# American Institute of Mathematical Sciences

September  2016, 5(3): 463-474. doi: 10.3934/eect.2016014

## Recasting a Brinkman-based acoustic model as the damped Burgers equation

 1 University of New Orleans, Department of Physics, New Orleans, LA 70148, United States, United States

Received  April 2016 Revised  May 2016 Published  August 2016

In order to gain a better understanding of the behavior of finite-amplitude acoustic waves under a Brinkman-based poroacoustic model, we make use of approximations and transformations to recast our model equation into the damped Burgers equation. We examine two special case solutions of the damped Burgers equation: the approximate solution to the damped Burgers equation and the boundary value problem given an initial sinusoidal signal. We study the effects of varying the Darcy coefficient, Reynolds number, and coefficient of nonlinearity on these solutions.
Citation: David Rossmanith, Ashok Puri. Recasting a Brinkman-based acoustic model as the damped Burgers equation. Evolution Equations & Control Theory, 2016, 5 (3) : 463-474. doi: 10.3934/eect.2016014
##### References:
 [1] J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics,, Quarterly of Applied Mathematics, 9 (1951), 225. Google Scholar [2] D. G. Crighton, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11. doi: 10.1146/annurev.fl.11.010179.000303. Google Scholar [3] P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009,, Mechanics Research Communications, 73 (2016), 127. doi: 10.1016/j.mechrescom.2016.02.014. Google Scholar [4] P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena,, Mathematics and Computers in Simulation (MATCOM), 80 (2009), 202. doi: 10.1016/j.matcom.2009.06.004. Google Scholar [5] S. V. Korsunskii, Propagation of nonlinear magnetoacoustic waves in electrically conductlnq dissipative media with drag,, Sov. Phys. Acoust., 37 (1991), 373. Google Scholar [6] W. Malfiet, Approximate solution of the damped Burgers equation,, Journal of Physics A: Mathematical and General, 26 (1993). Google Scholar [7] D. A. Nield and A. Bejan, Convection in Porous Media,, $2^{nd}$ edition, (1999). doi: 10.1007/978-1-4757-3033-3. Google Scholar [8] L. E. Payne, J. Rodrigues and B. Straughan, Effect of anisotropic permeability on Darcy's law,, Mathematical Methods in the Applied Sciences, 24 (2001), 427. doi: 10.1002/mma.228. Google Scholar [9] D. A. Rossmanith and A. Puri, The role of Brinkman viscosity in poroacoustic propagation,, International Journal of Non-Linear Mechanics, 67 (2014), 1. doi: 10.1016/j.ijnonlinmec.2014.07.002. Google Scholar [10] D. A. Rossmanith and A. Puri, Non-linear evolution of a sinusoidal pulse under a Brinkman-based poroacoustic model,, International Journal of Non-Linear Mechanics, 78 (2016), 53. doi: 10.1016/j.ijnonlinmec.2015.09.014. Google Scholar [11] S. I. Soluyan and R. V. Khokhlov, Finite amplitude acoustic waves in a relaxing medium,, Sov. Phys. Acoust., 8 (1962), 170. Google Scholar [12] P. A. Thompson, Compressible-fluid Dynamics,, McGraw-Hill, (1972). Google Scholar

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##### References:
 [1] J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics,, Quarterly of Applied Mathematics, 9 (1951), 225. Google Scholar [2] D. G. Crighton, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11. doi: 10.1146/annurev.fl.11.010179.000303. Google Scholar [3] P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910-2009,, Mechanics Research Communications, 73 (2016), 127. doi: 10.1016/j.mechrescom.2016.02.014. Google Scholar [4] P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena,, Mathematics and Computers in Simulation (MATCOM), 80 (2009), 202. doi: 10.1016/j.matcom.2009.06.004. Google Scholar [5] S. V. Korsunskii, Propagation of nonlinear magnetoacoustic waves in electrically conductlnq dissipative media with drag,, Sov. Phys. Acoust., 37 (1991), 373. Google Scholar [6] W. Malfiet, Approximate solution of the damped Burgers equation,, Journal of Physics A: Mathematical and General, 26 (1993). Google Scholar [7] D. A. Nield and A. Bejan, Convection in Porous Media,, $2^{nd}$ edition, (1999). doi: 10.1007/978-1-4757-3033-3. Google Scholar [8] L. E. Payne, J. Rodrigues and B. Straughan, Effect of anisotropic permeability on Darcy's law,, Mathematical Methods in the Applied Sciences, 24 (2001), 427. doi: 10.1002/mma.228. Google Scholar [9] D. A. Rossmanith and A. Puri, The role of Brinkman viscosity in poroacoustic propagation,, International Journal of Non-Linear Mechanics, 67 (2014), 1. doi: 10.1016/j.ijnonlinmec.2014.07.002. Google Scholar [10] D. A. Rossmanith and A. Puri, Non-linear evolution of a sinusoidal pulse under a Brinkman-based poroacoustic model,, International Journal of Non-Linear Mechanics, 78 (2016), 53. doi: 10.1016/j.ijnonlinmec.2015.09.014. Google Scholar [11] S. I. Soluyan and R. V. Khokhlov, Finite amplitude acoustic waves in a relaxing medium,, Sov. Phys. Acoust., 8 (1962), 170. Google Scholar [12] P. A. Thompson, Compressible-fluid Dynamics,, McGraw-Hill, (1972). Google Scholar
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