Journal ArticleDOI
Generalized Burgers equation for plane waves
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In this paper, the generalized Burgers equation is generalized by replacing the thermoviscous term Aut’t with an operator L(u), which represents the effect of attenuation and dispersion, even if known only empirically.Abstract:
Burgers’ equation, an equation for plane waves of finite amplitude in thermoviscous fluids, is generalized by replacing the thermoviscous term Aut’t’ (A is the thermoviscous coefficient, u particle velocity, and t’ retarded time) with an operator L(u). This operator represents the effect of attenuation and dispersion, even if known only empirically. Specific forms of L(u) are given for thermoviscous fluids, relaxing fluids, and fluids for which viscous and thermal boundary layers are important. A method for specifying L(u) when the attenuation and dispersion properties are known only empirically is described. A perturbation solution of the generalized Burgers equation is carried out to third order. An example is discussed for the case α2=2α1, where α1 and α2 are the small‐signal attenuation coefficients at the fundamental and second‐harmonic frequencies, respectively. The growth/decay curve of the second harmonic component is given both with and without the inclusion of dispersion. Dispersion causes a sma...read more
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Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids
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Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method
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Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency.
Wen Chen,Sverre Holm +1 more
TL;DR: A linear integro-differential equation wave model was developed for the anomalous attenuation by using the space-fractional Laplacian operation, and the strategy is then extended to the nonlinear Burgers equation.
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Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian
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A review of the physical properties and biological effects of the high amplitude acoustic fields used in extracorporeal lithotripsy
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