Finite-Amplitude Standing Acoustic Waves in a Cubically Nonlinear Medium

Document type: Journal Articles
Article type: Original article
Peer reviewed: Yes
Full text:
Author(s): Oleg Rudenko, Claes Hedberg, Bengt Enflo
Title: Finite-Amplitude Standing Acoustic Waves in a Cubically Nonlinear Medium
Translated title: Stående vågor av finit amplitud i ett kubiskt olinjärt medium
Journal: Acoustical Physics
Year: 2007
Volume: 53
Issue: 4
Pagination: 455-464
ISSN: 1063-7710
Publisher: MAIK Nauka Interperiodica
ISI number: 000248046600005
Organization: Blekinge Institute of Technology
Department: School of Engineering - Dept. of Mechanical Engineering (*** Error ***)
School of Engineering S- 371 79 Karlskrona, School of Engineering S- 371 79 Karlskrona
+46 455 38 50 00
Authors e-mail:,,
Language: English
Abstract: The behavior of the wave field in a resonator containing a cubically nonlinear medium is studied. The field is constructed as a linear superposition of two counter-propagating and strongly distorted waves. As distinct from a quadratic nonlinear medium the waves traveling in opposite directions are connected through their averaged (over the period) intensities. Both free and forced standing waves are studied. Profiles of discontinuous vibrations containing shock fronts of both compression and rarefaction are constructed. Resonant curves depicting the dependence of mean intensity on the difference between the frequency of vibration of the boundary and the natural frequency of one of the resonator’s mode are calculated. The structure of temporal profiles of strongly distorted forced waves is analyzed. It is shown, that shocks can appear only if the difference between the mean intensity and the discrepancy takes on definite negative values. The discontinuities are studied as jumps between the different solutions of a nonlinear integro-differential equation degenerating at weak dissipation to a third order algebraic equation with an undetermined coefficient. The dependence of the intensity of shocked vibrations on the frequency of vibration of the boundary is found. Nonlinear saturation is shown to appear: the intensity of wave field inside the resonator does not depend on the amplitude of boundary vibration when the amplitude is large. If the amplitude tends to infinity, the intensity tends to its limiting value determined by the nonlinear absorption at shock fronts. This maximum can be reached by smooth increase in frequency above the linear resonance. A hysteresis area and bistability appears, in analogy with the nonlinear resonance phenomena in localized vibration systems described by ordinary differential equations.
Summary in Swedish: Resultat av stående akustiska vågor i en kubisk olinjär resonator presenteras.
Subject: Mechanical Engineering\General
Keywords: nonlinear acoustics, standing waves, acoustic resonator, cubic nonlinearity