E. A. Cox

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.

Nonlinear waves in materials with mixed nonlinearity

Abstract The behaviour of nonlinear waves propagating in materials exhibiting mixed nonlinearity is examined. The effects of geometric spreading, diffraction and caustic formation are considered and model nonlinear evolution equations are derived. Appropriate shock conditions are constructed.

Non-classical kinematic shocks in suspensions of particles in fluids

SummaryThe properties of weakly nonlinear kinematic waves in suspensions are investigated under the assumption that the particle concentration deviates only slightly from the value at the inflexion point of the drift flux curve. Special emphasis is placed on the conditions for the existence of an internal dissipative-dispersive shock structure. The resulting shock admissibility criteria are found to be significantly different from those following from standard theories of kinematic waves. Most interesting, the analysis shows that non-classical shocks which emanate rather than absorb characteristics may be admissible under certain conditions.

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.

Near-critical hydraulic flows in two-layer fluids

This paper deals with the propagation of nearly resonant gravity waves in two-layer flows over a bottom topography assuming that both fluids are incompressible and inviscid. Evolution equations are derived for weakly nonlinear surface-layer and internal-layer waves in the hydraulic limit of infinite wavelength. Special emphasis is placed on the flow regime where the quadratic nonlinear parameter associated with internal-layer waves is small or vanishes. For example, this is the case for all possible density ratios if the velocities in both layers are equal and if the interface height is close to one-half the total fluid-layer height. The waves then exhibit so-called mixed nonlinearity leading in turn to the formation of positive and negative hydraulic jumps. Considerations based on a model equation for the internal dissipative-dispersive structure of hydraulic jumps indicate that the admissibility of discontinuities in this regime depends strongly on the relative magnitudes of dispersion and dissipation. Surprisingly, these admissible hydraulic jumps may violate the wave-speed-ordering relationship which requires that the upstream wave speed does not exceed the propagation speed of the discontinuity. An important example is provided by the inviscid hydraulic jump, which has been known for some time, although its non-classical nature, in that it transmits rather than absorbs waves, has apparently not been recognized before.

Propagation of weakly nonlinear waves in stratified media having mixed nonlinearity

The evolution of small-amplitude finite-rate waves in fluids having high specific heats is studied adopting the assumption that the unperturbed state varies in the propagation direction. It is shown that this not only leads to quantitative changes of the results holding for homogeneous media but also gives rise to new phenomena. Most interesting, shocks are found to terminate at a finite distance from the origin if the fundamental derivative changes sign along the propagation path

Waves and nonclassical shocks in a scalar conservation law with nonconvex

Abstract. The propagation of waves in the nonlinear equation¶¶

Resonant gas oscillations exhibiting mixed nonlinearity

\frac{\partial V}{\partial t} + \left(\Gamma V + \frac{\Lambda V^{2}}{2}\right) \frac{\partial V}{\partial\hat{x}} = \hat{\nu}\frac{\partial^{2}V}{\partial\hat{x}^{2}} + \hat{\beta}\frac{\partial^{3}V}{\partial\hat{x}^{3}}, \quad \hat{\nu}>0, \quad\hat{\beta}\Lambda>0,

The Evolution of Resonant Acoustic Oscillations with Damping

¶¶generates undercompressive shocks in the hyperbolic limit with dispersion and dissipation balanced. These shocks are undercompressive in type and the diversity of phenomena possible is illustrated for three different initial conditions: a propagating shock through a wave fan, a square pulse and a periodic pulse constructed from constant states. The rich variety of wave phenomena exhibited:- shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations produce waves that have no counterpart in classical shock theories. A mechanism for the formation of a nonclassical shock from a classical shock by wavefan interaction is presented.

Classical and Non-Classical Hydraulic Jumps in Two-Layer Fluids

A closed tube containing a BZT-fluid is driven by an applied velocity near and at resonant frequencies. A small-rate theory is shown to predict the existence of stable periodic expansion and compression shocks in a resonant frequency band. The significant effects of wave interaction and thermoviscous damping are demonstrated.