# 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 Generalized Eigenfunction Expansion for Water Waves or by using Memory Effect Function or by the Laplace Transform for Water Waves

## 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.

## Equations of motion in the Time Domain

### Two Dimensional Equations for fixed bodies in the time domain

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 $\displaystyle{ \Omega }$, the boundary of the fluid domain which touches the fixed bodies by $\displaystyle{ \partial\Omega }$, and the free surface by $\displaystyle{ F. }$ The $\displaystyle{ x }$ and $\displaystyle{ z }$ coordinates are such that $\displaystyle{ x }$ is pointing in the horizontal direction and $\displaystyle{ z }$ is pointing in the vertical upwards direction (we denote $\displaystyle{ \mathbf{x}=\left( x,z\right) ). }$ The free surface is at $\displaystyle{ z=0 }$ and the sea floor is at $\displaystyle{ z=-h }$. The fluid motion is described by a velocity potential $\displaystyle{ \Phi }$ and free surface by $\displaystyle{ \zeta }$.

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

$\displaystyle{ \Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega. }$

At the bottom surface we have no flow

$\displaystyle{ \partial_{n}\Phi=0,\ \ z=-h. }$

At the free surface we have the kinematic condition

$\displaystyle{ \partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in \partial\Omega_{F}, }$

and the dynamic condition (the linearized Bernoulli equation)

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

The body boundary condition for a fixed body is

$\displaystyle{ \partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega, }$

The initial conditions are

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

### Two dimensional equations for a floating body

We now consider the equations for a floating two-dimensional structure. The equations of motion in the time domain are Laplace's equation through out the fluid

$\displaystyle{ \Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega. }$

At the bottom surface we have no flow

$\displaystyle{ \partial_{n}\Phi=0,\ \ z=-h. }$

At the free surface we have the kinematic condition

$\displaystyle{ \partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in \partial\Omega_{F}, }$

and the dynamic condition (the linearized Bernoulli equation)

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

The body boundary condition for a floating body is given in terms of the 3 rigid body motions, namely surge, heave and pitch which are indexed as $\displaystyle{ \nu=1,3,5 }$ in order to be consistent with the three-dimensional problem. We have a kinematic condition

$\displaystyle{ \partial_{n}\Phi=\sum_{\nu}\partial_t \xi_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega_{B}, }$

where $\displaystyle{ \xi_{\nu} }$ is the motion of the $\displaystyle{ \mu }$th mode and $\displaystyle{ \mathbf{n}_{\nu} }$ is the normal associated with this mode. Note that we define all normal derivatives to point out of the fluid. The dynamic condition is the equation of motion for the structure:

$\displaystyle{ \sum_{\nu} M_{\mu\nu}\partial_t^2 \xi_{\nu}=-\rho\iint_{\partial\Omega_{B}}\partial_t\Phi n_{\mu}\, dS - \sum_{\nu} C_{\mu\nu}\xi_{\nu},\quad \textrm{for} \qquad \mu=1,3,5, }$

In this equation, $\displaystyle{ M_{\mu\nu} }$ are the elements of the mass matrix

$\displaystyle{ \mathbf{M}=\left[ \begin{matrix} M & 0 & M(z^c-Z^R) \\ 0 & M & -M(x^c-X^R) \\ M(z^c-Z^R)& -M(x^c-X^R) & I^b_{11}+I^b_{33} \end{matrix} \right] , }$

for the structure and $\displaystyle{ c_{\mu\nu} }$ are the elements of the buoyancy matrix

$\displaystyle{ \mathbf{C}=\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & \rho g W & -\rho g I^A_{1} \\ 0 & -\rho g I^A_{1} & \rho g (I^A_{11}+I^V_3)-Mg(z^c-Z^R) \end{matrix} \right]. }$

The terms $\displaystyle{ I^b_{11} }$, $\displaystyle{ I^b_{33} }$ are the moments of inertia of the body about the $\displaystyle{ x }$ and $\displaystyle{ z }$ axes and the terms $\displaystyle{ I_1^{A} }$, $\displaystyle{ I^{A}_{11} }$ are the first and second moments of the waterplane (the waterplane area is denoted $\displaystyle{ W }$) about the $\displaystyle{ x }$-axis (see Chapter 7, Mei 1983). In addition, $\displaystyle{ (x^c,z^c) }$ and $\displaystyle{ (X^R,Z^R) }$ are the positions of the centre of mass and centre of rotation of the body and $\displaystyle{ I^{V}_{3} }$ is $\displaystyle{ z }$-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive $\displaystyle{ x }$ direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

The initial conditions are

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

and The initial generalised displacements $\displaystyle{ \xi_{\mu} }$ and velocities $\displaystyle{ \dot{\xi}_{\mu} }$ of the body must be specified for all modes $\displaystyle{ \mu }$

### Floating body constrained to move in heave

The simplest type of floating body problem concerns the motion of a body constrained to move in heave in two-dimensions. Apart from the boundary condition on the structure surface which becomes

$\displaystyle{ \partial_n \Phi = \partial_t\xi_{3}\mathbf{n}_{3}, }$

where $\displaystyle{ \xi_{3} }$ is the heave displacement of the structure, the equations governing the motion of the fluid remain the same. The equation of motion for a body constrained to move in heave only is

$\displaystyle{ M\partial_t^2 \xi_{3}=-\rho\iint_{\partial\Omega_{\mathrm{B}}}\partial_t\Phi n_{3}\, dS- \rho g W\xi_{3}. }$

which is a significant simplification compared to the general case - the mass matrix and buoyancy matrix are replaced simply by the mass $\displaystyle{ M }$ and the hydrostatic term $\displaystyle{ \rho g W }$. The initial conditions for the fluid and the structure ($\displaystyle{ \xi_{3}(0) }$, $\displaystyle{ \partial_t{\xi}_{3}(0) }$) must also be prescribed to complete the problem specification.

## Pages in category "Time-Dependent Linear Water Waves"

The following 3 pages are in this category, out of 3 total.