WikiWaves - User contributions [en]

Linear Plane Progressive Regular Waves

2008-08-19T13:15:10Z

Adipro: /* Expression in terms of velocity potential */

= Introduction =

Regular time-harmonic linear plane progressive waves are the fundamental building block in describing the propagation of surface wave disturbances in a deterministic and stochastic setting and in predicting the response of floating structures in a seastate.

= Equations =

Regular time-harmonic linear plane progressive waves satisfy the boundary-value problem (in water of constant [[Finite Depth]])
<center><math> \begin{cases}
\frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} = 0, \qquad z=0 \\
\nabla^2\phi_1 = 0, \qquad -h < z < 0 \\
\frac{\partial\phi_1}{\partial z} = 0, \qquad z=-h
\end{cases} </math></center>
The last condition imposes a zero normal flux condition across a sea floor of constant depth, <math>h</math>. It turns out that if the water is sufficiently deep the wave does not feel any effect from the sea floor and we can make the [[Infinite Depth]] assumption.

= Propagating Wave =

[[Image:Phase_speed.jpg|thumb|right|600px|Propagating Wave]]

There are a number of ways to derive the propagating wave solution, but the simplest is to define the free surface elevation <u>apriori</u> as
<center><math> \xi (x,t) = A \cos ( \omega t - k x) = A \mathbf{Re} \{ e^{-ikx+i\omega t} \} </math></center>
where A is an apriori known wave amplitude and the wave frequency and wave number <math> (\omega, k) </math> pair have the same definitions as in all wave propagation problems, namely:

<center><math> T = \frac{2\pi}{\omega} = \mbox{Wave Period} </math></center>
<center><math> \lambda = \frac{2\pi}{k} = \mbox{Wave Length} </math></center>
<center><math> C= \frac{\omega}{k} = \mbox{Wave Phase Velocity} </math></center>
The relation between <math> \omega </math> and <math> k </math> is known as the dispersion relation often written in the form
<center><math> \omega = f (k) \,. </math></center>
<math> f(k) </math> depends on the physics of the wave propagation problem under study, in the case of sound propagation
<center><math> \omega = {C} k \equiv f (k) </math></center>
where <math> {C} </math> is the speed of sound. The above relation suggests that all sound waves in a medium with isotropic properties propagate with the same phase speed regardless of their frequency of wave length. These are unknown as non-dispersive waves.

Surface ocean waves are dispersive since <math> f(k) </math> is a non linear function of <math> k </math> as we will shortly show.

= Expression in terms of velocity potential =

By virtue of the definition of the wave elevation of a plane progressive regular wave, we seek a compatible definition of the respective velocity potential. Let:
<center><math> \phi_1 = A \mathbf{Re} \{ \phi (x,z) e^{i\omega t} \} </math></center>

The problem reduces to the definition of <math> \phi(x,z) </math> and the derivation of the appropriate dispersion relation between <math> \omega </math> and <math> k </math> so that the linear boundary value problem is satisfied. Remember that the equation requires that we consider the fluid potential rather than simply the free surface.

Before proceeding with the algebra, certain underlying principles are always at work:

* Linear system theory states that when the input signal is <math> e^{i \omega t} </math> the output signal must also be harmonic and with the same frequency.

* We assume that a solution always exists, otherwise the statement of the physical and/or mathematical boundary value problem is flawed. If we can find a solution in most cases it is the solution. So simply try out solutions that may make sense from the physical point of view.

* If the boundary value problem is satisfied by a complex velocity potential then it is also satisfied by its real and imaginary parts.

In our case we will first derive the boundary value problem satisfied by the complex potential <math> \phi(x,z) </math> and then we will try the plausible representation:

<center><math> \phi (x,z) = \psi (z) e^{- i k x} \,</math></center>

It follows upon substitution in the boundary value problem satisfied by <math> \phi_1 (x,z,t) </math> that <math> \phi (x,z) </math> is subject to:

<center><math> \begin{cases}
- \omega^2 \phi + g \phi_z = 0, \qquad z=0 \\
\nabla^2\phi = \phi_{xx} + \phi_{zz} = 0, \qquad -h<z<0 \\
\phi_z = 0, \qquad z=-h
\end{cases} </math></center>

Allowing for:

<center><math> \phi (x,z) = \psi(z) e^{-ikx} \,</math></center>

It follows that <math> \psi(z) </math> is subject to:

<center><math> \begin{cases}
- \omega^2 \psi + g \psi_z = 0, \qquad z=0 \\
\psi_{zz} - k^2\psi = 0, \qquad -h<z<0 \\
\psi_z = 0, \qquad z=-h
\end{cases} </math></center>

We can verify by simple substitution that
<center><math> \psi(z) = \frac{ig}{\omega} \frac{\cosh k(z+h)}{\cosh k h} </math></center>
satisfies the field equation <math> \psi_{zz} - k^2 \psi = 0 </math>, the seafloor condition <math> \psi_z=0, z=-h </math> and the free surface condition, only when
<center><math> \omega^2 = g k \tanh kh \, </math></center>
or
<center><math> \omega = \{ gk \tanh k h \}^{1/2} \, </math></center>
So by enforcing the free surface condition we have derived the [[Dispersion Relation for a Free Surface]] in [[Finite Depth]].

