# zbMATH — the first resource for mathematics

A quasi-analytical shock solution for general nonlinear progressive waves. (English) Zbl 1231.35163
Summary: Many physical phenomena are concerned with the propagation of weak nonlinear waves that can be modeled under the form of a generalized Burgers equation. Physical examples include nonlinearities that can be either quadratic (nonlinear acoustical waves in fluids or longitudinal waves in solids), cubic (nonlinear shear waves in isotropic soft solids), or nonpolynomial (Buckley-Leverett equation for diphasic fluids, models for car traffic, Hertz contact in granular media). A new weak shock formulation of the generalized Burgers equation using an intermediate variable called “potential” is proposed. This formulation is a generalization to nonquadratic nonlinearities of the method originally proposed by J.M. Burgers himself [Nederl. Akad. Wet., Proc., Ser. B 57, 159–169 (1954; Zbl 0058.12402)] for his own equation, and later applied to sonic boom applications by W.D. Hayes, R.C. Haefeli and H.E. Kulsrud [“Sonic boom propagation in a stratified atmosphere with computer program”, Aeronautical Research Associates of Princeton, Technical report, NASA CR-1299 (1969)]. It is an elegant way to locate the position of a shock. Its numerical implementation is almost exact, except for an interpolation of Poisson’s solution that can be performed at any order. It is also numerically efficient. As it is exact, a single iteration is sufficient to propagate the information at any distance. It automatically manages waveform distortion, formation of shock waves, and shock wave evolution and merging. The theoretical formulation and the principle of the algorithm are detailed and illustrated by various examples of applications.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 74J30 Nonlinear waves in solid mechanics 35L67 Shocks and singularities for hyperbolic equations 35L70 Second-order nonlinear hyperbolic equations
HE-E1GODF
Full Text:
##### References:
 [1] Baskar, S.; Coulouvrat, F.; Marchiano, R., Nonlinear reflection of grazing acoustical shock waves: unsteady transition from von Neumann to Mach to Snell-Descartes reflections, J. fluid mech., 575, 27-55, (2007) · Zbl 1147.76589 [2] Buckley, S.E.; Leverett, M.C., Mechanism of fluid displacement in sands, Trans. am. inst. MIN. eng., 146, 107-116, (1942) [3] Burgers, J.M., Further statistical problems connected with the solution of a simple nonlinear partial differential equation, Proc. kon. nederlandse akad. Van wet. ser. B, 57, 159-169, (1954) · Zbl 0058.12402 [4] Catheline, S.; Gennisson, J.-L.; Tanter, M.; Fink, M., Observation of shock transverse waves in elastic media, Phys. rev. lett., 91, 16, 164301, (2003), 1-4 [5] Coulouvrat, F.; Marchiano, R., Nonlinear fresnel diffraction of weak shock waves, J. acoust. soc. am., 114, 1749-1757, (2003) [6] Courant, R.; Friedrichs, K.O., Supersonic flow and shock waves, (1948), Springer Berlin · Zbl 0041.11302 [7] Godlewski, E.; Raviart, P.-A., Numerical approximation of hyperbolic systems of conservation laws, (1996), Springer New York · Zbl 1063.65080 [8] Hamilton, M.F.; Blackstock, D.T., Nonlinear acoustics, (1998), Academic Press San Diego · Zbl 0744.73019 [9] W.D. Hayes, R.C. Haefeli, H.E. Kulsrud, Sonic boom propagation in a stratified atmosphere with computer program, Technical report, NASA CR-1299, 1969. [10] Hill, T.G.; Knopoff, L., Propagation of shock-waves in one-dimensional crystal lattices, J. geophys. res., 85, 7025-7030, (1997) [11] Hirsch, C., Numerical computation of internal and external flows (vol. 2). computational methods for inviscid and viscous flows, (1984), John Wiley & Sons Chichester [12] Krehl, P., History of shock waves, (), 1-142 · Zbl 0845.01023 [13] Landau, L., On shock waves at large distances from the place of their origin, J. phys. USSR, 9, 496-500, (1945) [14] Lee-Bapty, I.P.; Crighton, D.G., Nonlinear wave motion governed by the modified Burgers equation, Philos. trans. R. soc. lond. ser. A, 323, 173-209, (1987) · Zbl 0643.35091 [15] Leveque, R.J., Numerical methods for conservation laws, (1992), Birkhauser Basel · Zbl 0847.65053 [16] Leveque, R.J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press Cambridge · Zbl 1010.65040 [17] Marchiano, R.; Coulouvrat, F.; Grenon, R., Numerical simulation of shock wave focusing at fold caustics, with application to sonic boom, J. acoust. soc. am., 114, 1758-1771, (2003) [18] Marchiano, R.; Coulouvrat, F.; Thomas, J.-L., Nonlinear focusing of acoustic shock waves at a caustic cusp, J. acoust. soc. am., 117, 566-577, (2005) [19] Marchiano, R.; Coulouvrat, F.; Thomas, J.-L., Numerical investigation of the properties of nonlinear acoustical vortices through weakly heterogeneous media, Phys. rev. E, 77, 016605, (2008), 1-11 [20] Naugolnykh, K.; Ostrovsky, L., Nonlinear wave processes in acoustics, (1998), Cambridge University Press Cambridge · Zbl 0908.76003 [21] Peyret, R.; Taylor, T.D., Computational methods for fluid flow, (1983), Springer New York · Zbl 0514.76001 [22] Pierce, A.D., Acoustics: an introduction to its physical principles and applications, (1989), Acoustical Society of America New York [23] Poisson, S.D., Mémoire sur la théorie du son, J. ecole polytech. (Paris), 7, 319-392, (1808) [24] Rénier, M.; Barrière, Ch.; Royer, D., Optical measurements of the self-demodulated displacement and its interpretation in terms of radiation pressure, J. acoust. soc. am., 121, 3341-3348, (2007) [25] Tanter, M.; Thomas, J.-L.; Coulouvrat, F.; Fink, M., On the breaking of time reversal invariance in nonlinear acoustics, Phys. rev. E, 64, 016602, (2001), 7 [26] Thompson, P.A., Fundamental derivative in gas dynamics, Phys. fluids, 14, 1843-1849, (1971) · Zbl 0236.76053 [27] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics: A practical introduction, (1999), Springer Berlin · Zbl 0923.76004 [28] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001 [29] Zabolotskaya, E.A.; Hamilton, M.F.; Ilinskii, Y.A.; Douglas Meegan, G., Modeling of nonlinear shear waves in soft solids, J. acoust. soc. am., 116, 2807-2813, (2004) [30] Zabolotskaya, E.A.; Khokhlov, R.V., Quasi-plane waves in the non-linear acoustics of confined beams, Sov. phys. acoust., 15, 35-40, (1969)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.