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Summary Two popular and computationally inexpensive class of methods for approximating the propagation of surface waves over two-dimensional variable bathymetry are ‘step approximations’ and ‘depth-averaged models’. In the former, the bathymetry is discretised into short sections of constant depth connected by vertical steps. Scattering across the bathymetry is calculated from the product of $2 \times 2$ transfer matrices whose entries encode scattering properties at each vertical step taken in isolation from all others. In the latter, a separable depth dependence is assumed in the underlying velocity field and a vertical averaging process is implemented leading to a second-order ordinary differential equation (ODE). In this article, the step approximation is revisited and shown to be equivalent to an ODE describing a depth-averaged model in the limit of zero-step length. The ODE depends on how the solution to the canonical vertical step problem is approximated. If a shallow water approximation is used, then the well-known linear shallow water equation results. If a plane-wave variational approximation is used, then a new variant of the mild-slope equations is recovered.

The theoretical limit for absorption of energy in monochomatic water waves of wavelength $\lambda$ by axisymmetric wave energy converters operating in rigid-body motion was established in the 1970s. The maximum mean power generated by a device absorbing due to heave motion is equivalent to that contained in $\lambda/2\upi$ length of incident wave crest. For devices absorbing through surge and/or pitch motions the so-called capture width doubles to $\lambda/\upi$. For devices absorbing in both heave and surge/pitch the capture width increases further to $3\lambda/2\upi$. In this paper it is demonstrated it is theoretically possible to extend the capture width for axisymmetric wave energy converters without bound through the use of generalised (non-rigid body) modes of motion. This concept will be applied to vertical cylinders whose surface is surrounded by an array of narrow vertical absorbing paddles.A continuum approximation is made to the paddle motion which simplifies theproblem and allows strategies to be developed for setting the springs and dampers that control the power absorption. Results demonstratethe main result: a cylinder of fixed size can absorb as much power asdemanded from a plane incident wave although there are practical considerations which are also discussed. The continuum approximation is tested against a discrete paddle simulation for accuracy.

In this paper, a semi-analytical model based on linear potential flow theory and an eigenfunction expansion method is developed to study wave scattering by an array of structured cylinders in water of finite depth. Each cylinder is formed by a closely spaced array of thin vertical plates, between which fluid can flow, extending through the depth. In order to consider the wave attenuation and energy dissipation in narrow gaps between the thin vertical plates, a damping mechanism is introduced at the surface of the fluid occupied by the structured cylinders. In addition to a direct calculation of the energy dissipation, an indirect method based on Kochin functions is derived with the employment of energy identities. The present model is shown to be in excellent agreement with both the published data and those obtained by using different methods. The validated model is then applied to study the effect of a pair of structured cylinders on wave focusing/blocking, scattered far-field amplitude and wave power dissipation. Results show that wave focusing/blocking can be achieved by the appropriate choice of plate alignment. The structured cylinders hold profound potential for wave power extraction.

An extension to the classical shallow-water equation (SWE) is derived that exactly satisfies the bed condition and can be regarded as an approximation to wave scattering at the next order in the small parameter $(h/\unicode[STIX]{x1D706})^{2}$ (depth to wavelength ratio squared). In the frequency domain, the extended SWE shares the same simple structure as the standard SWE with coefficients modified by terms relating to the bed variation. In three dimensions the governing equation demonstrates that variable topography gives rise to anisotropic effects on wave scattering not present in the standard SWE, with consequences for the design of water wave metamaterials. Numerical examples illustrate that approximations to wave scattering using the extended SWE are significantly improved in comparison with the standard SWE.

A linear theory, based on wide-spacing and high-frequency approximations, is developed to describe resonant behaviour in two-dimensional water-wave problems involving a freely floating half-immersed cylinder in the presence of a vertical rigid wall. The theory is not able to describe the lowest-frequency resonance, but otherwise yields explicit approximations for the locations of resonances in the complex plane and for their corresponding residues. Two problems are investigated in detail: the time-domain motion following a vertical displacement of the cylinder from equilibrium, and the time-harmonic motion of a cylinder that is excited by an incident plane wave.

It is shown that localised wave motions (often referred to edge waves or trapped modes) are capable of being supported by two complementary arrangements involving floating ice on water: a finite width ice floe of constant thickness floating on open water; and an open water channel – or lead – embedded in an ice-covered ocean. The search for such solutions is motivated by a simple observation, evidently not made before, that wavelengths of propagating waves in thin ice floe can be either greater or less than those of the same frequency on an unloaded water surface depending onphysical parameters in the problem. The existence of edge waves are confirmed by accurate computations of solutions to integral equations derived from the underlying boundary-value problems using Fourier transform methods.

