Professor Mike Meylan

The present study investigates the scattering of flexural gravity waves due to uneven bottom topography in the context of wave blocking. Emphasis is given to analyzing the effects of multiple propagating wave modes on the solution procedures. The mathematical model is developed for two scenarios: a bottom step and a submerged rectangular breakwater. For the bottom step case, the complete solution in terms of the velocity potential is obtained using the eigenfunction expansion method. Subsequently, the solution associated with the wave transformation by the bottom step is extended to the case of a submerged rectangular breakwater using symmetry characteristics of the velocity potential. The energy balance relation is derived in both cases using the conservation of energy flux in the presence of multiple propagating wave modes. Wave blocking occurs for four different frequencies in both the cases of the bottom step and the submerged breakwater due to variations in water depth. This makes the problem more complex as, depending on the frequency, multiple propagating wave modes can exist in either the reflected region, the transmitted region, or both. The transmitted wave amplitude associated with the lower wavenumber within the blocking frequencies exceeds unity, and this excess energy is balanced by the corresponding energy transfer rate. Additionally, removable discontinuities are observed at the blocking frequencies in the scattering coefficients, where group velocity ceases. In the context of floating ice sheets, the deflection is analyzed in the time domain for frequencies within and outside the blocking limits.

This study examines the formation of triads of flexural gravity wave in a homogeneous fluid within the context of blocking dynamics due to the presence of shear current. This study will enable us to understand the distribution of wave energy on an ice-covered sea surface. New classes of triads for flexural gravity waves are introduced depending on the direction of wave propagation with following and opposing currents. The study reveals that triad formation occurs due to the interaction of flexural gravity waves irrespective of the presence of compression and current, which has not been found in the case of free surface gravity waves. In addition, at most, three triads are formed in the case of flexural gravity waves in the presence of following and opposing currents prior to the threshold of blocking. In contrast, at least three triads are formed for any frequency within the primary and secondary blocking limits for certain values of compressive force and current speed. On the other hand, 11 triads are formed in the presence of uniform current speed as well as in the case of linear shear current with constant vorticity for a certain frequency within the blocking limit for higher values of compressive force and current speed.

The time-domain motion of a finite depth ocean subject to an arbitrary (in both time and space) imposed displacement of the bottom is studied under the assumption of linear theory. This solution provides results for this limiting case which may be helpful for benchmarking. The focus is on the numerical simulation of the near-field waves with application to the simulation of tsunami waves. The fluid domain is assumed two-dimensional, and the effect of compressibility is included. The time-domain solution is built from the frequency domain solution taking a Fourier series expansion of the bottom motion. This expansion allows complex displacements to be simulated. The solution in the frequency domain is expressed as a sum over modes. The time-domain solution is calculated by numerical evaluation of the Fourier transform in time, allowing arbitrary time-dependent motion. This code is extremely efficient and highly accurate, and there is no time¿stepping so that errors do not accumulate in time. The eigenfunction expansion method to obtain the velocity potential for a flat ocean bottom case is independently derived. A shallow water limit for all the above cases is provided, giving a method to check the correctness of the numerical solution. Separate treatment for all the situations under the compressible assumption is also performed. The horizontal and vertical particle velocities are graphically presented for the time-harmonic oscillation. Time-dependent surface wave propagation is computed to show the initiation of tsunami waves in the deep ocean and their subsequent propagation. The calculations presented here allow for the simulation of tsunami wave generation and to investigate various effects, including the role of acoustic gravity waves. It is shown that the compressibility is not always significant, but that when the water is either sufficiently deep or the rise sufficiently rapid, acoustic gravity waves are produced. It is shown that, in this case, the ocean surface undergoes a rapid oscillation and that this may be a method to detect tsunamis.

