Category:Time-Dependent Linear Water Waves

Generally the focus of research is on the Frequency Domain Problem. The time-domain problem can be solved by Generalised Eigenfunction Expansion or by an Expansion over the Resonances or using Memory Effect Function.

Introduction

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by Kennard 1949 and McIver 1994) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

We consider a two-dimensional fluid domain of constant depth, which contains a finite number of fixed bodies of arbitrary geometry. We denote the fluid domain by [math]\displaystyle{ \Omega }[/math], the boundary of the fluid domain which touches the fixed bodies by [math]\displaystyle{ \partial\Omega }[/math], and the free surface by [math]\displaystyle{ F. }[/math] The [math]\displaystyle{ x }[/math] and [math]\displaystyle{ z }[/math] coordinates are such that [math]\displaystyle{ x }[/math] is pointing in the horizontal direction and [math]\displaystyle{ z }[/math] is pointing in the vertical upwards direction (we denote [math]\displaystyle{ \mathbf{x}=\left( x,z\right) ). }[/math] The free surface is at [math]\displaystyle{ z=0 }[/math] and the sea floor is at [math]\displaystyle{ z=-h }[/math]. The fluid motion is described by a velocity potential [math]\displaystyle{ \Phi }[/math] and free surface by [math]\displaystyle{ \zeta }[/math].

The equations of motion in the time domain are Laplace's equation through out the fluid

[math]\displaystyle{ \Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega. }[/math]

At the bottom surface we have no flow

[math]\displaystyle{ \partial_{n}\Phi=0,\ \ z=-h. }[/math]

At the free surface we have the kinematic condition

[math]\displaystyle{ \partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in \partial\Omega_{F}, }[/math]

and the dynamic condition (the linearized Bernoulli equation)

[math]\displaystyle{ \partial_{t}\Phi = -g\zeta ,\ \ z=0,\ x\in \partial\Omega_{F}. }[/math]

The body boundary condition for a fixed body is

[math]\displaystyle{ \partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega, }[/math]

The initial conditions are

[math]\displaystyle{ \left.\zeta\right|_{t=0} = \zeta_0(x)\,\,\,\mathrm{and}\,\,\, \left.\partial_t\zeta\right|_{t=0} = \partial_t\zeta_0(x). }[/math]

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{\*\generator Msftedit 5.41.21.2508;}\viewkind4\uc1d\f0\fs20=Review of time-domain models=

==Frequency domain definitions==(1)

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from [math]\displaystyle{ t= -\infty }[/math] . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using

[math]\displaystyle{ (2) \begin{split} \Phi(\mbf{x},t)=Re \{\phi(\mbf{x},\omega) e^{-i\omega t}\} \ V(\mbf{x},t)=Re \{ v(\mbf{x},\omega) e^{-i\omega t}\} \end{split} }[/math]

and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only. However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

• take a Laplace transform of the time-domain potential

[math]\displaystyle{ \hat{\phi}(\mbf{x},s)=\int^{\infty}_{0}\Phi(\mbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s \gt 0; }[/math]

• apply the change of variables [math]\displaystyle{ s=-i\omega }[/math] so that

[math]\displaystyle{ \phi(\mbf{x},\omega)=\hat{\phi}(\mbf{x},-i\omega). }[/math]

• noting that [math]\displaystyle{ \phi(\mbf{x},-\omega)=\bar{\phi}(\mbf{x},\omega) }[/math], the inverse Fourier transform is given by

[math]\displaystyle{ (3) \Phi(\mbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mbf{x},\omega)e^{-i\omega t}\,d\omega }[/math]

 where the path of integration must pass over any singularities of [math]\displaystyle{ \phi }[/math] that lie on the real axis and it has been assumed that there is no motion prior to [math]\displaystyle{ t=0 }[/math], i.e. [math]\displaystyle{ \Phi(\mbf{x},t)=0 }[/math] for [math]\displaystyle{ t\lt 0 }[/math]. 
 