The resulting plane progressive wave velocity potential takes the form:
<center><math> \phi_1 (x,z,t) = A \ \mathbf{Re} \{ \frac{ig}{\omega} \frac{\cosh k (z+h)}{\cosh k h} e^{ - i k x + i \omega t} \} </math></center>

= Displacement and Pressure =

Verify that upon substitution
<center><math> \xi_1 = A \ \mathbf{Re} \{ e^{-ikx+i\omega t} \} = - \frac{1}{g} \left.\frac{\partial\phi_1}{\partial t} \right|_{z=0} </math></center>

The corresponding flow velocity at some point <math> \vec x = (x,z) </math> in the fluid domain or on <math> z=0, z=-h </math> is simply given by
<center><math> \vec V_1 = \nabla \phi_1 </math></center>

The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic, is
<center><math> P_1 = \rho \frac{\partial \phi_1}{\partial t} = \mathbf{Re} \{ \rho g A \frac{\cosh k (z+h)}{\cosh k h } e^{-ikx + i\omega t} \} </math></center>

From the Lagrangian kinematic relation
<center><math> \frac{d \vec \xi_1}{d t} = \vec V ( \vec \xi_1, t) = \nabla \phi_1 ( \vec xi_1, t) </math></center>

We may obtain ordinary differential equations governing <math> \vec \xi_1 (t) </math>. Marking a paticular particle with the fluid at rest, so that <math> \vec \xi_1 (0) = \vec x</math>, we may write:

<center><math> \vec \xi_1 (t) = \vec x + \vec {\Delta \xi (t)} </math></center>

Where <math> \vec x </math> is the particle position at rest and <math> \vec {\Delta\xi} </math> is its displacement due to the "arrival" of a plane progressive wave. Upon substitution in the equation of motion

<center><math> \frac{d \vec {\Delta\xi}}{dt} = \nabla \phi_1 (\vec x + \vec{\Delta\xi}, t) </math></center>

Since <math> \frac{d \vec x}{dt} = 0 </math>. Keeping therms of <math> O(\epsilon) </math> on both sides, it follows that

<center><math> \frac{d \vec{\Delta\xi}}{dt} = \nabla \phi_1 ( \vec x, t) + O (\epsilon^2) </math></center>

This equation when forced by the velocity vector that corresponds to the plane progressive wave solution derived above, leads to a harmonic solution for the particle displaced trajectories <math> \vec{\Delta\xi(t)} = (\Delta\xi_1, \Delta\xi_3) </math> which are circular.

If second-order effects are included, the particles under a plane progressive waves also undergo a steady-state drift known as the stokes drift. It can be easily modeled based on the approach described above by substituting second-order effects consistently into the right-hand side of the equation of motion (see Mh).

= [[Dispersion Relation for a Free Surface]] in [[Infinite Depth| deep]] and [[Shallow Depth| shallow waters]] =

In [[Finite Depth]]
<center><math> \omega^2 = gk\tanh kh \,</math></center>
which is a nonlinear algebraic equation for <math> \omega</math> as a function of <math>k</math> which has a unique positive real solution as can be shown graphically. It also has imaginary roots which are important in many application (see [[Dispersion Relation for a Free Surface]] and [[:Category:Eigenfunction Matching Method| Eigenfunction Matching Method]])
The unique positive real root <math> k </math> can only be found numerically. Yet it always exists and the iterative methods that may be implemented always converge rapidly.
In [[Infinite Depth|deep water]], <math> h \to \infty </math>
and therefore
<center><math> \tanh kh \to 1 </math></center>
which in turn implies the deep water dispersion relation <math> \omega^2 = g k </math>.
The phase speed is given by
<center><math> C = \frac{\omega}{k} = \frac{\omega}{\omega^2/g} = \frac{g}{\omega} = \frac{g}{2\pi/T} = \frac{gT}{2\pi} </math></center>
So the speed of the crest of a wave with period T=10 secs is approximately <math> 15.6 \frac{m}{s} </math> or about 30 knots!

Often we need a quick estimate of the length of a deed water wave the period of which we can measure accurately with a stop watch. We proceed as follows:

<center><math> C = \frac{\omega}{k} = \frac{\lambda}{T} \ \Longrightarrow \ \frac{\lambda}{T} = \frac{g}{\omega} = \frac{g}{2\pi/T} \ \Longrightarrow \ \lambda = \frac{gT^2}{2\pi} \simeq T^2 + \frac{1}{2} T^2 </math></center>
(by definition the phase speed is the ration of the wave length over the period, or the time it takes for a crest to travel that distance). So the wave length of a deep water wave in m is approximately the square of its period is seconds plus half that amount. So a wave with period <math>T=10</math> secs is about <math>150 m</math> long.

In the limit of [[Shallow Depth]] <math> kh \to 0</math>
which in turn implies that
<center><math> \tanh kh \simeq kh </math></center>
It therefore follows that
<center><math> \omega^2 = gk (kh) \ \Longrightarrow \ \frac{\omega^2}{k^2} = gh </math></center>
or
<center><math> \frac{\omega}{k} = C = \sqrt{gh} </math></center>

Thus, according to linear theory shallow water waves become non dispersive as is the case with acoustic waves. Unfortunately, nonlinear effects become more important as waves propagate from deep to shallow water (because the wave amplidute rises). Solitons and wave breaking are some manifestations of nonlinearity.

The transition from deep to finite depth wave effects occurs for values of <math> kh \le \pi </math>. This is because
<center><math> \tanh \pi \simeq 1 </math></center>
and for <math> kh = \pi \ \Longrightarrow \ \frac{2\pi h}{\lambda} = \pi \ \Longrightarrow \ \frac{h}{\lambda} = \frac{1}{2} </math>
so for <math> \frac{h}{\lambda} > \frac{1}{2} </math> or <math> kh > \pi </math> we are effectively dealing with [[Infinite Depth]]. This means that for most of the world ocean and wave conditions the water depth may be approximate as infinite.

