Laplace Transform for Water Waves

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Frequency domain definitions

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{ \Phi(\mathbf{x},t)=\mathrm{Re} \{\phi(\mathbf{x},\omega) e^{-i\omega t}\} }[/math] [math]\displaystyle{ V(\mathbf{x},t)=\mathrm{Re} \{ v(\mathbf{x},\omega) e^{-i\omega t}\} }[/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}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, \mathrm{d}t ,\quad \textrm{Re } s \gt 0; }[/math]

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

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

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

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

If [math]\displaystyle{ \omega }[/math] is real then [math]\displaystyle{ \phi(\mathbf{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(\mathbf{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{ \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{\partial\phi_{\mu}}{\partial 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} \,\mathrm{d}S }[/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} \,\mathrm{d}S }[/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} \,\mathrm{d}S. }[/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 which determines frequency domain solutions for prescribed structures and structure motions.