An experimental study is conducted to investigate the frequency dependent decay of free-surface water waves in a sloshing tank with partially-submerged floating plastic balls on the free-surface. The present study is motivated by the need to understand the possible mechanisms for ocean wave decay in the marginal ice zone. Laboratory experiments are performed to estimate the wave decay rate due to floating balls representing ice floes. The decay rates with the balls are sufficiently greater than the decay rates without the balls and we assume the effect of the balls dominates the decay. The temporal decay rates of the free-surface wave amplitudes are measured for a small excitation amplitude and different water-depths. The decay rates for the first and third modes of sloshing are extracted, even when these two modes are combined. It is shown that the decay rates obtained from the present study match with the exponent three power-law dependence on wave frequency as observed for the decay rates in the marginal ice zone. This matching of exponent suggests that the same mechanism may be responsible for both types of decay.

The throughput of a belt conveyor system is the primary design parameter when considering a new installation, determined by the cross-section of the material on the belt, coupled with the belt speed. Optimizing this area not only improves efficiency, but also minimizes capital costs through the optimal selection of equipment. Whilst speed-related optimization has seen considerable attention, the cross-sectional area has largely been neglected due to existing design constraints of the system. Conventional belt conveyors typically utilize a 3-idler troughing configuration, which forms a trapezoidal cross-section with a parabolic surcharge. The rigidity of this support directly limits the geometry of the cross-section that may be considered. A new conveyor system developed at the University of Newcastle supports the conveyor belt by a rail-based carriage, with no relative movement between the belt and carriage. This configuration allows the cross-section of the belt to be freely optimized in order to maximize the material throughput for a given belt width, or alternatively to minimize the belt width for a given throughput. This article utilizes the calculus of variations to optimize the form of this cross section, and demonstrates that an increase in throughput of up to 30% is possible, compared to troughed installations.

A hydroelastic model is developed to study the interaction of linear long gravity waves with a very large floating flexible plate resting on a viscoelastic foundation, which is based on the Kelvin-Voigt model. Flexural gravity wave blocking occurs for specific values of the compressive force in the absence of viscous damping. During wave blocking, the group velocity vanishes, and mode swapping occurs. Within wave blocking and plate buckling limit in the presence of compressive force, three distinct propagating modes occur in the absence of viscous damping. Moreover, the study reveals that irrespective of the values of viscous damping constant, the blocking/buckling points shift to a higher wavenumber with an increase in the value of elastic foundation constant. On the other hand, the flexural gravity wave modes become complex in the presence of a viscoelastic foundation. The complex wave modes are classified as predominant progressive wave modes and rapidly decaying modes. In the presence of viscous damping, wave blocking does not happen before the buckling limit of the compressive force. However, the phase velocity vanishes, and the group velocity becomes continuous irrespective of the value of non-zero viscous damping at the buckling limit for the compressive force. The detailed behavior of the roots of the dispersion equation and the mode shapes are illustrated through contour plots and by analyzing the roots' loci. Furthermore, plate deflections are exhibited for different wave and structural parameters.

The motion of a circular elastic plate floating on the surface is investigated in the timedomain. The solution is found from the single frequency solutions, and the method to solve for the circular plate is given using the eigenfunction matching method. Simple plane incident waves with a Gaussian profile in wavenumber space are considered, and a more complex focused wave group is considered. Results are given for a range of plate and incident wave parameters. Code is provided to show how to simulate the complex motion.

Two- and three-dimensional models are proposed for ocean-wave attenuation due to scattering by ice floes in the marginal ice zone, in which the attenuation rate depends on the horizontal size of the individual floes. The scattering models are shown to reproduce the behaviour of wave attenuation over short wave periods. However, it is shown that scattering alone cannot explain the observed asymptotic dependence of attenuation at long wave periods. Based on these findings, it is proposed that attenuation models consist of a scattering component supplemented by an empirical damping term based on measurements, so that attenuation over all periods is correctly modelled. Computer code to calculate wave attenuation through a field of ice floes is provided in the supplementary material.