-----
This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/C88824A3-CBFC-4857-A3DD-D463461C8B97/0/lecture3.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

[[Category:Linear Water-Wave Theory]]

Linear Plane Progressive Regular Waves

2008-08-19T13:12:04Z

Adipro: /* Expression in terms of velocity potential */

= Introduction =

Regular time-harmonic linear plane progressive waves are the fundamental building block in describing the propagation of surface wave disturbances in a deterministic and stochastic setting and in predicting the response of floating structures in a seastate.

= Equations =

Regular time-harmonic linear plane progressive waves satisfy the boundary-value problem (in water of constant [[Finite Depth]])
<center><math> \begin{cases}
\frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} = 0, \qquad z=0 \\
\nabla^2\phi_1 = 0, \qquad -h < z < 0 \\
\frac{\partial\phi_1}{\partial z} = 0, \qquad z=-h
\end{cases} </math></center>
The last condition imposes a zero normal flux condition across a sea floor of constant depth, <math>h</math>. It turns out that if the water is sufficiently deep the wave does not feel any effect from the sea floor and we can make the [[Infinite Depth]] assumption.

= Propagating Wave =

[[Image:Phase_speed.jpg|thumb|right|600px|Propagating Wave]]

There are a number of ways to derive the propagating wave solution, but the simplest is to define the free surface elevation <u>apriori</u> as
<center><math> \xi (x,t) = A \cos ( \omega t - k x) = A \mathbf{Re} \{ e^{-ikx+i\omega t} \} </math></center>
where A is an apriori known wave amplitude and the wave frequency and wave number <math> (\omega, k) </math> pair have the same definitions as in all wave propagation problems, namely:

<center><math> T = \frac{2\pi}{\omega} = \mbox{Wave Period} </math></center>
<center><math> \lambda = \frac{2\pi}{k} = \mbox{Wave Length} </math></center>
<center><math> C= \frac{\omega}{k} = \mbox{Wave Phase Velocity} </math></center>
The relation between <math> \omega </math> and <math> k </math> is known as the dispersion relation often written in the form
<center><math> \omega = f (k) \,. </math></center>
<math> f(k) </math> depends on the physics of the wave propagation problem under study, in the case of sound propagation
<center><math> \omega = {C} k \equiv f (k) </math></center>
where <math> {C} </math> is the speed of sound. The above relation suggests that all sound waves in a medium with isotropic properties propagate with the same phase speed regardless of their frequency of wave length. These are unknown as non-dispersive waves.

Surface ocean waves are dispersive since <math> f(k) </math> is a non linear function of <math> k </math> as we will shortly show.

= Expression in terms of velocity potential =

By virtue of the definition of the wave elevation of a plane progressive regular wave, we seek a compatible definition of the respective velocity potential. Let:
<center><math> \phi_1 = A \mathbf{Re} \{ \phi (x,z) e^{i\omega t} \} </math></center>

The problem reduces to the definition of <math> \phi(x,z) </math> and the derivation of the appropriate dispersion relation between <math> \omega </math> and <math> k </math> so that the linear boundary value problem is satisfied. Remember that the equation requires that we consider the fluid potential rather than simply the free surface.

Before proceeding with the algebra, certain underlying principles are always at work:

* Linear system theory states that when the input signal is <math> e^{i \omega t} </math> the output signal must also be harmonic and with the same frequency.

* We assume that a solution always exists, otherwise the statement of the physical and/or mathematical boundary value problem is flawed. If we can find a solution in most cases it is the solution. So simply try out solutions that may make sense from the physical point of view.

* If the boundary value problem is satisfied by a complex velocity potential then it is also satisfied by its real and imaginary parts.

In our case we will first derive the boundary value problem satisfied by the complex potential <math> \phi(x,z) </math> and then we will try the plausible representation:

<center><math> \phi (x,z) = \psi (z) e^{- i k x} \,</math></center>

It follows upon substitution in the boundary value problem satisfied by <math> \phi_1 (x,z,t) </math> that <math> \phi (x,z) </math> is subject to:

<center><math> \begin{cases}
\omega^2 \phi + g \phi_z = 0, \qquad z=0 \\
\nabla^2\phi = \phi_{xx} + \phi_{zz} = 0, \qquad -h<z<0 \\
\phi_z = 0, \qquad z=-h
\end{cases} </math></center>

Allowing for:

<center><math> \phi (x,z) = \psi(z) e^{-ikx} \,</math></center>

It follows that <math> \psi(z) </math> is subject to:

<center><math> \begin{cases}
- \omega^2 \psi + g \psi_z = 0, \qquad z=0 \\
\psi_{zz} - k^2\psi = 0, \qquad -h<z<0 \\
\psi_z = 0, \qquad z=-h
\end{cases} </math></center>

We can verify by simple substitution that
<center><math> \psi(z) = \frac{ig}{\omega} \frac{\cosh k(z+h)}{\cosh k h} </math></center>
satisfies the field equation <math> \psi_{zz} - k^2 \psi = 0 </math>, the seafloor condition <math> \psi_z=0, z=-h </math> and the free surface condition, only when
<center><math> \omega^2 = g k \tanh kh \, </math></center>
or
<center><math> \omega = \{ gk \tanh k h \}^{1/2} \, </math></center>
So by enforcing the free surface condition we have derived the [[Dispersion Relation for a Free Surface]] in [[Finite Depth]].

