View source for Category:Time-Dependent Linear Water Waves

[[Category:Linear Water-Wave Theory]]

{{incomplete pages}}

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.

[[Fourier Transform Solution for an Incident Wave Packet]]

== Equations of motion in the Time Domain ==

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

{{Equations for fixed bodies in the time domain}}

=== Two dimensional equations for a floating body ===
We now consider the equations for a floating two-dimensional structure.
{{equations of motion time domain without body condition}}

{{two dimensional floating body time domain}}

The initial conditions are 
{{initial free surface time domain}} 
and
{{initial conditions floating body}}

=== 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
<center><math>
\partial_n \Phi = \partial_t\xi_{3}\mathbf{n}_{3},
</math></center>
where <math>\xi_{3}</math> 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
<center><math>
M\partial_t^2 \xi_{3}=-\rho\iint_{\partial\Omega_{\mathrm{B}}}\partial_t\Phi n_{3}\, dS- \rho g W\xi_{3}.
</math></center>
which is a significant simplification compared to the general case - the mass matrix and buoyancy matrix are replaced simply by the mass <math>M</math> and the hydrostatic term <math>\rho g W</math>. The initial conditions for the fluid and the structure (<math>\xi_{3}(0)</math>, <math>\partial_t{\xi}_{3}(0)</math>) must also be prescribed to complete the problem specification.