IJPAM: Volume 42, No. 2 (2008)

NONLINEAR INTERACTION AND RESONANCE OF
COUNTERPROPAGATING WAVES

Andres Braunbrück$^1$, Arvi Ravasoo$^2$
$^{1,2}$Centre for Nonlinear Studies
Institute of Cybernetics at Tallinn
University of Technology
Akadeemia Tee 21, Tallinn, 12618, ESTONIA
$^1$e-mail: andres@cens.ioc.ee
$^2$e-mail: arvi@ioc.ee


Abstract.A nonlinear partial differential equation with variable coefficients that describes wave motion in inhomogeneous material is considered. By assuming that variable material properties (the density and the elastic coefficient) have a weak variation the equation of motion is solved by making use of the perturbation method and the solution is sought in a series with a small parameter $\varepsilon$. The solution describes initial stage of nonlinear propagation, interaction and reflection of longitudinal waves in weakly inhomogeneous elastic material. The main features of wave interaction are illustrated by comparing solutions in homogeneous and inhomogeneous material, respectively. The analytical solution is studied numerically with the view to clarify the boundary oscillations in terms of the wave induced stress. Extensive numerical computations indicate that the influence of the parameters that describe weak inhomogeneous properties of the material on the amplitude-frequency dependence is close to linear.

Received: August 17, 2007

AMS Subject Classification: 35L70, 81Q15

Key Words and Phrases: nonlinear wave interaction, wave resonance, inhomogeneous material

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 42
Issue: 2