The transmission of acoustic waves along a two-dimensional waveguide whichis coupled through an opening in its wall to a rectangular cavity resonator is considered. The resonator acts as a classical band-stop filter, significantlyreducing acoustic transmission across a range of frequencies. Assuming wavefrequencies below the first waveguide cut-off, the solution for the reflectedand transmitted wave amplitudes is formulated exactly within the frameworkof inviscid linear acoustics. The main aim of the paper is to develop anapproximation in closed form for reflected and transmitted amplitudes whenthe gap in the thin wall separating the waveguide and the cavity resonatoris assumed to be small. This approximation is shown to accurately capturethe effect of all cavities resonances, not just the fundamental Helmholtz resonance. It is envisaged this formula (and more generally the mathematicalapproach adopted) could be used in the development of acoustic metamaterialdevices containing resonator arrays.

The problem of the transmission of plane waves along a uniform parallel-walledwaveguide or channel through a periodically-spaced array of thin screens with gaps placed across the channel is investigated. Of particular interest is the existence of frequencies at which energy it totally transmitted – shown to occur when there are two or more screens – and how they depend upon the size of the gaps in the screen. Attention focuses on small gaps where the phenomenon of total transmission persists and where an approximate analysis of a system of full coupled integrals equations is reduced to simple linear difference equations which can be solved in closed form. Results show comparisons between exact computations, the small-gap computations and corresponding wide-spacing approximations. A discussion of total reflection is also included when the gaps are no longer assumed to be small.

The paper describes a process which allows a vertical circular cylinder subject to plane monochromatic surface gravity waves to appear invisible to the far-field observer. This is achieved by surrounding the cylinder with an annular region of variable bathymetry. Two approaches are taken to investigate this effect. First a mild-slope approximation is applied to the governing linearised three-dimensional water wave equations to formulate a depth-averaged two-dimensional wave equation with varying wavenumber over the variable bathmetry. This is then solved by formulating a domain integral equation, solved numerically by discretisation. For a given set of geometrical and wave parameters, the bathymetry is selected by a numerical optimisation process and it is shown that the scattering cross-section is reduced towards zero with increasing refinement of the bathymetry. A fully three-dimensional boundary-element method, based on the WAMIT solver (see www.wamit.com) but adapted here to allow for depressions in the bed, is used to assess the accuracy of the mild-slope results and then further numerically optimise the bathymetry towards a cloaking structure. Numerical results provide strong evidence that perfect cloaking is possible for the fully three-dimensional problem. One practical application of the results is that cloaking implies a reduced mean drift force on the cylinder.

In this paper connections are made between the solutions of two water wave scattering problems, namely the diffraction of oblique waves by a thin vertical barrier with gaps and the complementary problem where the barriers are interchanged with the gaps. It is shown that the potential everywhere for the barrier problem is expressible in terms of the potential for the gap problem and a connection potential also associated with gaps in barriers. As a result the reflection coefficients are also shown to be connected. The theory is illustrated in two ways. First, by analytically deriving Ursell's (Q. J. Mech. Appl. Math. 1 (1947)) explicit result for a surface-piercing barrier in infinite depth from Dean's (Proc. Camb. Phil. Soc. 41 (1945)) explicit result for a submerged barrier in infinite depth. Secondly, numerical results for complementary arrangements of barriers and gaps in finite depth and under oblique wave incidence are presented.This article is dedicated to the memory of Professor Fritz Ursell who died in 2012.

A novel design concept for a wave energy converter (WEC) is presented and analysed. Its purpose is to balance the theoretical capacity for power absorption against engineering design issues which plague many existing WEC concepts. The WEC comprises a fully submerged buoyant circular cylinder tethered to the sea bed by a simple mooring system which permits coupled surge and roll motions of the cylinder. Inside the cylinder a mechanical system of pendulums rotate with power generated by the relative rotation rates of the pendulums and the cylinder. The attractive features of this design include: making use of the mooring system as a passive component of the power take off (PTO); using a submerged device to protect it from excessive forces associated with extreme wave conditions; locating the PTO within the device and using a PTO mechanism which does not need to be constrained; exploiting multiple resonances of the system to provide a broad-banded response. A mathematical model is developed which couples the hydrodynamic waves forces on the device with the internal pendulums under a linearized framework. For a cylinder spanning a wave tank (equivalent to a two-dimensional assumption) maximum theoretical power for this WEC device is limited to 50 % maximum efficiency. However, numerical results show that a systematically optimized system can generate theoretical efficiencies of more than 45 % over a 6 s range of wave period containing most of the energy in a typical energy spectrum. Furthermore, three-dimensional results for a cylinder of finite length provide evidence that a cylinder device twice the length of its diameter can produce more than its own length in the power of an equivalent incident wave crest.