If [math]\displaystyle{ \omega }[/math] is real then [math]\displaystyle{ \phi(\mbf{x},\omega) }[/math] is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that [math]\displaystyle{ \Phi(\mbf{x},t)=0 }[/math] for [math]\displaystyle{ t\lt 0 }[/math] , no singularities can exist in [math]\displaystyle{ \textrm{Im } \omega\gt 0 }[/math]. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of [math]\displaystyle{ \omega }[/math]. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from [math]\displaystyle{ t= -\infty }[/math], the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential [math]\displaystyle{ \phi^{S} }[/math] and a radiation potential [math]\displaystyle{ \phi^{R} }[/math]. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is

[math]\displaystyle{ (4) \phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu} }[/math]

where [math]\displaystyle{ u_{\mu} }[/math] is the complex amplitude of the generalised velocity in the [math]\displaystyle{ \mu }[/math] direction and [math]\displaystyle{ \phi_{\mu} }[/math] describes the fluid response due to the forced oscillations in mode [math]\displaystyle{ \mu }[/math] with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode [math]\displaystyle{ \mu }[/math] the boundary condition on the structure will be

[math]\displaystyle{ \frac{tial\phi_{\mu}}{tial n}=n_{\mu}, }[/math]

because the total velocity is

[math]\displaystyle{ v(\omega)=\sum_{\mu}v_{\mu}n_{\mu} }[/math]

where [math]\displaystyle{ n_{\mu} }[/math] is the [math]\displaystyle{ \mu }[/math] component of the generalised normal and [math]\displaystyle{ v_{\mu} }[/math] is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the [math]\displaystyle{ \mu }[/math] direction due to the fluid response to the forced oscillations is

[math]\displaystyle{ (5) F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS }[/math]

and the force due to the diffraction of an incident wave by the fixed structure is

[math]\displaystyle{ (6) F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS }[/math]

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as [math]\displaystyle{ \sum_{\nu} v_{\nu}f_{\nu\mu} }[/math] where

[math]\displaystyle{ f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS. }[/math]

It is conventional to decompose the radiation force into a term featuring the added mass matrix [math]\displaystyle{ a_{\alpha\beta} }[/math] and a term featuring the damping matrix [math]\displaystyle{ b_{\alpha\beta} }[/math] as follows

[math]\displaystyle{ (7) f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega}) }[/math]

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency [math]\displaystyle{ \omega }[/math] and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:[math]\displaystyle{ \backslash\backslash }[/math]www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==(8)

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by~\citeasnoun{pmciver2005}, i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is [math]\displaystyle{ v_{\mu}(\omega) }[/math] then the Fourier transform of the acceleration is given by

[math]\displaystyle{ \int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0) }[/math]

the time-derivative of the potential obeys a similar relation

[math]\displaystyle{ \int_{0}^{\infty}\frac{tial\Phi}{tial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mbf{x},0). }[/math]

Therefore, the initial conditions of the structure [math]\displaystyle{ (X_{\mu}(0),V_{\mu}(0)) }[/math] and of the potential [math]\displaystyle{ \Phi(\mbf{x},0) }[/math] will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before~(4) then the Fourier transform of the equation of motion~(\ref{linearisedmotion}) for the structure is

[math]\displaystyle{ (9) \begin{split} M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]& =-\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+\ i\omega \sum_{\nu}&(f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)- \sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega) \end{split} }[/math]

for [math]\displaystyle{ \mu=1,\ldots,6 }[/math], where [math]\displaystyle{ f^{A}_{\mu}(\omega) }[/math] is the Fourier transform of the applied force [math]\displaystyle{ F_{\mu}(t) }[/math] in equation~(\ref{linearisedmotion}). Although it is assumed that [math]\displaystyle{ \Phi(\mbf{x},t)=0 }[/math] for [math]\displaystyle{ t\lt 0 }[/math], for a non-zero initial velocity [math]\displaystyle{ \lim_{t^{+}\rightarrow 0}\Phi(\mbf{x},t)\neq 0 }[/math] because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by