A solution method is developed for a linear model of ice shelf flexural vibrations in response to ocean waves, in which the ice shelf thickness and seabed beneath the ice shelf vary over distance, and the ice shelf/sub-ice-shelf cavity are connected to the open ocean. The method combines a decomposition of the ice shelf displacement profile at a prescribed frequency of motion into mode shapes of free vibrations, a finite-element method for the cavity water motion and a non-local operator to connect to the open ocean. An investigation is conducted into the effects of ice shelf thickening, seabed shoaling and the grounding-line conditions on time-harmonic ice shelf vibrations, induced by regular incident waves in the swell regime. Furthermore, results are given for ice shelf vibrations in response to irregular incident waves by superposing time-harmonic responses, and ocean-to-ice-shelf transfer functions are derived. The findings add to evidence that ice shelves experience appreciable flexural vibrations in response to swell, and that ice shelf thickening and seabed shoaling can have a considerable influence on predictions of how ice shelves respond to ocean waves.

A plate-wave energy converter (pWEC) moored in front of a floating stationary breakwater is considered. The pWEC is composed of a submerged flexible plate with piezoelectric layers bonded to both faces of it. Hence the elastic motion of the plate excited by water waves can be transformed into useful electricity due to the piezoelectric effect. To evaluate the performance of the breakwater-attached pWEC in terms of wave power absorption and wave attenuation, a hydroelastic model based on linear potential flow theory and the eigenfunction matching method is developed with the electromechanical and the hydrodynamic problems of the pWEC coupled together. The pWEC can be either simply supported or clamped at the edge. A multi-parameter analysis is carried out with the employment of the present model. Effects of the width, submergence and edge types of the plate, together with the scales of the breakwater, including its width and draft, on wave power absorption and wave attenuation, are examined. As the pWEC moves towards a deeper position, the main peaks of the frequency response of the wave power absorption efficiency become lower and narrower. In contrast, its effect on wave attenuation is¿limited.

Bragg scattering of long flexural gravity waves due to an array of submerged trenches is studied under the shallow water approximation and small amplitude structural response in the presence of lateral compressive force. The group velocity vanishes at two different points in the frequency space for specific values of the compressive force, which are referred to as primary and secondary blocking points. Between these two blocking points, three propagating modes exist for each frequency, of which two are associated with the positive group velocity and one with the negative group velocity. Of the three propagating wave modes, the contribution to the energy relation by the lowest wavenumber is predominant near the secondary blocking frequency. In contrast, the higher wavenumber is dominant in the proximity of the primary blocking frequency. The study reveals the occurrence of Bragg scattering of flexural gravity waves in the presence of compressive force for more than two submerged trenches, which is analogous to that of surface gravity waves. However, within the blocking limits of the compressive force, the superposition of multiple propagating wave modes and the change in the incident wave mode contribute to certain irregularities and an increase in wave amplitude in the Bragg reflection pattern. The response amplitude of the structure and the pulse rate increase with an increase in the number of trenches.

The effects of instrument noise on estimating the spectral attenuation rates of ocean waves in sea ice are explored using synthetic observations in which the true attenuation rates are known explicitly. The spectral shape of the energy added by noise, relative to the spectral shape of the true wave energy, is the critical aspect of the investigation. A negative bias in attenuation that grows in frequency is found across a range of realistic parameters. This negative bias decreases the observed attenuation rates at high frequencies, such that it can explain the rollover effect commonly reported in field studies of wave attenuation in sea ice. The published results from five field experiments are evaluated in terms of the noise bias, and a spurious rollover (or flattening) of attenuation is found in all cases. Remarkably, the wave heights are unaffected by the noise bias, because the noise bias occurs at frequencies that contain only a small fraction of the total energy.

The role of a floating elastic plate on the hydrodynamic instability of a gravity-driven flow down an inclined plane is examined in the linear and nonlinear regimes. Linear instability of the system with respect to infinitesimal disturbances is captured using normal-mode analysis. The critical conditions for instability are obtained analytically utilizing the small film aspect ratio. The bifurcation of the nonlinear evolution equation is analyzed using weakly nonlinear stability analysis. The time evolution of the surface elevation is analyzed using the nonlinear analysis. The Orr¿Sommerfeld boundary value problem corresponding to the perturbed flow is derived, and it is solved numerically using the spectral collocation method. The behavior of the marginal stability curves and temporal growth of the unstable waves are portrayed for a range of dimensionless flow parameters. Moreover, the pressure acting on the surface are calculated and analyzed for various structural parameters, applicable to different flow configurations. The study reveals that the structural parameters such as rigidity and mass per unit length play a crucial role in suppressing and facilitating the unstable surface waves of the flow. However, the compressive force acting on the plate results in the flexural destabilization. Thus, the plate parameters are more efficient in damping the shorter wave disturbances. Numerical observations imply that the floating elastic plate assists in stabilizing the free-falling flow and dampens the high amplitude waves.

