Summary
The evolution of small amplitude disturbances in a bounded medium, under fixed and nearly fixed end conditions, is considered. The various physical effects accounted for are amplitude dispersion, frequency dispersion and dissipation due to both radiation of energy out of the medium and rate-dependence of the medium. In a nonlinear geometrical acoustics theory the transport equations which determine the signal carried by a component wave have the form of a simple wave equation, Korteweg-de Vries equation, damped simple wave equation and Burgers' equation.
Zusammenfassung
Die Entwicklung von Störungen kleiner Amplitude in einem begrenzten Medium wird untersucht, mit festen und nahezu festen Endbedingungen. Die berücksichtigten physikalischen Effekte sind Amplituden-Dispersion, Frequenz-Dispersion und Dissipation sowohl durch Abstrahlung von Energie aus dem Medium wie auch durch die Deformationsgeschwindigkeit im Medium. In der nicht-linearen geometrischen Akustik ist die Transportgleichung, welche das von einer Wellenkomponente übertragene Signal bestimmt, die einfache Wellengleichung, bezw. die Korteweg-de Vries-Gleichung, die gedämpfte einfache Wellengleichung und die Burgers-Gleichung.
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References
W. Chester,Resonant Oscillations in Closed Tubes, J. Fluid Mech.18, 44–64 (1964).
S. C. Chikwendu andJ. Kevorkian,A Perturbation Method for Hyperbolic Equations with Small Nonlinearities. SIAM J. Appl. Math.,22, 235–258 (1972).
J. D. Cole,Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, Mass., (1968).
M. D. Kruskal andN. J. Zabusky,Stroboscopic Perturbation Procedure for Treating a Class of Nonlinear Wave Equations, J. Math. Phys.,5, 231–244 (1964).
R. W. Lardner,The Formation of Shock Waves in Krylov-Bogoliubov Solutions of Hyperbolic Partial Differential Equations, J. Sound and Vibrations,39, 489–502 (1975).
R. W. Lardner,The Development of Plane Shock Waves in Nonlinear Viscoelastic Media, Proc. Roy. Soc. Lond.,A347, 329–344 (1976).
P. D. Lax,Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations, J. Math. Phys.5, 611–613 (1964).
J. A. Morrison,Comparison of the Modified Method of Averaging and the Two Variable Expansion Procedure, SIAM Review,8, 66–85 (1966)
M. P. Mortell,Nonlinear Periodic Waves in a Bounded Medium: A Regular Perturbation Approach. Lehigh University Technical Report No. CAM-110-10 (1970).
M. P. Mortell,Resonant Oscillations: A Regular Perturbation Approach, J. Math. Phys.,12, 1069–1075 (1971).
M. P. Mortell andB. R. Seymour,Pulse Propagation in a Nonlinear Viscoelastic Rod of Finite Length, SIAM J. Appl. Math.,22, 209–224 (1972).
M. P. Mortell andE. Varley,Finite Amplitude Waves in Bounded Media: Nonlinear Free Vibrations of an Elastic Panel, Proc. Roy. Soc. Lond.A318, 169–196 (1970).
M. Oikawa andN. Yajima,Interaction of Solitary Waves—A Perturbation Approach to Nonlinear Systems, J. Phys. Soc. Japan,34, 1093–1099 (1973).
H. Segur,The Korteweg-de Vries Equation and Water Waves, Solutions of the Equation Part 1, J. Fluid Mech.,59, 721–736, (1973).
B. R. Seymour andM. P. Mortell,Resonant Acoustic Oscillations with Damping: Small Rate Theory, J. Fluid Mech.,58, 353–374 (1973).
B. R. Seymour andE. Varley,High Frequency Periodic Disturbances in Dissipative Systems. I. Small Amplitude Finite Rate Theory, Proc. Roy. Soc. Lond.,A314, 387–415 (1970).
G. B. Whitham,Linear and Nonlinear Waves, Wiley-Interscience, New York (1974).
N. J. Zabusky, fromNonlinear Partial Differential Equations, W. F. Ames, (Ed.), Academic Press, New York, 1967.
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Mortell, M.P. The evolution of nonlinear standing waves in bounded media. Journal of Applied Mathematics and Physics (ZAMP) 28, 33–46 (1977). https://doi.org/10.1007/BF01590706
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DOI: https://doi.org/10.1007/BF01590706