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Fourier Decomposition of a Plane Nonlinear Sound Wave Developing from a Sinusoidal Source

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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 given by the Cole-Hopf transformation is a quotient between two Fourier series. Two approximate Fourier series representations of this solution are known: Fubini's [1] solution, neglecting dissipation and valid at short distance from the sound source, and Fay's 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(-ɛσ), where ɛ is a dimensionless measure of dissipation and σ is a dimensionless measure of distance from the boundary. Curves of the Fourier coefficients as functions of σ are given for σ > 0.9. They join smoothly to Fubini's solution (valid for σ < 1 and corrected for dissipation) and to Fay's solution (valid for σ ≫ 1). Maxima for the Fourier coefficients of the higher harmonics are given as functions of σ. These maxima lie in a region where neither Fubini's nor Fay's solution can be used.
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Document Type: Research Article

Publication date: March 1, 2001

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  • Acta Acustica united with Acustica, published together with the European Acoustics Association (EAA), is an international, peer-reviewed journal on acoustics. It publishes original articles on all subjects in the field of acoustics, such as general linear acoustics, nonlinear acoustics, macrosonics, flow acoustics, atmospheric sound, underwater sound, ultrasonics, physical acoustics, structural acoustics, noise control, active control, environmental noise, building acoustics, room acoustics, acoustic materials, acoustic signal processing, computational and numerical acoustics, hearing, audiology and psychoacoustics, speech, musical acoustics, electroacoustics, auditory quality of systems. It reports on original scientific research in acoustics and on engineering applications. The journal considers scientific papers, technical and applied papers, book reviews, short communications, doctoral thesis abstracts, etc. In irregular intervals also special issues and review articles are published.
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