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

21.3 Progressive Waves in Fluids 625Analytical studies have been made of cylindrical explosions and line sourceenergy releases (Lin, 1954; Plooster, 1970, 1971). Lin was working on the problemof shock generation by meteors or missiles, whereas Plooster was studying thedevelopment and decay of cylindrical shocks with more realistic initial energydistributions and atmospheres. A strong shock estimate of the decay of cylindricalwaves from an instantaneous line source in air (γ = 1.4) at standard pressure isgiven byp = 0.216Er 2Thus, a cylindrical decay proportional to r –2 is characteristic of cylindrical strongshocks.Nonlinearity in SolidsThe propagation of ultrasonic waves of finite amplitudes in a crystal of cubicsymmetry can be analyzed with a nonlinear differential equation similar to thatused for fluids.The propagation of a finite-amplitude ultrasonic wave in any direction in anisotropic solid is given by∂ 2 ξρ 0∂t = κ ∂ 2 ξ2 2∂a + (3κ 2 2 + k 3 ) ∂2 ξ ∂ξ(21.24)∂a 2 ∂awhere ξ is the particle displacement and a is the distance measured in the directionof propagation. κ 2 and κ 3 represent elastic constants that depend on the directionof propagation in the solid.The solution of the nonlinear equation (21.24) is made by assuming that the waveis initially sinusoidal at a = 0, with ξ = A 1 sin (ka – ωt). On this assumption wecan obtain a perturbation solution in the formξ = A 1 sin(ka − ωt) + β A2 2 k2 acos 2(ka − ωt) +···. (21.25)4where β is the nonlinear parameter. The negative ratio of the coefficient of thenonlinear term in Equation (21.24)2β =−(3 + κ 3k 2)is the quantity to be determined from measurement. From 2β we can establish κ 3since κ 2 = ρc0 2 is known.Equation (21.25) indicates that an initially sinusoidal ultrasonic wave generatesa second harmonic as it propagates. In order to measure the nonlinear parameter2β, the absolute value of the fundamental displacement amplitude A 1 needs to bemeasured and also that of the second harmonicA 2 = β A2 1 k2 a4