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  • 1. Enflo, Bengt
    et al.
    Hedberg, Claes
    Fourier decomposition of a plane nonlinear sound wave developing from a sinusoidal source2001In: Acustica, ISSN 0001-7884, Vol. 87, no 2, p. 163-169Article in journal (Refereed)
    Abstract [en]

    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-Hopftransformation is 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 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. Curves which join smoothly to Fubini's solution (valid up to slightly before shock formation) and to Fay's solution (valid for approximately three shock formation distances). Maxima for the Fourier coefficients of the higher harmonics are given. These maxima are situated in a region where neither Fubini's nor Fay's solution is valid.

  • 2. Hedberg, Claes
    et al.
    Gurbatov, Sergey
    Nonlinear crosstransformation of amplitude-frequency modulation of quasi-monochromatic acoustic signals1998In: Acustica, ISSN 0001-7884, Vol. 84, no 3, p. 414-424Article in journal (Refereed)
    Abstract [en]

    This work is devoted to the investigation of evolution of intense quasi-harmonic signals in the case of infinite acoustic Reynolds numbers. The consideration is based on the zero viscocity limit solution of the Burgers equation, which reduces the Cole-Hopf solution to a "maximum" principle. This limit solution permits an easy way to get the profile of the waves, postition of shocks and their velocities at arbitrary times. The process of transformation of an initial quasi-monochromatic wave into s sawtooth wave is considered. It is shown that the nonlinearity leads to suppression of the initial amplitude modulation and to the transformation of the initial frequency modulation inot a shock amplitude modulation. The amplitude of the low frequency component generated by a quasi-mono-chromatic wave is found. It is shown that the interaction of this component with high frequency waves leads to phase modulation, which increases with distance. The amplitudes of the new components of the spectrum are found. Is is show n that when the value of phase modulation is small, the amplitudes of the satellites do not depend on the distance or the number of harmonics of the primary wave.

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