The resulting plane progressive wave velocity potential takes the form:
<center><math> \phi_1 (x,z,t) = A \ \mathbf{Re} \{ \frac{ig}{\omega} \frac{\cosh k (z+h)}{\cosh k h} e^{ - i k x + i \omega t} \} </math></center>

= Displacement and Pressure =

Verify that upon substitution
<center><math> \xi_1 = A \ \mathbf{Re} \{ e^{-ikx+i\omega t} \} = - \frac{1}{g} \left.\frac{\partial\phi_1}{\partial t} \right|_{z=0} </math></center>

The corresponding flow velocity at some point <math> \vec x = (x,z) </math> in the fluid domain or on <math> z=0, z=-h </math> is simply given by
<center><math> \vec V_1 = \nabla \phi_1 </math></center>

The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic, is
<center><math> P_1 = \rho \frac{\partial \phi_1}{\partial t} = \mathbf{Re} \{ \rho g A \frac{\cosh k (z+h)}{\cosh k h } e^{-ikx + i\omega t} \} </math></center>

From the Lagrangian kinematic relation
<center><math> \frac{d \vec \xi_1}{d t} = \vec V ( \vec \xi_1, t) = \nabla \phi_1 ( \vec xi_1, t) </math></center>

We may obtain ordinary differential equations governing <math> \vec \xi_1 (t) </math>. Marking a paticular particle with the fluid at rest, so that <math> \vec \xi_1 (0) = \vec x</math>, we may write:

<center><math> \vec \xi_1 (t) = \vec x + \vec {\Delta \xi (t)} </math></center>

Where <math> \vec x </math> is the particle position at rest and <math> \vec {\Delta\xi} </math> is its displacement due to the "arrival" of a plane progressive wave. Upon substitution in the equation of motion

<center><math> \frac{d \vec {\Delta\xi}}{dt} = \nabla \phi_1 (\vec x + \vec{\Delta\xi}, t) </math></center>

Since <math> \frac{d \vec x}{dt} = 0 </math>. Keeping therms of <math> O(\epsilon) </math> on both sides, it follows that

<center><math> \frac{d \vec{\Delta\xi}}{dt} = \nabla \phi_1 ( \vec x, t) + O (\epsilon^2) </math></center>

This equation when forced by the velocity vector that corresponds to the plane progressive wave solution derived above, leads to a harmonic solution for the particle displaced trajectories <math> \vec{\Delta\xi(t)} = (\Delta\xi_1, \Delta\xi_3) </math> which are circular.

If second-order effects are included, the particles under a plane progressive waves also undergo a steady-state drift known as the stokes drift. It can be easily modeled based on the approach described above by substituting second-order effects consistently into the right-hand side of the equation of motion (see Mh).

= [[Dispersion Relation for a Free Surface]] in [[Infinite Depth| deep]] and [[Shallow Depth| shallow waters]] =

In [[Finite Depth]]
<center><math> \omega^2 = gk\tanh kh \,</math></center>
which is a nonlinear algebraic equation for <math> \omega</math> as a function of <math>k</math> which has a unique positive real solution as can be shown graphically. It also has imaginary roots which are important in many application (see [[Dispersion Relation for a Free Surface]] and [[:Category:Eigenfunction Matching Method| Eigenfunction Matching Method]])
The unique positive real root <math> k </math> can only be found numerically. Yet it always exists and the iterative methods that may be implemented always converge rapidly.
In [[Infinite Depth|deep water]], <math> h \to \infty </math>
and therefore
<center><math> \tanh kh \to 1 </math></center>
which in turn implies the deep water dispersion relation <math> \omega^2 = g k </math>.
The phase speed is given by
<center><math> C = \frac{\omega}{k} = \frac{\omega}{\omega^2/g} = \frac{g}{\omega} = \frac{g}{2\pi/T} = \frac{gT}{2\pi} </math></center>
So the speed of the crest of a wave with period T=10 secs is approximately <math> 15.6 \frac{m}{s} </math> or about 30 knots!

Often we need a quick estimate of the length of a deed water wave the period of which we can measure accurately with a stop watch. We proceed as follows:

<center><math> C = \frac{\omega}{k} = \frac{\lambda}{T} \ \Longrightarrow \ \frac{\lambda}{T} = \frac{g}{\omega} = \frac{g}{2\pi/T} \ \Longrightarrow \ \lambda = \frac{gT^2}{2\pi} \simeq T^2 + \frac{1}{2} T^2 </math></center>
(by definition the phase speed is the ration of the wave length over the period, or the time it takes for a crest to travel that distance). So the wave length of a deep water wave in m is approximately the square of its period is seconds plus half that amount. So a wave with period <math>T=10</math> secs is about <math>150 m</math> long.

In the limit of [[Shallow Depth]] <math> kh \to 0</math>
which in turn implies that
<center><math> \tanh kh \simeq kh </math></center>
It therefore follows that
<center><math> \omega^2 = gk (kh) \ \Longrightarrow \ \frac{\omega^2}{k^2} = gh </math></center>
or
<center><math> \frac{\omega}{k} = C = \sqrt{gh} </math></center>

Thus, according to linear theory shallow water waves become non dispersive as is the case with acoustic waves. Unfortunately, nonlinear effects become more important as waves propagate from deep to shallow water (because the wave amplidute rises). Solitons and wave breaking are some manifestations of nonlinearity.

The transition from deep to finite depth wave effects occurs for values of <math> kh \le \pi </math>. This is because
<center><math> \tanh \pi \simeq 1 </math></center>
and for <math> kh = \pi \ \Longrightarrow \ \frac{2\pi h}{\lambda} = \pi \ \Longrightarrow \ \frac{h}{\lambda} = \frac{1}{2} </math>
so for <math> \frac{h}{\lambda} > \frac{1}{2} </math> or <math> kh > \pi </math> we are effectively dealing with [[Infinite Depth]]. This means that for most of the world ocean and wave conditions the water depth may be approximate as infinite.