The problem of ice shelf vibrations forced by incident waves from the open ocean is considered, using a linear hydroelastic model involving combined thin-plate and potential-flow theories. Complex resonances are shown to generate near-resonant ice shelf responses. An efficient algorithm is developed to capture the complex resonances, based on a homotopy, and initialized with the real-valued eigenfrequencies of a problem in which the ice shelf and subshelf water cavity are uncoupled from the open ocean. It is shown the complex resonances may be used to approximate the reflection coefficient in the frequency domain and the large-time ice shelf response to an incident wave packet. Shelf thickening is shown to have a major effect on complex resonances at midrange frequencies, resulting in difficulty exciting certain near resonances in transient problems.

The time-dependent motion of flexural waves generated by the vibration of an elastic plate resting on a variable Winkler foundation with compression is studied. The physical problem is analysed under the assumption of small amplitude structural response in two dimensions. The dispersion equation is analysed in detail, and points of wave blocking are shown to exist. The time-dependent propagation is simulated for the case of both constant and variable properties. For the case of varying properties, it is shown that the wave amplitude near the blocking point for this class of flexural waves satisfies the hyper-Airy differential equation, which is solved analytically in terms of the fourth-order Airy function. Further, the asymptotic solution of the hyper-Airy differential equation is derived by employing the WKB method to match the far-field solution with the near-field solution. The results obtained analytically are compared with a novel time-domain simulation based on a spectral method.

This study investigates the use of a nonlinear model based on the penalisation approach to couple fluid flow and porous media flow. The problem is formulated using a unified Brinkman equation on the domain with a nonlinear permeability which is given a function of porosity, which in turn is governed by an advection equation. The permeability is assumed to be governed by the Kozeny¿Carman equation which relates the permeability with the average grain size and porosity. The model is solved using an adaptive finite element method in space and the method of characteristics in time. Finally, numerical examples are provided to illustrate the model and discuss possible extensions.

We reformulate a three-dimensional theory for wave-ice interactions for flexible ice floes and present a comparison of this new formulation with selected recent parameterizations for the scattering of ocean surface waves due to individual ice floes. The formulation is based on single flow scattering and a transport equation for energy, which fits with the paradigm used in wave prediction code. These parameterizations are implemented as source terms in the action balance equation for a modern version of the phase-averaging wave model WAVEWATCH III® (denoted WW3). In this comparison, a simple experiment is performed with regularly distributed ice floes in a marginal ice zone. With the new wave-ice formulation, results show that attenuation in the direction of propagation is less intense than for the other considered formulations, scattering is more isotropic, and the wave energy is attenuated in the region of the original spectral peak. Thus, a new spectral peak is developed, which is shifted to higher frequencies. The wave scattering and subsequent attenuation are related to the floe response amplitude and the dimensions of the ice floes.

We outline a method to compute the solution in the frequency¿domain for scattering in a waveguide by exploiting symmetry. The method is illustrated by considering a simple scattering example, where soft hard boundary conditions are alternated. We show how the straightforward mode matching or eigenfunction matching solution can be easily converted to scattering and transmission matrices when symmetry is exploited. We then show how the solution for two scatterers can be found explicitly, using symmetry which allows validation of our subsequent solution by scattering matrices. We also give a series of identities which the scattering matrix must satisfy for further numerical validation. Using these frequency¿domain solutions we compute the time-domain scattering by incident Gaussian wave¿packets.