[math]\displaystyle{ \rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS=a(\infty)V(0) }[/math]

(see \citeasnoun{ccmei2}, \S~8.12.1)

where [math]\displaystyle{ a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega) }[/math] is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form

[math]\displaystyle{ \begin{split} \sum_{\nu}&\{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega) \ =& -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)\end{split} }[/math]

for [math]\displaystyle{ \mu=1,\ldots,6 }[/math]. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by

[math]\displaystyle{ (10) v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}. }[/math]

As described by \citeasnoun{mciver2006}, the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient [math]\displaystyle{ f_{\mu\nu}(\omega) }[/math] will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see \S~6 of \citeasnoun{mciver2006}) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving equation~(10) for [math]\displaystyle{ v(\omega) }[/math] then the time-domain velocity can be recovered using the inverse transform

[math]\displaystyle{ V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt. }[/math]

Integro-differential equation method

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by [math]\displaystyle{ \delta(t-\tau)V_{\alpha}(\tau) }[/math] where [math]\displaystyle{ \dot{X_{\alpha}}=V_{\alpha} }[/math]. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function [math]\displaystyle{ V_{\alpha}(t) }[/math] an integral of the impulse response over the range [math]\displaystyle{ -\infty\lt \tau\lt \infty }[/math] must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

[math]\displaystyle{ (11) \mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau }[/math]

where [math]\displaystyle{ m_{\beta\alpha}(\infty) }[/math] is an infinite frequency added mass coefficient and [math]\displaystyle{ L_{\beta\alpha}(t-\tau) }[/math] is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. [math]\displaystyle{ V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \} }[/math], the impulse response function [math]\displaystyle{ L_{\beta\alpha} }[/math] can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

[math]\displaystyle{ \begin{split} a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \ b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau \end{split} }[/math]

and so if the added mass or damping coefficients are known for all frequencies [math]\displaystyle{ 0\leq\omega\leq\infty }[/math], the impulse response or memory function [math]\displaystyle{ L_{\alpha\beta}(t) }[/math] can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

[math]\displaystyle{ \left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t), }[/math]

for [math]\displaystyle{ \alpha=1,\ldots,6 }[/math], where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency [math]\displaystyle{ \omega }[/math] on the interval [math]\displaystyle{ [0,\infty) }[/math]. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

Generalised eigenfunction expansion method

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the  governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator [math]\displaystyle{ \mbf{B} }[/math] which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by [math]\displaystyle{ \mbf{B} }[/math] so that [math]\displaystyle{ \mbf{B}\Psi=\Phi }[/math] where [math]\displaystyle{ \Psi=\Phi }[/math] on the free-surface [math]\displaystyle{ F }[/math]. If [math]\displaystyle{ D }[/math] denotes the total fluid domain, then potential [math]\displaystyle{ \Phi }[/math] satisfies

[math]\displaystyle{ \begin{matrix} \nabla^{2}\Phi&=0 \quad\textrm{in } D, \ \frac{tial\Phi}{tial n} &= 0 \quad\textrm{on } tial D,\ \Phi&=\Psi \quad\textrm{on } F \end{matrix} }[/math]

where [math]\displaystyle{ tial D }[/math] includes the bed and the structure surface. The Dirichlet to Neumann map [math]\displaystyle{ tial_{\mbf{n}}\mbf{B} }[/math] defined by

[math]\displaystyle{ (12) tial_{\mbf{n}}\mbf{B}\Psi=\frac{tial\Phi}{tial z}, \quad x\in F, }[/math]

then recovers the corresponding normal derivative to [math]\displaystyle{ \Psi }[/math] on the free surface.

