TITLE

Fourier decomposition of a plane nonlinear sound wave and transition from Fubini’s to Fay’s solution of Burgers’ equation

AUTHOR(S)
Enflo, Bengt O.; Hedberg, Claes M.
PUB. DATE
July 2000
SOURCE
AIP Conference Proceedings;2000, Vol. 524 Issue 1, p117
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
Burgers' equation describes plane sound wave propagation through a thermoviscous fluid. If the boundary condition at the sound source is given as a pure sine wave, the exact solution is given by the Cole-Hopf transformation as a quotient between two Fourier series. Two approximate Fourier series representations of this solution are known: Fubini's (1935) solution, neglecting dissipation and valid at short distance from the sound source, and Fay's (1931) solution, valid far from the source. In the present investigation a linear system of equations is found, from which the coefficients in a series expansion of each Fourier coefficient can be derived one by one. The Fourier coefficients turn out to be power series in exp(-es), where e is a dimensionless measure of dissipation and s is a dimensionless measure of distance from the boundary. Curves of the Fourier coefficients as functions of s are given for s > 0.9. They join smoothly to Fubini's solution (valid for s < 1 and corrected for dissipation) and to Fay's solution (valid for s ??? 1 ). Maxima for the Fourier coefficients of the higher harmonics as functions of s are given. These maxima are situated in a region where neither Fubini's nor Fay's solution is valid.
ACCESSION #
5665319

 

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