The frequency-domain and time-domain response of a floating ice shelf to wave forcing are calculated using the finite element method. The boundary conditions at the front of the ice shelf, coupling it to the surrounding fluid, are written as a special non-local linear operator with forcing. This operator allows the computational domain to be restricted to the water cavity beneath the ice shelf. The ice shelf motion is expanded using the in vacuo elastic modes and the method of added mass and damping, commonly used in the hydroelasticity of ships, is employed. The ice shelf is assumed to be of constant thickness while the fluid domain is allowed to vary. The analysis is extended from the frequency domain to the time domain, and the resonant behaviour of the system is studied. It is shown that shelf submergence affects the resonant vibration frequency, whereas the corresponding mode shapes are insensitive to the submergence in constant depth. Further, the modes are shown to have a property of increasing node number with increasing frequency.

We present solutions to both trifurcated and pentafurcated spaced waveguides using the mode matching (or eigenfunction expansion) method. While the trifurcated problem with mean fluid flow has been solved previously using the Wiener¿Hopf technique, we solve this problem to validate and demonstrate our method. We then show how we can easily generalize the method to the pentafurcated problem that has not been solved previously. We observe that mode matching method is easier to derive and generalize than the Wiener¿Hopf technique. We also investigate the numerical solution in detail for various geometries to model practical exhaust systems. Copyright © 2015 John Wiley & Sons, Ltd.

The surge, heave and pitch motions of two solitary, thin, floating disks, extracted from laboratory wave basin experiments are presented. The motions are forced by regular incident waves, for a range of wave amplitudes and frequencies. One disk has a barrier attached to its edge to stop the incident waves from washing across its upper surface. It is shown that the motions of the disk without the barrier are smaller than those of the disk with the barrier. Moreover, it is shown that the amplitudes of the motions, relative to the incident amplitude, decrease with increasing incident wave amplitude for the disk without a barrier and for short incident wavelengths. Two theoretical models of the disk motions are considered. One is based on slope-sliding theory and the other on combined linear potential-flow and thin-plate theories. The models are shown to have almost the same form in the long-wavelength regime. The potential-flow/thin-plate model is shown to capture the experimentally measured disk motions with reasonable accuracy.

The sea state of the Beaufort and Chukchi seas is controlled by the wind forcing and the amount of ice-free water available to generate surface waves. Clear trends in the annual duration of the open water season and in the extent of the seasonal sea ice minimum suggest that the sea state should be increasing, independent of changes in the wind forcing. Wave model hindcasts from four selected years spanning recent conditions are consistent with this expectation. In particular, larger waves are more common in years with less summer sea ice and/or a longer open water season, and peak wave periods are generally longer. The increase in wave energy may affect both the coastal zones and the remaining summer ice pack, as well as delay the autumn ice-edge advance. However, trends in the amount of wave energy impinging on the ice-edge are inconclusive, and the associated processes, especially in the autumn period of new ice formation, have yet to be well-described by in situ observations. There is an implicit trend and evidence for increasing wave energy along the coast of northern Alaska, and this coastal signal is corroborated by satellite altimeter estimates of wave energy.

The advent of pixelated detectors for time-of-flight neutron transmission experiments has raised significant interest in terms of the potential for tomographic reconstructions of triaxial strain distributions. A recent publication by Lionheart and Withers [WRB Lionheart and PJ Withers, "Diffraction tomography of strain", Inverse Problems, v31:045005, 2015] has demonstrated that reconstruction is not possible in the general sense; however, various special cases may exist. In this paper, we outline a process by which it is possible to tomographically reconstruct average triaxial elastic strains within individual particles in a granular assembly from a series of Bragg edge strain measurements. This algorithm is tested on simulated data in two and three dimensions and is shown to be capable of rejecting Gaussian measurement noise. Sources of systematic error that may present problems in an experimental implementation are briefly discussed.