-----
This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/C88824A3-CBFC-4857-A3DD-D463461C8B97/0/lecture3.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

[[Category:Linear Water-Wave Theory]]

Linear Plane Progressive Regular Waves

2008-08-19T13:08:20Z

Adipro: /* Expression in terms of velocity potential */

= Introduction =

Regular time-harmonic linear plane progressive waves are the fundamental building block in describing the propagation of surface wave disturbances in a deterministic and stochastic setting and in predicting the response of floating structures in a seastate.

= Equations =

Regular time-harmonic linear plane progressive waves satisfy the boundary-value problem (in water of constant [[Finite Depth]])
<center><math> \begin{cases}
\frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} = 0, \qquad z=0 \\
\nabla^2\phi_1 = 0, \qquad -h < z < 0 \\
\frac{\partial\phi_1}{\partial z} = 0, \qquad z=-h
\end{cases} </math></center>
The last condition imposes a zero normal flux condition across a sea floor of constant depth, <math>h</math>. It turns out that if the water is sufficiently deep the wave does not feel any effect from the sea floor and we can make the [[Infinite Depth]] assumption.

= Propagating Wave =

[[Image:Phase_speed.jpg|thumb|right|600px|Propagating Wave]]

There are a number of ways to derive the propagating wave solution, but the simplest is to define the free surface elevation <u>apriori</u> as
<center><math> \xi (x,t) = A \cos ( \omega t - k x) = A \mathbf{Re} \{ e^{-ikx+i\omega t} \} </math></center>
where A is an apriori known wave amplitude and the wave frequency and wave number <math> (\omega, k) </math> pair have the same definitions as in all wave propagation problems, namely:

<center><math> T = \frac{2\pi}{\omega} = \mbox{Wave Period} </math></center>
<center><math> \lambda = \frac{2\pi}{k} = \mbox{Wave Length} </math></center>
<center><math> C= \frac{\omega}{k} = \mbox{Wave Phase Velocity} </math></center>
The relation between <math> \omega </math> and <math> k </math> is known as the dispersion relation often written in the form
<center><math> \omega = f (k) \,. </math></center>
<math> f(k) </math> depends on the physics of the wave propagation problem under study, in the case of sound propagation
<center><math> \omega = {C} k \equiv f (k) </math></center>
where <math> {C} </math> is the speed of sound. The above relation suggests that all sound waves in a medium with isotropic properties propagate with the same phase speed regardless of their frequency of wave length. These are unknown as non-dispersive waves.

Surface ocean waves are dispersive since <math> f(k) </math> is a non linear function of <math> k </math> as we will shortly show.

= Expression in terms of velocity potential =

By virtue of the definition of the wave elevation of a plane progressive regular wave, we seek a compatible definition of the respective velocity potential. Let:
<center><math> \phi_1 = A \mathbf{Re} \{ \phi (x,z) e^{i\omega t} \} </math></center>

The problem reduces to the definition of <math> \phi(x,z) </math> and the derivation of the appropriate dispersion relation between <math> \omega </math> and <math> k </math> so that the linear boundary value problem is satisfied. Remember that the equation requires that we consider the fluid potential rather than simply the free surface.

Before proceeding with the algebra, certain underlying principles are always at work:

* Linear system theory states that when the input signal is <math> e^{i \omega t} </math> the output signal must also be harmonic and with the same frequency.

* We assume that a solution always exists, otherwise the statement of the physical and/or mathematical boundary value problem is flawed. If we can find a solution in most cases it is the solution. So simply try out solutions that may make sense from the physical point of view.

* If the boundary value problem is satisfied by a complex velocity potential then it is also satisfied by its real and imaginary parts.

In our case we will first derive the boundary value problem satisfied by the complex potential <math> \phi(x,z) </math> and then we will try the plausible representation:

<center><math> \phi (x,z) = \psi (z) e^{- i k x} \,</math></center>

It follows upon substitution in the boundary value problem satisfied by <math> \phi_1 (x,z,t) </math> that <math> \phi (x,z) </math> is subject to:

<center><math> \begin{cases}
\omega^2 \phi + g \phi_z = 0, \qquad z=0 \\
\nabla^2\phi = \phi_{xx} + \phi_{zz} = 0, \qquad -h<z<0 \\
\phi_z = 0, \qquad z=-h
\end{cases} </math></center>

Allowing for:

<center><math> \phi (x,z) = \psi(z) e^{-ikx} \,</math></center>

It follows that <math> \psi(z) </math> is subject to:

<center><math> \begin{cases}
- \omega^2 \psi + g \psi_z = 0, \qquad z=0 \\
\psi_{zz} - k^2\psi = 0, \qquad -h<z<0 \\
\psi_z = 0, \qquad z=-h
\end{cases} </math></center>

We can verify by simple substitution that
<center><math> \psi(z) = \frac{ig}{\omega} \frac{\cosh k(z+h)}{\cosh k h} </math></center>
satisfies the field equation <math> \psi_{zz} - k^2 \psi = 0 </math>, the seafloor condition <math> \psi_z=0, z=-h </math> and the free surface condition, only when
<center><math> \omega^2 = g k \tanh kh \, </math></center>
or
<center><math> \omega = \{ gk \tanh h k h \}^{1/2} \, </math></center>
So by enforcing the free surface condition we have derived the [[Dispersion Relation for a Free Surface]] in [[Finite Depth]].