Under the assumptions of the linearized theory of small-amplitude water waves, it is proved that plane waves, normally incident upon a semi-immersed cylinder of uniform circular cross-section floating freely on the surface of a fluid of infinite depth, are capable of being totally reflected. Numerically, this is shown to occur at a single non-dimensional frequency. This remarkable result is used to construct examples of motion-trapped modes, involving pairs of freely floating cylinders moving either in phase or out of phase. The former case is equivalent to having a motion-trapped mode for a single such cylinder next to a rigid vertical wall. In the latter out-of-phase case, the pair of cylinders move as if they form the wetted sections of a single rigidly connected catamaran structure.

A freely floating motion-trapping structure can be defined as one or more rigid bodies floating on the surface of a fluid which extends to infinity in at least one direction, whose free motion under its natural hydrostatic restoring force is coupled to that of the surrounding fluid in such a way that no waves are radiated to infinity. The resulting local time-harmonic oscillation of the structure and the surrounding fluid is called a motion-trapped mode. Such a structure would, if displaced slightly from its equilibrium position and released, ultimately oscillate indefinitely at the trapped-mode frequency. Previous examples of motion-trapping structures have been devised using an inverse approach in which the shape of pairs of such structures is determined implicitly by sketching certain streamlines. In this paper an alternative direct approach to the construction of motion-trapping structures in the form of a pair of identical floating cylinders of rectangular cross-section in two dimensions is presented. It is also shown that a thick-walled axisymmetric heaving circular cylinder can act as a motion-trapping structure.

The scattering of water waves by a long array of rigid vertical circular cylinders is analysed under the usual assumptions of linear theory. Our primary goal is to show how solutions obtained for semi-infinite arrays can be combined to provide accurate and numerically efficient solutions to problems involving long, but finite, arrays. The particular diffraction problem considered here has been chosen both for its theoretical interest and its applicability. The design of offshore structures supported by cylindrical columns is commonplace and understanding how the multiple interactions between the waves and the supports affect the field is clearly important. The theoretical interest comes from the fact that, for wavelengths greater than twice the geometric periodicity, the associated infinite array can support Rayleigh–Bloch surface waves that propagate along the array without attenuation. For a long finite array we expect to see these surface waves travelling back and forth along the array and interacting with the ends. For particular sets of parameters, near trapping has previously been observed and we provide, for the first time, a quantitative explanation of this phenomenon based on the excitation and reflection of surface waves by the ends of the finite array.

A motion trapping structure can be defined as a freely floating structure under natural or externally applied restoring forces, on or below the free surface of a heavy fluid extending to infinity in at least one direction, which generates a persistent local time-harmonic oscillation of the fluid of finite energy at a particular frequency, due to its own motion at that frequency. Such an oscillation is termed a motion-trapped mode. In this paper it is shown, using accurate numerical computations, that a submerged circular cylinder making forced time-harmonic two-dimensional heave or sway motions of small amplitude in a fluid of either finite or infinite depth, can create a local flow field in which no waves radiate to infinity at particular frequencies and depths of submergence of the cylinder. By tethering such a buoyant cylinder to the bottom of a fluid of finite depth, using a vertical inelastic mooring line, it is shown, by suitable choice of buoyancy and length of tether, how the cylinder, moving freely under its mooring forces, can operate as a motion trapping structure. Such a cylinder would, if displaced from its equilibrium position and released, ultimately oscillate indefinitely at the trapped mode frequency. This simple geometry is the first example of a submerged isolated motion trapping structure free to move under its natural mooring forces.

