Michael P. Mortell

The evolution of resonant water-wave oscillations

This paper is concerned with the evolution of small-amplitude, long-wavelength, resonantly forced oscillations of a liquid in a tank of finite length. It is shown that the surface motion is governed by a forced Korteweg—de Vries equation. Numerical integration indicates that the motion does not evolve to a periodic steady state unless there is dissipation in the system. When there is no dissipation there are cycles of growth and decay reminiscent of Fermi–Pasta–Ulam recurrence. The experiments of Chester & Bones (1968) show that for certain frequencies more than one periodic solution is possible. We illustrate the evolution of two such solutions for the fundamental resonance frequency.

Pulse Propagation in a Nonlinear Viscoelastic Rod of Finite Length

The problem of impact on a, nonlinear viscoelastic rod of finite length is considered. It is shown that, in the high frequency or geometrical acoustics limit, the disturbance in the rod may be represented as the superposition of two modulated simple waves traveling in opposite directions which do not interact in the body of the material. Based on this result the impact and initial boundary value problems are reduced to finding the solution of a nonlinear difference equation, which is solved exactly. We examine the competing effects of amplitude dispersion, which may produce shocks, and dissipation and give conditions under which each dominates. The corresponding results for an inhomogeneous rod are also given.

Nonlinear resonant oscillations in open tubes

A gas in a tube, one end of which is open, is driven by a periodic applied velocity or pressure at or near a resonant frequency. Damping is introduced into the system by radiation of energy through the open end. It is shown that shocks are possible at an open end and that there is a critical level of damping which ensures a continuous gas response for all frequencies. At the critical level the amplitude of the response is O (e 1/3 ), where e is the amplitude of the input, and it is bounded by the amplitude predicted by linear theory. There is agreement with the qualitative experimental results available.

Mechanism of Synchronization in Frequency Dividers

We investigate the mechanism leading to synchronization in injection-locked frequency dividers using methods of asymptotic analysis. We introduce a response function which allows for qualitative evaluation and intuitive interpretation of the locking phenomenon. We show that the linear asymptotic approximation predicts the locking intervals with high accuracy for a class of models and parameter sets reported in the literature. The accuracy of the approach is evaluated by comparing the theoretical prediction with numerical results. We use phase space analysis to study the case where the limit cycle is dominated by a strongly anharmonic oscillation.

Nonlinear resonant oscillations in closed tubes of variable cross-section

An axisymmetric tube with a variable cross-sectional area, closed at both ends, containing a polytropic gas is oscillated parallel to its axis at or near a resonant frequency. The resonant gas oscillations in an equivalent tube of constant cross-section contain shocks. We show how cone, horn and bulb resonators produce shockless periodic outputs. The output consists of a dominant fundamental mode, where its amplitude and detuning are connected by a cubic equation - the amplitude-frequency relation. For the same gas, a cone resonator exhibits a hardening behaviour, while a bulb resonator may exhibit a hardening or softening behaviour. These theoretical results agree qualitatively with available experimental results and are the basis for resonant macrosonic synthesis (RMS).

The evolution of resonant oscillations in closed tubes

SummaryAn inviscid gas is contained in a tube which is closed at one end and is excited by an oscillating piston at the other end. The motion of the gas is assumed to be one-dimensional and isentropic. The evolution of the gas motion, from the initial rest state to the final periodic state, is examined. The motion is characterised by a similarity parameter. It is shown how shocks form and are sustained within the resonant band, while outside the resonant band shocks form but decay to leave essentially the linear solution.ZusammenhangEin Rohr mit einem geschlossenen Ende ist mit einem reibungsfreien Gas gefüllt und wird durch einen schwingenden Kolben am anderen Ende des Rohres angeregt. Angenommen wird eine eindimensionale isentrope Bewegung des Gases. Es wird die Entwicklung der Bewegung im Gas untersucht, ausgehend vom ursprünglichen Zustand der Ruhe bis zum Erreichen des periodischen Bewegungszustandes. Die Bewegung wird durch einen Ähnlichkeitsparameter gekennzeichnet. Es wird angezeigt wie Stöße sich formen und innerhalb des Resonanzbandes erhalten bleiben, während außerhalb des Resonanzgebietes Stöße zunächst auch entstehen, aber später abklingen, so daß im wesentlichen die lineare Lösung übrigbleibt.

A finite-rate theory of quadratic resonance in a closed tube

Shock waves have been observed travelling in a closed gas-filled tube when the gas is excited by a piston operating at half the fundamental frequency of the tube. Linear theory predicts a continuous periodic solution, while its first correction in a regular expansion is unbounded at such a quadratic resonant frequency. To take account of the intrinsic nonlinearity of travelling waves, a finite-rate theory of resonance is necessary. The periodic motion is then calculated from discontinuous solutions of a functional equation. Two of the three weak-shock conditions and the entropy condition are inherent in the functional equation, and hence the addition of the equal-area rule to fit shocks ensures uniqueness of the solutions.

On chaotic wave patterns in periodically forced steady-state KdVB and extended KdVB equations

The periodically forced KdVB and extended KdVB equations are considered. We investigate the structure of the totality of steady profiles. The existence of profiles that are close to any shuffling of two basic profiles is proved, and hence the existence of spatially chaotic and recurrent solutions. The proofs are based on topological degree theory to analyse chaotic behaviour. These proofs combine ideas suggested by P. Zgliczyński (Zgliczyński 1996 Topol. Methods Nonlinear Anal. 8, 169–177) with the method of topological shadowing. The results are also applicable to the classical problem of a quite general model of a forced nonlinear oscillator with viscous damping.

The evolution of a finite rate periodic oscillation

Abstract Finite rate oscillations of a gas in a closed tube arise when the amplitude of the applied periodic piston velocity is small while its acceleration is unrestricted. The asymptotic form of the periodic motion for large acceleration is given. The evolution to the final periodic motion from the initial state of rest is constructed for a finite rate oscillation. Exact results for a piecewise linear piston velocity are used to illustrate the solutions.

Discontinuous Solutions of a Measure-Preserving Mapping

Area-preserving mappings have received some attention in the recent physics literature (e.g., Chirikov [11]). One reason is that although they arise in deterministic problems, they may exhibit multivalued solutions which appear chaotic. Similar mappings have also arisen in nonlinear acoustics, in the study of the nonlinear oscillatory motion of a gas in a closed tube. In this context the multivalued solutions correspond to the formation of shock waves in the flow. They are made single-valued by inserting discontinuities satisfying the weak shock conditions. For piecewise linear forcing functions we calculate exact discontinuous solutions which contain an infinite number of discontinuities on the unit interval. Two classes of exact solutions are presented, corresponding to first and second order fixed points of the mapping.