An approach for tomographic reconstruction of three-dimensional strain distributions from Bragg-edge neutron transmission strain images is outlined and investigated. This algorithm is based on the link between Bragg-edge strain measurements and the Longitudinal Ray Transform, which has been shown to be sensitive only to boundary displacement. By exploiting this observation we provide a method for reconstructing boundary displacement from sets of Bragg-edge strain images. In the case where these displacements are strictly the result of externally applied tractions, corresponding internal strain fields can then be found through traditional linear-static finite element methods. This approach is tested on synthetic data in two-dimensions, where the rate of convergence in the presence of measurement noise and beam attenuation is examined.

A laboratory experimental model of an incident ocean wave interacting with an ice floe is used to validate the canonical, solitary floe version of contemporary theoretical models of wave attenuation in the ice-covered ocean. Amplitudes of waves transmitted by the floe are presented as functions of incident wave steepness for different incident wavelengths. The model is shown to predict the transmitted amplitudes accurately for low incident steepness but to overpredict the amplitudes by an increasing amount, as the incident wave becomes steeper. The proportion of incident wave energy dissipated by the floe in the experiments is shown to correlate with the agreement between the theoretical model and the experimental data, thus implying that wave-floe interactions increasingly dissipate wave energy as the incident wave becomes steeper. Key Points Wave scattering theory alone is not sufficient to predict attenuation of waves Wave energy is not conserved during wave-ice interactions Turbulent bores at the floes front and rear edges induce dissipation

An experimental model of transmission of ocean waves by an ice floe is presented. Thin plastic plates with different material properties and thicknesses are used to model the floe. Regular incident waves with different periods and steepnesses are used, ranging from gently-sloping to storm-like conditions. A wave gauge is used to measure the water surface elevation in the lee of the floe. The depth of wave overwash on the floe is measured by a gauge in the centre of the floe's upper surface. Results show transmitted waves are regular for gently-sloping incident waves but irregular for storm-like incident waves. The proportion of the incident wave transmitted is shown to decrease as incident wave steepness increases, and to be at its minimum for an incident wavelength equal to the floe length. Further, a trend is noted for transmission to decrease as the mean wave height in the overwash region increases.

An experimental model is used to validate a theoretical model of a sea ice floe's flexural motion, induced by ocean waves. A thin plastic plate models the ice floe in the experiments. Rigid and compliant plastics and two different thicknesses are tested. Regular incident waves are used, with wavelengths less than, equal to, and greater than the floe length, and steepnesses ranging from gently sloping to storm-like. Results show the models agree well, despite the overwash phenomenon occurring in the experiments, which the theoretical model neglects.

A theoretical model and an experimental model of surge motions of an ice floe due to regular waves are presented. The theoretical model is a modified version of Morrison's equation, valid for small floating bodies. The experimental model is implemented in a wave basin at a scale 1:100, using a thin plastic disc to model the floe. The processed experimental data display a regime change in surge amplitude when the incident wavelength is approximately twice the floe diameter. It is shown that the theoretical model is accurate in the high-wavelength regime, but highly inaccurate in the lowwavelength regime.

Wave attenuation coefficients (a, m^-1) were calculated from in situ data transmitted by custom wave buoys deployed into the advancing pancake ice region of the Weddell Sea. Data cover a 12day period as the buoy array was first compressed and then dilated under the influence of a passing low-pressure system. Attenuation was found to vary over more than 2 orders of magnitude and to be far higher than that observed in broken-floe marginal ice zones. A clear linear relation between a and ice thickness was demonstrated, using ice thickness from a novel dynamic/thermodynamic model. A simple expression for a in terms of wave period and ice thickness was derived, for application in research and operational models. The variation of a was further investigated with a two-layer viscous model, and a linear relation was found between eddy viscosity in the sub-ice boundary layer and ice thickness.

The density of states, which measures the density of the spectrum, is evaluated for a platonic crystal (periodically structured elastic plate) using the Green's function approach. Results are presented not only for the standard density of states, but also for the mutual, local, and spectral density of states. These other state functions provide a pathway to the standard density of states and characterize the radiative and other properties of the crystal. This is the first known examination of the density of states for a platonic crystal and extends the existing Green's function approach for photonic crystals to thin, elastic plates. As a motivating example the theory is applied to the problem of a square array of pins embedded in a thin plate. The density of states functions for an empty lattice (a uniform plate) are also presented in order to give a clear illustration of the steps in the derivation. Careful numerical calculations are given which reveal the complex behavior of the crystal, including intervals of suppressed density of states. These results are compared to calculations for a finite crystal with an interior source, and the behaviors of the finite and infinite systems are shown to be connected through the density of states.