The two-dimensional problem of the free sloshing of an inviscid fluid in a vertically walled tank with an arbitrary bed shape is solved at both first and second order in the Stokes expansion of the velocity potential. The approach employed at both orders uses Green's functions for a flat bed in conjunction with the Cauchy–Riemann equations to derive integral equations for the tangential flux along the varying bed. The first- and second-order potentials everywhere in the fluid may then be related to these fluxes. Significant analytic progress is made with the calculation of various contributions to the integral equations at second order. The equations at first and second order are ultimately solved using a variational principle equivalent to the Galerkin method, giving efficient and accurate results. In particular, the work involved in determining the second-order solution is no more intensive than in solving the first-order problem. The first-order solution is shown to reproduce known results for specific bed shapes. The method is applied to a range of bed shapes and the second-order correction to the free-surface elevation is illustrated.

An investigation is carried out into the effect on wave propagation of an ice sheet of varying thickness floating on water of varying depth, in three dimensions. By deriving a variational principle equivalent to the governing equations of linear theory and invoking the mild-slope approximation in respect of the ice thickness and water depth variations, a simplified form of the problem is obtained from which the vertical coordinate is absent. Two situations are considered: the scattering of flexural-gravity waves by variations in the thickness of an infinite ice sheet and by depth variations; and the scattering of free-surface gravity waves by an ice sheet of finite extent and varying thickness, again incorporating arbitrary topography. Numerical methods are devised for the two-dimensional versions of these problems and a selection of results is presented. The variational approach that is developed can be used to implement more sophisticated approximations and is capable of producing the solution of full linear problems by taking a large enough basis in the Rayleigh-Ritz method. It is also applicable to other situations that involve wave scattering by a floating elastic sheet.

An explicit solution is provided for the scattering of an obliquely incident flexural-gravity wave by a narrow straight-line crack separating two semi-infinite thin elastic plates floating on water of finite depth. By first separating the solution into the sum of symmetric and antisymmetric parts it is shown that a simple form for each part can be derived in terms of a rapidly convergent infinite series multiplied by a fundamental constant of the problem. This constant is simply determined by applying an appropriate edge condition. Curves of reflection and transmission coefficients are presented, showing how they vary with plate properties and angle of incidence. It is also shown that in the absence of incident waves and for certain relations between their wavelength and frequency, symmetric edge waves exist which travel along the crack and decay in a direction normal to the crack.

The behaviour of water waves over periodic beds is considered in a two-dimensional context and using linear theory. Three cases are investigated: the scattering of waves by a finite section of periodic topography; the Bloch problem for infinite periodic topography; and sloshing motions over periodic topography confined between vertical boundaries. Connections are established between these problems. A transfer matrix method incorporating evanescent modes is developed for the scattering problem, which reduces the computation to that required for a single period, without compromising full linear theory. The problem of the existence of Bloch waves can also be posed on a single period, leading to a close relationship between it and the scattering problem. Sloshing motions over periodic beds, which may be regarded as special cases of the Bloch problem, are also found to have a significant connection with wave scattering. Integral equations methods allied to the Galerkin approximation are used to resolve the three problems numerically. In particular, the full linear solution for Bragg resonance is presented, allowing the accuracy of existing approximations to this phenomenon to be assessed. The selection of results given illustrates the main features of the work.

An investigation is made into the trapping of surface gravity waves by totally submerged three-dimensional obstacles and strong numerical evidence of the existence of trapped modes is presented. The specific geometry considered is a submerged elliptical torus. The depth of submergence of the torus and the aspect ratio of its cross-section are held fixed and a search for a trapped mode is made in the parameter space formed by varying the radius of the torus and the frequency. A plane wave approximation to the location of the mode in this space is derived and an integral equation and a side condition for the exact trapped mode are obtained. Each of these conditions is satisfied on a different line in the plane and the point at which the lines cross corresponds to a trapped mode. Although it is not possible to locate this point exactly, because of numerical error, existence of the mode may be inferred with confidence as small changes in the numerical results do not alter the fact that the lines cross.If the torus makes small vertical oscillations, it is customary to try to express the fluid velocity as the gradient of the so-called heave potential, which is assumed to have the same time dependence as the body oscillations. A necessary condition for the existence of this potential at the trapped mode frequency is derived and numerical evidence is cited which shows that this condition is not satisfied for an elliptical torus. Calculations of the heave potential for such a torus are made over a range of frequencies, and it is shown that the force coefficients behave in a singular fashion in the vicinity of the trapped mode frequency. An analysis of the time domain problem for a torus which is forced to make small vertical oscillations at the trapped mode frequency shows that the potential contains a term which represents a growing oscillation.