The resulting plane progressive wave velocity potential takes the form:
<center><math> \phi_1 (x,z,t) = A \ \mathbf{Re} \{ \frac{ig}{\omega} \frac{\cosh k (z+h)}{\cosh k h} e^{ - i k x + i \omega t} \} </math></center>

= Displacement and Pressure =

Verify that upon substitution
<center><math> \xi_1 = A \ \mathbf{Re} \{ e^{-ikx+i\omega t} \} = - \frac{1}{g} \left.\frac{\partial\phi_1}{\partial t} \right|_{z=0} </math></center>

The corresponding flow velocity at some point <math> \vec x = (x,z) </math> in the fluid domain or on <math> z=0, z=-h </math> is simply given by
<center><math> \vec V_1 = \nabla \phi_1 </math></center>

The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic, is
<center><math> P_1 = \rho \frac{\partial \phi_1}{\partial t} = \mathbf{Re} \{ \rho g A \frac{\cosh k (z+h)}{\cosh k h } e^{-ikx + i\omega t} \} </math></center>

From the Lagrangian kinematic relation
<center><math> \frac{d \vec \xi_1}{d t} = \vec V ( \vec \xi_1, t) = \nabla \phi_1 ( \vec xi_1, t) </math></center>

We may obtain ordinary differential equations governing <math> \vec \xi_1 (t) </math>. Marking a paticular particle with the fluid at rest, so that <math> \vec \xi_1 (0) = \vec x</math>, we may write:

<center><math> \vec \xi_1 (t) = \vec x + \vec {\Delta \xi (t)} </math></center>

Where <math> \vec x </math> is the particle position at rest and <math> \vec {\Delta\xi} </math> is its displacement due to the "arrival" of a plane progressive wave. Upon substitution in the equation of motion

<center><math> \frac{d \vec {\Delta\xi}}{dt} = \nabla \phi_1 (\vec x + \vec{\Delta\xi}, t) </math></center>

Since <math> \frac{d \vec x}{dt} = 0 </math>. Keeping therms of <math> O(\epsilon) </math> on both sides, it follows that

<center><math> \frac{d \vec{\Delta\xi}}{dt} = \nabla \phi_1 ( \vec x, t) + O (\epsilon^2) </math></center>

This equation when forced by the velocity vector that corresponds to the plane progressive wave solution derived above, leads to a harmonic solution for the particle displaced trajectories <math> \vec{\Delta\xi(t)} = (\Delta\xi_1, \Delta\xi_3) </math> which are circular.

If second-order effects are included, the particles under a plane progressive waves also undergo a steady-state drift known as the stokes drift. It can be easily modeled based on the approach described above by substituting second-order effects consistently into the right-hand side of the equation of motion (see Mh).

= [[Dispersion Relation for a Free Surface]] in [[Infinite Depth| deep]] and [[Shallow Depth| shallow waters]] =

In [[Finite Depth]]
<center><math> \omega^2 = gk\tanh kh \,</math></center>
which is a nonlinear algebraic equation for <math> \omega</math> as a function of <math>k</math> which has a unique positive real solution as can be shown graphically. It also has imaginary roots which are important in many application (see [[Dispersion Relation for a Free Surface]] and [[:Category:Eigenfunction Matching Method| Eigenfunction Matching Method]])
The unique positive real root <math> k </math> can only be found numerically. Yet it always exists and the iterative methods that may be implemented always converge rapidly.
In [[Infinite Depth|deep water]], <math> h \to \infty </math>
and therefore
<center><math> \tanh kh \to 1 </math></center>
which in turn implies the deep water dispersion relation <math> \omega^2 = g k </math>.
The phase speed is given by
<center><math> C = \frac{\omega}{k} = \frac{\omega}{\omega^2/g} = \frac{g}{\omega} = \frac{g}{2\pi/T} = \frac{gT}{2\pi} </math></center>
So the speed of the crest of a wave with period T=10 secs is approximately <math> 15.6 \frac{m}{s} </math> or about 30 knots!

Often we need a quick estimate of the length of a deed water wave the period of which we can measure accurately with a stop watch. We proceed as follows:

<center><math> C = \frac{\omega}{k} = \frac{\lambda}{T} \ \Longrightarrow \ \frac{\lambda}{T} = \frac{g}{\omega} = \frac{g}{2\pi/T} \ \Longrightarrow \ \lambda = \frac{gT^2}{2\pi} \simeq T^2 + \frac{1}{2} T^2 </math></center>
(by definition the phase speed is the ration of the wave length over the period, or the time it takes for a crest to travel that distance). So the wave length of a deep water wave in m is approximately the square of its period is seconds plus half that amount. So a wave with period <math>T=10</math> secs is about <math>150 m</math> long.

In the limit of [[Shallow Depth]] <math> kh \to 0</math>
which in turn implies that
<center><math> \tanh kh \simeq kh </math></center>
It therefore follows that
<center><math> \omega^2 = gk (kh) \ \Longrightarrow \ \frac{\omega^2}{k^2} = gh </math></center>
or
<center><math> \frac{\omega}{k} = C = \sqrt{gh} </math></center>

Thus, according to linear theory shallow water waves become non dispersive as is the case with acoustic waves. Unfortunately, nonlinear effects become more important as waves propagate from deep to shallow water (because the wave amplidute rises). Solitons and wave breaking are some manifestations of nonlinearity.

The transition from deep to finite depth wave effects occurs for values of <math> kh \le \pi </math>. This is because
<center><math> \tanh \pi \simeq 1 </math></center>
and for <math> kh = \pi \ \Longrightarrow \ \frac{2\pi h}{\lambda} = \pi \ \Longrightarrow \ \frac{h}{\lambda} = \frac{1}{2} </math>
so for <math> \frac{h}{\lambda} > \frac{1}{2} </math> or <math> kh > \pi </math> we are effectively dealing with [[Infinite Depth]]. This means that for most of the world ocean and wave conditions the water depth may be approximate as infinite.