In this paper, we compute the band structure for a pinned elastic plate which is constrained at the points of a hexagonal lattice. Existing work on platonic crystals has been restricted to square and rectangular array geometries, and an examination of other Bravais lattice geometries for platonic crystals has yet to be made. Such hexagonal arrays have been shown to support Dirac cone dispersion at the center of the Brillouin zone for phononic crystals, and we demonstrate the existence of double Dirac cones for the first time in platonic crystals here. In the vicinity of these Dirac points, there are several complex dispersion phenomena, including a multiple interference phenomenon between families of waves which correspond to free space transport and those which interact with the pins. An examination of the reflectance and transmittance for large finite gratings arranged in a hexagonal fashion is also made, where these effects can be visualized using plane waves. This is achieved via a recurrence relation approach for the reflection and transmission matrices, which is computationally stable compared to transfer matrix approaches.© 2013 Taylor and Francis.

© 2014 © 2014 Taylor & Francis. A corrected version of the multipole solution for a thin plate perforated in a doubly periodic fashion is presented. It is assumed that free-edge boundary conditions are imposed at the edge of each cylindrical inclusion. The solution procedure given here exploits a well-known property of Bessel functions to obtain the solution directly, in contrast to the existing incorrect derivation. A series of band diagrams and an updated table of values are given for the resulting system (correcting known publications on the topic), which shows a spectral band at low frequency for the free-edge problem. This is in contrast to clamped-edge boundary conditions for the same biharmonic plate problem, which features a low-frequency band gap. The numerical solution procedure outlined here is also simplified relative to earlier publications, and exploits the spectral properties of complex-valued matrices to determine the band structure of the structured plate.

We present a higher-order method to calculate the motion of a floating, shallow draft, elastic plate of arbitrary geometry subject to linear wave forcing at a single frequency. The solution is found by coupling the boundary element and finite element methods. We use the same nodes, basis functions, and maintain the same order in both methods. Two equations are derived that relate the displacement of the plate and the velocity potential under the plate. The first equation is derived from the elastic plate equation. The discrete version of this equation is very similar to the standard finite element method elastic plate equation except that the potential of the water is included in a consistent manner. The second equation is based on the boundary integral equation which relates the displacement of the plate and the potential using the free-surface Green function. The discrete version of this equation, which is consistent with the order of the basis functions, includes a Green matrix that is analogous to the mass and stiffness matrices of the classical finite element method for an elastic plate. The two equations are solved simultaneously to give the potential and displacement. Results are presented showing that the method agrees with previous results and its performance is analysed. © 2004 Elsevier Ltd. All rights reserved.

A fully three-dimensional model for the motion and bending of a solitary ice floe due to wave forcing is presented. This allows the scattering and wave-induced force for a realistic ice floe to be calculated. These are required to model wave scattering and wave-induced ice drift in the marginal ice zone. The ice floe is modeled as a thin plate, and its motion is expanded in the thin plate modes of vibration. The modes are substituted into the integral equation for the water. This gives a linear system of equations for the coefficients used to expand the ice floe motion. Solutions are presented for the ice floe displacement, the scattered energy, and the time-averaged force for a range of ice floe geometries and wave periods. It is found that ice floe stiffness is the most important factor in determining ice floe motion, scattering, and force. However, above a critical value of stiffness the floe geometry also influences the scattering and force.

Under the action of floodwaters, rail ballast can be damaged and even wash away completely. We discuss here a modelling paradigm for this process, especially the solution of the free-boundary problem. Two methods of solution are presented, the first based on the boundary element method and the second based on conformal mappings. We show that for the conformal map, we can derive an explicit solution which involves only matrix multiplication by rewriting the systems of equations as an integral equation. Brief preliminary results are presented.