Integral-Equation Method for Burgers Equation with Geometrical-Spreading Effects

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This paper presents a new integral-equation method to solve Burgers equation for nonlinearacoustic waves in a duct spreading geometrically along its axis. The nonlinear integral eqtrationis derived by effecting the Cole-Hopf transforrnation to the Burgers equation and then by apply-lug the Green's function to the equation thus transformed. Under a step initial condition, twotnethods are denaonstrated to solve the integral equation, one being the method of perttrrbationexpansion and the other the unethod of successive approximation. They are exernplified by solv-ing evolution of an acoustic shock wave in a duct whose cross-sectional area changes abruptly inthe forna of a step function. It turns out that the tnethod of perturbation expansion gives riseto the non-uniformity to f'ail in describing a long-tirne and far-field behavior. By contrast, themethod of successive approximation solves the linearized integral equations by iterations infinitetimes to derive exact solvxtions if' the convergence is attained. It is revealed that the methodof successive approximation can yield a uniformly valid solution. Finally there are given somediscussions on the properties of the nonlinear integral eqtration and on a trial approach to exactsolutions by the method of successive approximation.
社団法人日本物理学会の論文
1997-09-15