-----
This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/C88824A3-CBFC-4857-A3DD-D463461C8B97/0/lecture3.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

[[Category:Linear Water-Wave Theory]]

Linear Plane Progressive Regular Waves

2008-08-19T13:01:31Z

Adipro: /* Propagating Wave */

= Introduction =

Regular time-harmonic linear plane progressive waves are the fundamental building block in describing the propagation of surface wave disturbances in a deterministic and stochastic setting and in predicting the response of floating structures in a seastate.

= Equations =

Regular time-harmonic linear plane progressive waves satisfy the boundary-value problem (in water of constant [[Finite Depth]])
<center><math> \begin{cases}
\frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} = 0, \qquad z=0 \\
\nabla^2\phi_1 = 0, \qquad -h < z < 0 \\
\frac{\partial\phi_1}{\partial z} = 0, \qquad z=-h
\end{cases} </math></center>
The last condition imposes a zero normal flux condition across a sea floor of constant depth, <math>h</math>. It turns out that if the water is sufficiently deep the wave does not feel any effect from the sea floor and we can make the [[Infinite Depth]] assumption.

= Propagating Wave =

[[Image:Phase_speed.jpg|thumb|right|600px|Propagating Wave]]

There are a number of ways to derive the propagating wave solution, but the simplest is to define the free surface elevation <u>apriori</u> as
<center><math> \xi (x,t) = A \cos ( \omega t - k x) = A \mathbf{Re} \{ e^{-ikx+i\omega t} \} </math></center>
where A is an apriori known wave amplitude and the wave frequency and wave number <math> (\omega, k) </math> pair have the same definitions as in all wave propagation problems, namely:

<center><math> T = \frac{2\pi}{\omega} = \mbox{Wave Period} </math></center>
<center><math> \lambda = \frac{2\pi}{k} = \mbox{Wave Length} </math></center>
<center><math> C= \frac{\omega}{k} = \mbox{Wave Phase Velocity} </math></center>
The relation between <math> \omega </math> and <math> k </math> is known as the dispersion relation often written in the form
<center><math> \omega = f (k) \,. </math></center>
<math> f(k) </math> depends on the physics of the wave propagation problem under study, in the case of sound propagation
<center><math> \omega = {C} k \equiv f (k) </math></center>
where <math> {C} </math> is the speed of sound. The above relation suggests that all sound waves in a medium with isotropic properties propagate with the same phase speed regardless of their frequency of wave length. These are unknown as non-dispersive waves.

Surface ocean waves are dispersive since <math> f(k) </math> is a non linear function of <math> k </math> as we will shortly show.

= Expression in terms of velocity potential =

By virtue of the definition of the wave elevation of a plane progressive regular wave, we seek a compatible definition of the respective velocity potential. Let:
<center><math> \phi_1 = A \mathbf{Re} \{ \phi (x,z) e^{i\omega t} \} </math></center>

The problem reduces to the definition of <math> \phi(x,z) </math> and the derivation of the appropriate dispersion relation between <math> \omega </math> and <math> k </math> so that the linear boundary value problem is satisfied. Remember that the equation requires that we consider the fluid potential rather than simply the free surface.

Before proceeding with the algebra, certain underlying principles are always at work:

* Linear system theory states that when the input signal is <math> e^{i \omega t} </math> the output signal must also be harmonic and with the same frequency.

* We assume that a solution always exists, otherwise the statement of the physical and/or mathematical boundary value problem is flawed. If we can find a solution in most cases it is the solution. so simply try out solutions that may make sense from the physical point of view.

* If the boundary value problem is satisfied by a complex velocity potential then it is also satisfied by its real and imaginary parts.

In our case we will first derive the boundary value problem satisfied by the complex potential <math> \phi(x,z) </math> and then we will try the plausible representation:

<center><math> \phi (x,z) = \psi (z) e^{- i k x} \,</math></center>

It follows upon substitution in the boundary value problem satisfied by <math> \phi_1 (x,z,t) </math> that <math> \phi (x,z) </math> is subject to:

<center><math> \begin{cases}
\omega^2 \phi + g \phi_z = 0, \qquad z=0 \\
\nabla^2\phi = \phi_{xx} + \phi_{zz} = 0, \qquad -h<z<0 \\
\phi_z = 0, \qquad z=-h
\end{cases} </math></center>

Allowing for:

<center><math> \phi (x,z) = \psi(z) e^{-ikx} \,</math></center>

It follows that <math> \psi(z) </math> is subject to:

<center><math> \begin{cases}
- \omega^2 \psi + g \psi_z = 0, \qquad z=0 \\
\psi_{zz} - k^2\psi = 0, \qquad -h<z<0 \\
\psi_z = 0, \qquad z=-h
\end{cases} </math></center>

We can verify by simple substitution that
<center><math> \psi(z) = \frac{ig}{\omega} \frac{\cosh k(z+h)}{\cosh k h} </math></center>
satisfies the field equation <math> \psi_{zz} - k^2 \psi = 0 </math>, the seafloor condition <math> \psi_z=0, z=-h </math> and the free surface condition, only when
<center><math> \omega^2 = g k \tanh kh \, </math></center>
or
<center><math> \omega = \{ gk \tanh h k h \}^{1/2} \, </math></center>
So by enforcing the free surface condition we have derived the [[Dispersion Relation for a Free Surface]] in [[Finite Depth]].

