Resonantly Interacting Weakly Nonlinear Hyperbolic Waves in the Presence of Shocks: A Single Space Variable in a Homogeneous, Time Independent Medium

We present an overview of averaging method for solving weakly nonlinear hyperbolic systems. An asymptotic solution is constructed, which is uniformly valid in the “large” domain of variables t + |x| ∼ O(ϵ –1). Using this method we obtain the averaged system, which disintegrates into independent equations for the nonresonant systems. A scheme for theoretical justification of such algorithms is given and examples are presented. The averaged systems with periodic solutions are investigated for the following problems of mathematical physics: shallow water waves, gas dynamics and elastic waves. In the resonant case the averaged systems must be solved numerically. They are approximated by the finite difference schemes and the results of numerical experiments are presented.

Download Full-text

Two-Wave Interactions for Weakly Nonlinear Hyperbolic Waves

Studies in Applied Mathematics ◽

10.1002/sapm1993883241 ◽

1993 ◽

Vol 88 (3) ◽

pp. 241-268 ◽

Cited By ~ 7

Author(s):

Yuanping He ◽

T. Bryant Moodie

Keyword(s):

Wave Interactions ◽

Weakly Nonlinear ◽

Hyperbolic Waves

Download Full-text

Asymptotic integration of systems of two wave equations with small parameter

Lietuvos matematikos rinkinys ◽

10.15388/lmr.b.2016.06 ◽

2016 ◽

Vol 57 ◽

Author(s):

Aleksandras Krylovas

Keyword(s):

Small Parameter ◽

Wave Equations ◽

Space Variable ◽

Initial Conditions ◽

Perturbation Technique ◽

Asymptotic Series ◽

Asymptotic Integration ◽

Weakly Nonlinear ◽

Multi Scale ◽

Scale Perturbation

In this article we consider a hyperbolic system of two weakly nonlinear equations. Coefficients of the equations and initial conditions are periodical functions of the space variable. A multi-scale perturbation technique and integrating along characteristics are used to construct asymptotic series without secular members. The scheme of asymptotic integration is applied to analysis of oscillations of nonlinear non-homogeneous strings.

Download Full-text

Weakly-nonlinear Hyperbolic Waves in BZT-fluids

Hyperbolic Problems: Theory, Numerics, Applications ◽

10.1007/978-3-0348-8724-3_7 ◽

1999 ◽

pp. 559-568

Author(s):

Alfred Kluwick

Keyword(s):

Weakly Nonlinear ◽

Bzt Fluids ◽

Hyperbolic Waves

Download Full-text

ASYMPTOTIC METHOD FOR APPROXIMATION OF RESONANT INTERACTION OF NONLINEAR MULTIDIMENSIONAL HYPERBOLIC WAVES

Mathematical Modelling and Analysis ◽

10.3846/1392-6292.2008.13.47-54 ◽

2008 ◽

Vol 13 (1) ◽

pp. 47-54 ◽

Cited By ~ 3

Author(s):

A. Krylovas

Keyword(s):

Asymptotic Method ◽

Hyperbolic Systems ◽

Resonant Interaction ◽

Method Of Averaging ◽

Weakly Nonlinear ◽

One Dimensional ◽

Hyperbolic Waves ◽

Wave Problems ◽

Nonlinear Hyperbolic Systems

A method of averaging along characteristics of weakly nonlinear hyperbolic systems, which was presented in earlier works of the author for one dimensional waves, is generalized for some cases of multidimensional wave problems. In this work we consider such systems and discuss a way to use the internal averaging along characteristics for new problems of asymptotical integration.

Download Full-text

Two-Wave Interactions for Weakly Nonlinear Hyperbolic Waves

Studies in Applied Mathematics ◽

10.1002/sapm1994913275 ◽

1994 ◽

Vol 91 (3) ◽

pp. 275-277

Author(s):

Yuanping He ◽

T. Bryant Moodie

Keyword(s):

Wave Interactions ◽

Weakly Nonlinear ◽

Hyperbolic Waves

Download Full-text

Nonlinear hyperbolic waves

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1988.0062 ◽

1988 ◽

Vol 417 (1853) ◽

pp. 299-308 ◽

Cited By ~ 6

Keyword(s):

Geometrical Optics ◽

Dimensional Theory ◽

Weakly Nonlinear ◽

One Dimensional ◽

Hyperbolic Waves ◽

Scale Length

A theory describing the propagation of nonlinear hyperbolic waves of any strength is developed. It is valid for small values of the wavelength, i. e. of the typical scale length of variation in the direction of propagation. At first the wave propagates along wave normals according to one-dimensional theory. It quickly splits up into a set of distinct waves, each of which soon becomes weak. The weak waves then propagate along rays according to weakly nonlinear geometrical optics.

Download Full-text