Wave motion in a medium with a cubic nonlinearity

Document type: Conference Papers
Peer reviewed: Yes
Full text:
Author(s): Claes Hedberg, Bengt Enflo, Oleg Rudenko
Title: Wave motion in a medium with a cubic nonlinearity
Translated title: Vågrörelse i ett medium med kubisk olinjaritet
Conference name: 4th Polyakov Readings
Year: 2006
Pagination: 65-74
City: S:t Petersburg
Organization: Blekinge Institute of Technology
Department: School of Engineering - Dept. Mathematics and Science (*** Error ***)
School of Engineering S- 371 79 Karlskrona
+46 455 38 50 00
Authors e-mail: claes.hedberg@bth.se
Language: English
Abstract: An example of wave motion in a medium with a cubic nonlinearity is a transverse finite amplitude wave in an isotropic solid.
The corresponding cubically nonlinear wave equation is derived with the nonlinearity expressed in terms of elastic
constants. This nonlinear wave equation with dissipation is studied for standing and propagating waves. For standing waves
in a resonator a simplified approach results in functional equations, from which frequency response curves are derived.
These curves show the dependence of the amplitude on the difference between one of the resonator's ekgenfrequencies and the
driving frequency. The frequency response curves are plotted for different values of the dissipation and are very different
for quadratic and cubic nonlinearities. In the propagating wave case an N-wave evolution is studied, described by a
modified Burgers' equation with a cubic nonlinearity. Approximate solutions to this equation are found for parts of the wave
profile not studied in detail before.
Subject: Mechanical Engineering\Structural Mechanics
Mechanical Engineering\Structural Dynamics
Keywords: cubic nonlinearity, nonlinear acoustic, nonlinear wave propagation