The evolution equation

[math]\displaystyle{ i\frac{tial \mbf{U}}{tial t} = \mathcal{A}\mbf{U} }[/math]

accounts for the linear time-dependent free-surface equations using the vector

[math]\displaystyle{ \mbf{U}= \begin{pmatrix} \tab \Psi \ \tab -i\eta \end{pmatrix}, }[/math]

with [math]\displaystyle{ \eta }[/math] denoting the time-dependent free-surface elevation, and the operator [math]\displaystyle{ \mathcal{A} }[/math] which is both unitary and self-adjoint given by

[math]\displaystyle{ \mathcal{A}= \begin{pmatrix} \tab 0 & g \ \tab tial_{\mbf{n}}\mbf{B} & 0 \ \end{pmatrix} }[/math]

A self-adjoint operator will possess an entirely real spectrum and the generalised eigenfunctions [math]\displaystyle{ \mbf{u}=(\psi,-i\zeta)^{T} }[/math], which are non-trivial solutions of

[math]\displaystyle{ (13) \mathcal{A}\mbf{u}=\omega\mbf{u}, }[/math]

are just frequency-domain scattering solutions. (On a technical note, the word `generalised' precedes eigenfunction because the energy is unbounded.) This can be shown easily by combining the scalar equations resulting from (13) so as to give the frequency-domain free-surface condition. For a given frequency [math]\displaystyle{ \omega }[/math], the free-surface condition in three-dimensions is satisfied by waves from an infinite number of directions and so for each [math]\displaystyle{ \omega }[/math] there is an infinite set of eigenfunctions [math]\displaystyle{ \mbf{u}_{n}=(1,\omega/g)^{T}\psi_{n}(\mbf{x},\omega) }[/math], with the [math]\displaystyle{ n^{th} }[/math] eigenfunction corresponding to incident waves of the form [math]\displaystyle{ J_{n}(k r) e^{in\theta} }[/math]. The general solution of the time-evolution equation is, from spectral theory,

[math]\displaystyle{ (14) \mbf{U}(\mbf{x},t)=\int^{\infty}_{-\infty}\left[ \sum_{n}f_{n}(\omega)\mbf{u}_{n}(\mbf{x},\omega) \right]e^{-i\omega t}\, d\omega }[/math]

where [math]\displaystyle{ f_{n}(\omega) }[/math] is determined by the initial conditions now expressed as [math]\displaystyle{ \mbf{U}(\mbf{x},0) }[/math]. To obtain this expression for [math]\displaystyle{ f_{n}(\omega) }[/math], apply the energy inner product to (14) evaluated at [math]\displaystyle{ t=0 }[/math]. The eigenfunctions satisfy the orthogonality condition

[math]\displaystyle{ (15) \lt \mbf{u}_{m}(\mbf{x},\omega),\mbf{u}_{n}(\mbf{x},\omega')\gt _{E}=\Lambda_{m}(\omega)\delta_{mn}\delta(\omega-\omega') }[/math]

where [math]\displaystyle{ \lt \gt _{E} }[/math] denotes a special energy inner product (defined with a [math]\displaystyle{ \mathcal{H} }[/math] subscript by \citeasnoun{meylaniwwwfb2008}) and it can be shown that

[math]\displaystyle{ \Lambda_{m}(\omega)=\frac{4\pi\omega^{2}}{g k}\frac{d\omega}{dk}. }[/math]

Therefore, it is straightforward to show that

[math]\displaystyle{ f_{n}(\omega)=\frac{1}{\Lambda_{n}(\omega)}\lt \mbf{U}(\mbf{x},0),\mbf{u}_{n}(\mbf{x},\omega)\gt _{E}. }[/math]

by evaluating the energy inner product [math]\displaystyle{ \lt \mbf{U}(\mbf{x},0),\mbf{u}_{n}(\mbf{x},\omega)\gt _{E} }[/math] using the orthogonality relation~(15) and the definition~(14). This expression can be further simplified using the definition of the energy product but details will not be provided here. Instead, it should be noted that an analytic expression for the general solution has been obtained. Thus, for a given scattering problem it is in theory possible to obtain the motion of the free-surface and the potential on the free-surface.

