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

21.3 Progressive Waves in Fluids 621where r represents the radial coordinate. The retarded time is nowτ = t − (r − r 0)c 0where r 0 is a reference distance, which could be the radius of the source, and ahas a value of unity for spherical waves, 1 / 2 for cylindrical waves, and 0 for planewaves (in which case r is replaced by x). Introducing the coordinate stretchingfunctionz = r 0 ln r r 0(spherical waves)and the spreading compensation function= 2( √ rr 0 − r 0 ) (cylindrical waves)w =( rr 0) au (21.9)into Equation (21.8) results in this equation being reduced to the plane wave form∂w∂z = β c02 w ∂w∂τThus, plane wave solutions may be extended to spherical and cylindrical wavesby replacing u and xwith w and z, respectively. The Fubini solution for sphericalwaves has the same form as that of Equation (21.7).One-dimensional propagation is applicable to ducts of slowly varying crosssection, such as horns or ray tubes. In such cases, the spreading compensationfunction is√w =AA 0uDissipative Function: Burgers EquationEquation (21.8) can be useful for a good variety of finite-amplitude problems, butthe distortion nearly always leads to the formation of shocks that are naturallydissipative. The losses must now be taken into account. The first truly successfulmodel was developed by Burgers (1948) originally to model turbulence, but italso turned out to be an excellent approximation of the equation describing finiteamplitudeplane waves of traveling in a thermoviscous fluid. In the form usablefor source problems, the Burgers equation is∂u∂x − β c02 u ∂u∂r =δ2c 3 0∂ 2 u∂t 2 (21.10)