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

66 3. Sound Wave Propagation and CharacteristicsInserting Equation (3.57) into Equation (3.56) yieldse k = 1 (∫ ) 2∇ pdt2 ρ 0For a general case of a sine spherical wave described byp = A sin(r − ct)we havee k = 12 ρ 0 c 2 p2 (3.58)From elementary thermodynamics, the change in energy per unit volume V 0 associatedwith the variation of density is given for a volume V of the fluid by∫ V0e p =− 1 pdV (3.59)V0 VThe negative sign indicates that the potential energy increases when compressionoccurs (i.e., when density increases) and decreases with rarefaction (when densitydecreases) under the impetus of an acoustic signal. In order to perform the integrationwe need to express all variables in terms of one variable, namely, instantaneouspressure p. From conservation of mass p 0 V 0 = pV = constant, and differentiatingyieldsFrom Equation (2.23),dV =− V ρdρ ≈−V0ρ 0dρdpdρ = γ p 0ρ 0= c 2for an isentropic process in an ideal gas. Eliminating dρ between the precedingtwo equations givesdV =− V0ρ 0 c dp 2which now can be inserted into Equation (3.59), which is then integrated from 0to p to yielde p = 1 (3.60)2 ρ 0 c 2The sum of Equations (3.58) and (3.60) constitutes the total instantaneous energydensity denoted by ee = 1 )2 ρ 0(u 2 + p2ρ0 2 (3.61)c2Because the particle speed and acoustic pressure are functions of both timeand space, the instantaneous energy density is not constant throughout the fluidp 2