Chiang C. Mei

Resonant reflection of surface water waves by periodic sandbars

One of the possible mechanisms of forming offshore sandbars parallel to a coast is the wave-induced mass transport in the boundary layer near the sea bottom. For this mechanism to be effective, sufficient reflection must be present so that the waves are partially standing. The main part of this paper is to explain a theory that strong reflection can be induced by the sandbars themselves, once the so-called Bragg resonance condition is met. For constant mean depth and simple harmonic waves this resonance has been studied by Davies (1982), whose theory, is however, limited to weak reflection and fails at resonance. Comparison of the strong reflection theory with Heathershaws (1982) experiments is made. Furthermore, if the incident waves are slightly detuned or slowly modulated in time, the scattering process is found to depend critically on whether the modulational frequency lies above or below a threshold frequency. The effects of mean beach slope are also studied. In addition, it is found for periodically modulated wave groups that nonlinear effects can radiate long waves over the bars far beyond the reach of the short waves themselves. Finally it is argued that the breakpoint bar of ordinary size formed by plunging breakers can provide enough reflection to initiate the first few bars, thereby setting the stage for resonant reflection for more bars.

Scattering of surface waves by rectangular obstacles in waters of finite depth

The scattering of infinitesimal surface waves normally incident on a rectangular obstacle in a channel of finite depth is considered. A variational formulation is used as the basis of numerical computations. Scattering properties for bottom and surface obstacles of various proportions, including thin barriers and surface docks, are presented. Comparison with experimental and theoretical results by other investigators is also made.

The Transformation of a Solitary Wave over an Uneven Bottom

Based on a set of approximate equations for long waves over an uneven bottom, numerical results show that as a solitary wave climbs a slope the rate of amplitude increase depends on the initial amplitude as well as on the slope. Results are also obtained for a solitary wave progressing over a slope onto a shelf. On the shelf a disintegration of the initial wave into a train of solitary waves of decreasing amplitude is found. Experimental evidence is also presented.

The effect of weak inertia on flow through a porous medium

Using the theory of homogenization we examine the correction to Darcys law due to weak convective inertia of the pore fluid. General formulae are derived for all constitutive coefficients that can be calculated by numerical solution of certain canonical cell problems. For isotropic and homogeneous media the correction term is found to be cubic in the seepage velocity, hence remains small even for Reynolds numbers which are not very small. This implies that inertia, if it is weak, is of greater importance locally than globally. Existing empirical knowledge is qualitatively consistent with our conclusion since the linear law of Darcy is often accurate for moderate flow rates.

Slow spreading of a sheet of Bingham fluid on an inclined plane

To study the dynamics of fluid mud with a high concentration of cohesive clay particles, we present a theory for a thin sheet of Bingham-plastic fluid flowing slowly on an inclined plane. The physics is discussed on the approximate basis of the lubrication theory. Because of the yield stress, the free surface need not be horizontal when the Bingham fluid is in static equilibrium, nor parallel to the plane bed when in steady flow. We then show that there is a variety of gravity currents that can advance at a constant speed and with the same profile. Experimental confirmation of one type is presented. By solving a nonlinear partial differential equation, transient flows due either to a steady upstream discharge or to the sudden release of a finite fluid mass on another fluid layer are studied. In the first case there is a mud front which ultimately propagates as a constant speed as a steady gravity current. In the second case, when the ambient layer is sufficiently shallow that there is no initial motion, the flow induced by the new fluid can terminate after the disturbance has travelled a finite distance. The extent of the final spread is examined. Disturbances due to an external pressure travelling parallel to the free surface are also examined. It is found in particular that a travelling localized pulse of pressure gradient not only generates a localized mud disturbance which travels along with the forcing pressure, but further leaves behind a permanent footprint.

A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation

In existing experiments it is known that the slow evolution of nonlinear deep-water waves exhibits certain asymmetric features. For example, an initially symmetric wave packet of sufficiently large wave slope will first lean forward and then split into new groups in an asymmetrical manner, and, in a long wavetrain, unstable sideband disturbances can grow unequally to cause an apparent downshift of carrier-wave frequency. These features lie beyond the realm of applicability of the celebrated cubic Schrodinger equation (CSE), but can be, and to some extent have been, predicted by weakly nonlinear theories that are not limited to slowly modulated waves (i.e. waves with a narrow spectral band). Alternatively, one may employ the fourth-order equations of Dysthe (1979), which are limited to narrow-banded waves but can nevertheless be solved more easily by a pseudospectral numerical method. Here we report the numerical simulation of three cases with a view to comparing with certain recent experiments and to complement the numerical results obtained by others from the more general equations.

On slowly varying Stokes waves.

A WKB-perturbation technique is applied to study the slow modulation of a Stokes wave train on the surface of water. It is found that new terms directly representing modulation rates must be included to extend the scope of validity of Whithams theory based on an averaged Lagrangian. Two examples are discussed. In the first, a monochromatic wave normally incident on a mild beach is studied and the local rate of depth variation is found to affect the wave phase. In the second, the ‘side-band instability’ problem of Benjamin & Feir is discussed from both linear and non-linear points of view.

Radiation and scattering of water waves by rigid bodies

Schwingers variational formulation is applied to the radiation of surface waves due to small oscillation of bodies. By means of Haskinds theorem the wave forces on a stationary body due to a plane incident wave are found using only far-field properties. Results for horizontal rectangular and vertical circular cylinders are presented.

Roll waves on a shallow layer of mud modelled as a power-law fluid

We give a theory of permanent roll waves on a shallow layer of fluid mud which is modelled as a power-law fluid. Based on the long-wave approximation, Karman’s momentum integral method is applied to derive the averaged continuity and the momentum equations. Linearized instability analysis of a uniform flow shows that the growth rate of unstable disturbances increases monotonically with the wavenumber, and therefore is insufficient to suggest a preferred wavelength for the roll wave. Nonlinear roll waves are obtained next as periodic shocks connected by smooth profiles with depth increasing monotonically from the rear to the front. Among all wavelengths only those longer than a certain threshold correspond to positive energy loss across the shock, and are physically acceptable. This threshold also implies a minimum discharge, viewed in the moving system, for the roll wave to exist. These facts suggest that a roll wave developed spontaneously from infinitesimal disturbances should have the shortest wavelength corresponding to zero dissipation across the shock, though finite dissipation elsewhere. The discontinuity at the wave front is a mathematical shortcoming needing a local requirement. Predictions for the spontaneously developed roll waves in a Newtonian case are compared with available experimental data. Longer roll waves, with dissipation at the discontinuous fronts, cannot be maintained if the uniform flow is linearly stable, when the fluid is slightly non-Newtonian. However, when the fluid is highly non-Newtonian, very long roll waves may still exist even if the corresponding uniform flow is stable to infinitesimal disturbances. Numerical results are presented for the phase speed, wave height and wavenumber, and wave profiles for a representative value of the flow index of fluid mud.

The damping of surface gravity waves in a bounded liquid

In deducing the viscous damping rate in surface waves confined by side walls, Ursell found in an example that two different calculations, one by energy dissipation within and the other by pressure working on the edge of the side-wall boundary layers, gave different answers. This discrepancy occurs in other examples also and is resolved here by examining the energy transfer in the neighbourhood of the free-surface meniscus. With due care near the meniscus a boundary-layer–Poincare method is employed to give an alternative derivation for the rate of attenuation and to obtain in addition the frequency (or wave-number) shift due to viscosity. Surface tension is not considered.