The scattering and trapping of water waves by three-dimensional submerged topography, infinite and periodic in one horizontal coordinate and of finite extent in the other, is considered under the assumptions of linearized theory. The mild-slope approximation is used to reduce the governing boundary value problem to one involving a form of the Helmholtz equation in which the coefficient depends on the topography and is therefore spatially varying.Two problems are considered: the scattering by the topography of parallel-crested obliquely incident waves and the propagation of trapping modes along the periodic topography. Both problems are formulated in terms of ‘domain’ integral equations which are solved numerically.Trapped waves are found to exist over any periodic topography which is ‘sufficiently’ elevated above the unperturbed bed level. In particular, every periodic topography wholly elevated above that level supports trapped waves. Fundamental differences are shown to exist between these trapped waves and the analogous Rayleigh–Bloch waves which exist on periodic gratings in acoustic theory.Results computed for the scattering problem show that, remarkably, there exist zeros of transmission at discrete wavenumbers for any periodic bed elevation and for all incident wave angles. One implication of this property is that total reflection of an incident wave of a particular frequency will occur in a channel with a single symmetric elevation on the bed. The zeros of transmission in the scattering problem are shown to be related to the presence of a ‘nearly trapped’ mode in the corresponding homogeneous problem.The scattering of waves by multiple rows of periodic topography is also considered and it is shown how Bragg resonance – well-established in scattering of waves by two-dimensional ripple beds – occurs in modes other than the input mode.

In this paper we investigate the existence of embedded trapped modes for symmetric obstacles which are placed on the centreline of a two-dimensional acoustic waveguide. Modes are sought which are antisymmetric about the centreline of the channel but which have frequencies that are above the first cut-off for antisymmetric wave propagation down the guide. In the terminology of spectral theory this means that the eigenvalue associated with the trapped mode is embedded in the continuous spectrum of the relevant operator. A numerical procedure based on a boundary integral technique is developed to search for embedded trapped modes for bodies of general shape. In addition two approximate solutions for trapped modes are found;the first is for long plates on the centreline of the channel and the second is for slender bodies which are perturbations of plates perpendicular to the guide walls. It is found that embedded trapped modes do not exist for arbitrary symmetric bodies but if an obstacle is defined by two geometrical parameters then branches of trapped modes may be obtained by varying both of these parameters simultaneously. One such branch is found for a family of ellipses of varying aspect ratio and size. The thin plates which are parallel and perpendicular to the guide walls are found to correspond to the end points of this branch.

The two-dimensional scattering of water waves over a finite region of arbitrarily varying topography linking two semi-infinite regions of constant depth is considered. Unlike many approaches to this problem, the formulation employed is exact in the context of linear theory, utilizing simple combinations of Green's functions appropriate to water of constant depth and the Cauchy–Riemann equations to derive a system of coupled integral equations for components of the fluid velocity at certain locations. Two cases arise, depending on whether the deepest point of the topography does or does not lie below the lower of the semi-infinite horizontal bed sections. In each, the reflected and transmitted wave amplitudes are related to the incoming wave amplitudes by a scattering matrix which is defined in terms of inner products involving the solution of the corresponding integral equation system.This solution is approximated by using the variational method in conjunction with a judicious choice of trial function which correctly models the fluid behaviour at the free surface and near the joins of the varying topography with the constant-depth sections, which may not be smooth. The numerical results are remarkably accurate, with just a two-term trial function giving three decimal places of accuracy in the reflection and transmission coefficents in most cases, whilst increasing the number of terms in the trial function results in rapid convergence. The method is applied to a range of examples.

Rayleigh–Bloch surface waves are acoustic or electromagnetic waves which propagate parallel to a two-dimensional diffraction grating and which are exponentially damped with distance from the grating. In the water-wave context they describe a localized wave having dominant wavenumber β travelling along an infinite periodic array of identical bottom-mounted cylinders having uniform cross-section throughout the water depth. A numerical method is described which enables the frequencies of the Rayleigh–Bloch waves to be determined as a function of β for an arbitrary cylinder cross-section. For particular symmetric cylinders, it is shown how a special choice of β produces results for the trapped mode frequencies and mode shapes in the vicinity of any (finite) number of cylinders spanning a rectangular waveguide or channel. It is also shown how one particular choice of β gives rise to a new type of trapped mode near an unsymmetric cylinder contained within a parallel-sided waveguide with locally-distorted walls. The implications for large forces due to incident waves on a large but finite number of such cylinders in the ocean is discussed.

