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

620 21. Nonlinear Acousticsmay be integrated at once to yield∂u∂x = 1 ∂u(21.5)c 0 ∂twhich is a first-order differential equation whose solution is u = f (t − x/c). 1Of the two types of problem approaches mentioned in the preceding section,most of the acoustic problems entail radiation, so source problems receive moreemphasis in this chapter.Plane WavesThe model equation for a source-generated finite plane wave in a lossless fluid is∂u∂x = β c02 u ∂u(21.6)∂tConsider a sinusoidal source excitation governed by u = U 0 sin ωt at x = 0. Thesolution to Equation (21.6) is the Fubini solution∑ 2u = U 0nnσ J n(nσ ) sin nωt (21.7)whereσ = βεκx = x¯x or ¯x = 1βεκxHere ¯x is the shock formation distance, ε = U 0 /c 0 , and κ = ω/c 0 is the wavenumber. The size of σ , the so-called dimensionless distortion range, indicates ameasure of the amount of distortion that has occurred. The value of σ = 1 indicatesshock formation.Other One-Dimensional Waves and Ray TheoryIn one-dimensional progressive waves that are not planar, geometric spreadingslows down the rate of distortion and is described mathematically by an extra termin the model equation (21.6). If the wave is spherical or cylindrical, model equation(21.6) becomes∂u∂r + a r = β c02 u ∂u(21.8)r1 We can simplify Equation (21.5) by transforming coordinates x, t to x, τ, where τ = x − c 0 /t, theretarded time. Then Equation (21.5) becomes∂u∂x = 0which provides the building block on which most model equations for more complex progressivewaves are based.