: In this paper we derive a method for finding small amplitude high frequency solutions to hyperbolic systems of quasilinear partial differential equations. Our solution is the sum of two parts: (i) a superposition of small amplitude high frequency waves; (ii) a slowly varying 'mean solution'. Each high frequency wave displays nonlinear distortion of the wave profile and shocks may form. Shock conditions are derived for conservative systems. Different high frequency waves do not interact provided the frequencies and wave numbers of two waves are not linearly related to those of a third. The mean solution is found by solving a linear partial differential equation. This method generalizes Whitham's nonlinearization technique 9 for single waves, to problems where many waves are present. We obtain these results by generalizing a scheme first proposed by Choquet-Bruhat 1 which employs the method of multiple scales. (Author)

Weakly nonlinear high frequency waves

: A gas in a tube is excited by a reciprocating piston operating at or near a resonant frequency. Damping is introduced into the system by three means: radiation of energy from one end of the tube, rate dependence of the gas, and boundary layer friction. A lumped damping coefficient is defined to account for the effects of all three. It is shown that in the small rate limit the signal in the periodic state suffers negligible distortion in one travel time, and hence its propagation according to acoustic theory is valid. The shape of the signal is determined by a nonlinear ordinary differential equation. The small rate condition provides a test of the applicability of the theory to given experimental conditions. The resonant band is analyzed for both damped and inviscid cases. The predictions of the theory are compared with experiment.

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Resonant Acoustic Oscillations with Damping: Small Rate Theory

Chapter 2 – Singular Surfaces and Waves

The equations governing the stretching of a nonlinear elastic panel of finite length are integrated in the limit when the strains are small but the strain rates are comparable with the natural frequency of the panel. Such a theory is needed to describe many small amplitude but nonlinear phenomena, such as near-resonant oscillations. It is shown that such a theory must be used when investigating the distortion and decay of free vibrations. These are discussed in detail. A brief discussion of the effect of friction on such vibrations is also given. The mathematical problem reduces to solving nonlinear difference equations and involves functions whose values repeat at a sequence of times which are separated by a time interval which depends on the value repeated. Although the problem is stated in the language of elasticity theory the results are directly applicable to nonlinear optical systems and gas flows in Kundt tubes.

Finite Amplitude Waves in Bounded Media. Nonlinear Free Vibrations of an Elastic Panel

An asymptotic approach is used to analyze the main features of weakly nonlinear waves propagating through an electrically conducting gas permeated by a transverse magnetic field. The analysis leads to an evolution equation, which characterizes the wave process in the high‐frequency domain. The growth equation for an acceleration front is recovered as a special case. The influence of the magnetic field on the decay behavior of a sawtooth profile, which is headed by a weak shock front and ends with a magnetosonic disturbance, is investigated. A remarkable difference between the plane and cylindrical wave profiles is noted; for instance, when the adiabatic index γ is 2, the field does not affect the decay behavior of plane waves but does affect cylindrical waves.

The progressive wave approach analyzing the decay of a sawtooth profile in magnetogasdynamics

A theory is developed describing finite amplitude, high frequency, periodic disturbances in dissipative systems. Although, for definiteness, the transmitting medium is taken to be a viscoelastic string, the results are applicable to nonlinear transmission lines and nonlinear dielectrics, as well as relaxing and reacting gas mixtures. Part I of the paper describes small amplitude but finite rate processes: part II will describe disturbances of unrestricted amplitude. It is shown that by interpreting high frequency waves as modulated simple waves with slowly changing Riemann invariants, the parameter expansion techniques of geometrical optics can be modified to include finite amplitude waves. The roles of the linear models of viscoelasticity and elasticity as well as that of nonlinear elasticity as approximations to the nonlinear viscoelastic model are elucidated.

E. Varley

Papers

Finite Amplitude Waves in Bounded Media. Nonlinear Free Vibrations of an Elastic Panel

High frequency, periodic disturbances in dissipative systems - I. Small amplitude, finite rate theory