Aspects of the solution to the linearized water-wave problem involving a pair of surface-piercing cylinders in oblique waves and infinite water depth are examined. In particular, the solution is proved to be unique for certain geometrical arrangements and wave parameters depending on whether the wave frequency is above or below the cut-off frequency. Outside these regions of uniqueness are constructed examples of non-uniqueness using ideas developed in McIver (1996) in the normal incidence case. Although non-uniqueness examples are obtained numerically, we are able to prove the existence of non-uniqueness under the assumption that the wave obliqueness is small.

The existence of trapped modes due to rigid obstacles placed symmetrically in between parallel walls having either Neumann or Dirichlet conditions imposed upon them are well known to occur for frequencies below the continuous spectrum or channel cut-off and for a range of geometric configurations. In this paper, we provide convincing numerical evidence for an additional isolated trapped mode of both Neumann and Dirichlet type embedded in the continuous spectrum (or above the channel cut-off) in the case of a rigid circular cylinder placed on the centre-plane of the channel. Thus,for each type of mode we give results showing that there is just one cylinder size and wave frequency at which the trapped mode occurs.

The trapped modes which can occur in a long narrow wave channel containing any number of different-sized bottom-mounted circular cylinders arbitrarily spaced along the centreline of the channel are considered. The modes, all of which are antisymmetric with respect to the centreplane of the channel are of two types: Neumann modes, in which the fluid has normal velocity zero on the channel walls corresponding to a localized sloshing near the cylinders, and Dirichlet modes, in which the dynamic pressure vanishes on the channel walls. These latter modes have no physical meaning in the water-wave context but have been observed in a related acoustic context where the same governing equations and boundary conditions apply.It is shown that in general there are [les ]N trapped modes for any configuration of N cylinders, the precise number depending critically on the geometry of the configuration. Both types are of importance in predicting the exciting forces on individual cylinders within a large but finite periodic arrangement of cylinders.

Scattering of waves by vertical barriers in infinite-depth water has received much attention due to the ability to solve many of these problems exactly. However, the same problems in finite depth require the use of approximation methods. In this paper we present an accurate method of solving these problems based on a Galerkin approximation. We will show how highly accurate complementary bounds can be computed with relative ease for many scattering problems involving vertical barriers in finite depth and also for a sloshing problem involving a vertical barrier in a rectangular tank.

This paper presents solutions to a number of problems posed for the out-of-plane displacement of infinite thin elastic plates that are rigidly pinned in periodic configurations, but that possess a finite number of ‘defects’. We begin by considering a single one-dimensional periodic array of pins. We derive an analytic solution for the displacement produced by the forced oscillation of the central pin in the array, and this solution is shown to be closely connected to the problem of scattering of plane waves by an array when a finite number of pins are removed. Attention then focuses on doubly periodic rectangular arrays of pinned points possessing defects. Central to approaching such problems is an understanding of Bloch–Floquet waves in periodic arrays in the absence of defects and a simple method is described for computing the associated dispersion surfaces. The solutions to three problems are then sought: the trapping of localised waves by a finite number of missing pins; trapping of waves by entire rows of missing pins; and the wave radiation pattern due to the forcing of a single pin. All problems are treated analytically using bounded Green's functions for thin elastic plates, a discrete Fourier transform solution method and simple, explicit and rapidly convergent evaluations of the one- and two-dimensional lattice sums that arise.

A new method is developed for approximating the scattering of linear surface gravity waves on water of varying quiescent depth in two dimensions. A conformal mapping of the fluid domain onto a uniform rectangular strip transforms steep and discontinuous bed profiles into relatively slowly varying, smooth functions in the transformed free-surface condition. By analogy with the mild-slope approach used extensively in unmapped domains, an approximate solution of the transformed problem is sought in the form of a modulated propagating wave which is determined by solving a second-order ordinary differential equation. This can be achieved numerically, but an analytic solution in the form of a rapidly convergent infinite series is also derived and provides simple explicit formulae for the scattered wave amplitudes. Small-amplitude and slow variations in the bedform that are excluded from the mapping procedure are incorporated in the approximation by a straightforward extension of the theory. The error incurred in using the method is established by means of a rigorous numerical investigation and it is found that remarkably accurate estimates of the scattered wave amplitudes are given for a wide range of bedforms and frequencies.