The singularity expansion method involves moving the path of integration in~(14) in the [math]\displaystyle{ \omega }[/math]-plane across the singularities in the lower half plane Im[math]\displaystyle{ \,\omega\leq0 }[/math] using the method of contour integration. Thus, the general solution will consist of contributions from the poles, any branch cuts, the path at infinity and the remainder of the path located below the poles in the [math]\displaystyle{ \omega }[/math]-plane. As explained by \citeasnoun{hazardSEM}, the contributions from infinity are assumed to be identically zero and the contribution from the remainder of the path decays faster than the pole contributions. Given that the contribution from the branch cuts are thought to be significant only for very large times an approximation to the general solution can be obtained for medium and large [math]\displaystyle{ t }[/math] because the contributions from the poles dominate for this range of times. Therefore, the SEM requires a knowledge of the pole structure of the integrand to approximate the complete integral. Furthermore, the behaviour of the eigenfunctions in the vicinity of the poles must be known in order to compute the residue of each pole. The scattering potential will satisfy

[math]\displaystyle{ (I+T(\omega))\phi^{S}=g_{I} }[/math]

where the form of the operator [math]\displaystyle{ T(\omega) }[/math] depends on the solution method chosen and [math]\displaystyle{ g_{I} }[/math] is determined by the incident wave. Given [math]\displaystyle{ g_{I} }[/math], the scattered field is

[math]\displaystyle{ \phi^{S}=(I+T(\omega))^{-1}g_{I} }[/math]

and the pole structure of [math]\displaystyle{ \phi_{S} }[/math] will be inherited from the operator [math]\displaystyle{ (I+T(\omega))^{-1} }[/math], referred to as the resolvent.

In the case where the resolvent operator is approximated by a matrix, it can be shown that the potential has the form

[math]\displaystyle{ (16) \phi_{n}(\mbf{x},\omega)\sim\frac{\alpha_{nj}v_{j}(\mbf{x})}{\omega-\omega_{j}} \textrm{ as } \omega\rightarrow\omega_{j} }[/math]

in the vicinity of the pole. The poles themselves are determined by locating values of [math]\displaystyle{ \omega }[/math] in the lower complex plane where the resolvent is not invertible. Thus, [math]\displaystyle{ v_{j} }[/math] is a generalised eigenfunction of [math]\displaystyle{ A_{j0}=I+T(\omega_{j}) }[/math]. The integral in equation~(14) can be approximated by closing the integration path in the lower half plane and then moving the integration path across a finite number of [math]\displaystyle{ P }[/math] poles and summing over the contributions of these poles. As described by~\citeasnoun{hazardSEM}, any branch cuts in the complex [math]\displaystyle{ \omega }[/math]-plane will only be significant at very large times and the integrals at infinity are expected to be zero. Therefore, the contribution of the [math]\displaystyle{ P }[/math] poles closest to the real-[math]\displaystyle{ \omega }[/math] axis (and hence with the smallest decay rates [math]\displaystyle{ e^{-Im(\omega_{j})} }[/math]) will dominate in the medium term. So, after neglecting other contributions to the integral and using~(16) to determine the residues of the poles, the potential on the free-surface will be given by

[math]\displaystyle{ (17) \Psi(\mbf{x},t)\approx -2\pi\sum_{j}\left[ \sum_{n}f_{n}(\omega_{j})\alpha_{nj}\right]v_{j}(\mbf{x},0)e^{-i\omega_{j}t}. }[/math]

Although full details of how this expression can be evaluated are not given here, examples of applications of this method are given by~\citeasnoun{meylaniwwwfb2007}, \citeasnoun{meylaniwwwfb2008} and \citeasnoun{meylan2002}. Each of these papers also contain more detailed instructions regarding the computational aspects of the method, such as determining the locations of the poles.

\citeasnoun{hazardSEM} notes that the SEM is a non-rigorous method and requires some heuristic arguments to justify its use. However, in the specific cases considered the results are generally accurate for a large range of times and much frequency-domain information regarding resonances is inherent in the time-domain solution. At present, the method only applies to scattering problems; however, it is hoped that it can be extended to radiation and coupled motion problems. It is not clear how the forces on the structure will be determined.

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