V(2.27)c 2 ∂t 2in the three-dimensional case. It is also a straightforward matter to derive the waveequation in terms of density, resulting in the following expressions:∂ 2 ρ∂x = 1 ∂ 2 ρ(2.28a)2 c 2 ∂t 2for the one-dimensional case and∇ 2 ρ = 1 ∂ 2 ρ(2.28b)c 2 ∂t 2for three dimensions.