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

622 21. Nonlinear Acousticswhereδ = υ[ν + (γ − 1) Pr] = diffusivity of soundυ = μ/ρ = kinematic viscosityμ = shear viscosity coefficientρ 0 = static densityν = 4 3 + μ B= viscosity numberμμ B = bulk viscosity numberPr = c pμ= Prandtl number, a property of the fluidk rc p = specific heat at constant pressure of the fluidk r = thermal conductivity of the fluidOn comparing Equations (21.8) and (21.10) we observe that the former equationdoes not have a dissipation term and it is often called a lossless Burgers equation.The Burgers equation is exactly integrable (Cole and Hopf), which renders ita very useful model. Applications of the Burgers equation have been made tosinusoidal source excitations. Although it is exact, the extraction of the solutionis quite complicated (Hopf, 1950; Blackstock, 1964). For distances larger than3 ¯x(σ >3), it reduces to the Fay solution (Fay, 1931)2 ∑u = u 0Ɣsin nωτsinh n(1 + σ )Ɣ(21.11)where Ɣ = βεκ/α = 1/(α ¯x) that characterizes the importance of nonlinear distortionto the absorption process and α = δω 2 /2c0 3 represents the small signalabsorption at source frequency.In addition to one-dimensional sound waves in thermoviscous fluids, the Burgersequation can also be applied to spherical and cylindrical waves, with the modelequation (21.8) now including the right-hand side of Equation (21.10), i.e.,∂u∂r + u r − βu ∂uc02 ∂r = δ ∂ 2 u2c02 (21.12)∂t 2But Equation (21.12) is not exactly integrable (i.e., no exact solution is known)and numerical methods of solution would therefore be required.Shock WavesSounds from impulsive sources strong enough to result in shock waves includeblast waves from explosions, thunder from lightning, aircraft sonic booms, ballisticmissiles, and N-waves from spark sources.The basic equations describing shock-wave propagation incorporate conservationof mass, Newton’s second law, conservation of energy, and an equation of