The resulting plane progressive wave velocity potential takes the form:
<center><math> \phi_1 (x,z,t) = A \ \mathbf{Re} \{ \frac{ig}{\omega} \frac{\cosh k (z+h)}{\cosh k h} e^{ - i k x + i \omega t} \} </math></center>

= Displacement and Pressure =

Verify that upon substitution
<center><math> \xi_1 = A \ \mathbf{Re} \{ e^{-ikx+i\omega t} \} = - \frac{1}{g} \left.\frac{\partial\phi_1}{\partial t} \right|_{z=0} </math></center>

The corresponding flow velocity at some point <math> \vec x = (x,z) </math> in the fluid domain or on <math> z=0, z=-h </math> is simply given by
<center><math> \vec V_1 = \nabla \phi_1 </math></center>

The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic, is
<center><math> P_1 = \rho \frac{\partial \phi_1}{\partial t} = \mathbf{Re} \{ \rho g A \frac{\cosh k (z+h)}{\cosh k h } e^{-ikx + i\omega t} \} </math></center>

From the Lagrangian kinematic relation
<center><math> \frac{d \vec \xi_1}{d t} = \vec V ( \vec \xi_1, t) = \nabla \phi_1 ( \vec xi_1, t) </math></center>

We may obtain ordinary differential equations governing <math> \vec \xi_1 (t) </math>. Marking a paticular particle with the fluid at rest, so that <math> \vec \xi_1 (0) = \vec x</math>, we may write:

<center><math> \vec \xi_1 (t) = \vec x + \vec {\Delta \xi (t)} </math></center>

Where <math> \vec x </math> is the particle position at rest and <math> \vec {\Delta\xi} </math> is its displacement due to the "arrival" of a plane progressive wave. Upon substitution in the equation of motion

<center><math> \frac{d \vec {\Delta\xi}}{dt} = \nabla \phi_1 (\vec x + \vec{\Delta\xi}, t) </math></center>

Since <math> \frac{d \vec x}{dt} = 0 </math>. Keeping therms of <math> O(\epsilon) </math> on both sides, it follows that

<center><math> \frac{d \vec{\Delta\xi}}{dt} = \nabla \phi_1 ( \vec x, t) + O (\epsilon^2) </math></center>

This equation when forced by the velocity vector that corresponds to the plane progressive wave solution derived above, leads to a harmonic solution for the particle displaced trajectories <math> \vec{\Delta\xi(t)} = (\Delta\xi_1, \Delta\xi_3) </math> which are circular.

If second-order effects are included, the particles under a plane progressive waves also undergo a steady-state drift known as the stokes drift. It can be easily modeled based on the approach described above by substituting second-order effects consistently into the right-hand side of the equation of motion (see Mh).

= [[Dispersion Relation for a Free Surface]] in [[Infinite Depth| deep]] and [[Shallow Depth| shallow waters]] =

In [[Finite Depth]]
<center><math> \omega^2 = gk\tanh kh \,</math></center>
which is a nonlinear algebraic equation for <math> \omega</math> as a function of <math>k</math> which has a unique positive real solution as can be shown graphically. It also has imaginary roots which are important in many application (see [[Dispersion Relation for a Free Surface]] and [[:Category:Eigenfunction Matching Method| Eigenfunction Matching Method]])
The unique positive real root <math> k </math> can only be found numerically. Yet it always exists and the iterative methods that may be implemented always converge rapidly.
In [[Infinite Depth|deep water]], <math> h \to \infty </math>
and therefore
<center><math> \tanh kh \to 1 </math></center>
which in turn implies the deep water dispersion relation <math> \omega^2 = g k </math>.
The phase speed is given by
<center><math> C = \frac{\omega}{k} = \frac{\omega}{\omega^2/g} = \frac{g}{\omega} = \frac{g}{2\pi/T} = \frac{gT}{2\pi} </math></center>
So the speed of the crest of a wave with period T=10 secs is approximately <math> 15.6 \frac{m}{s} </math> or about 30 knots!

Often we need a quick estimate of the length of a deed water wave the period of which we can measure accurately with a stop watch. We proceed as follows:

<center><math> C = \frac{\omega}{k} = \frac{\lambda}{T} \ \Longrightarrow \ \frac{\lambda}{T} = \frac{g}{\omega} = \frac{g}{2\pi/T} \ \Longrightarrow \ \lambda = \frac{gT^2}{2\pi} \simeq T^2 + \frac{1}{2} T^2 </math></center>
(by definition the phase speed is the ration of the wave length over the period, or the time it takes for a crest to travel that distance). So the wave length of a deep water wave in m is approximately the square of its period is seconds plus half that amount. So a wave with period <math>T=10</math> secs is about <math>150 m</math> long.

In the limit of [[Shallow Depth]] <math> kh \to 0</math>
which in turn implies that
<center><math> \tanh kh \simeq kh </math></center>
It therefore follows that
<center><math> \omega^2 = gk (kh) \ \Longrightarrow \ \frac{\omega^2}{k^2} = gh </math></center>
or
<center><math> \frac{\omega}{k} = C = \sqrt{gh} </math></center>

Thus, according to linear theory shallow water waves become non dispersive as is the case with acoustic waves. Unfortunately, nonlinear effects become more important as waves propagate from deep to shallow water (because the wave amplidute rises). Solitons and wave breaking are some manifestations of nonlinearity.

The transition from deep to finite depth wave effects occurs for values of <math> kh \le \pi </math>. This is because
<center><math> \tanh \pi \simeq 1 </math></center>
and for <math> kh = \pi \ \Longrightarrow \ \frac{2\pi h}{\lambda} = \pi \ \Longrightarrow \ \frac{h}{\lambda} = \frac{1}{2} </math>
so for <math> \frac{h}{\lambda} > \frac{1}{2} </math> or <math> kh > \pi </math> we are effectively dealing with [[Infinite Depth]]. This means that for most of the world ocean and wave conditions the water depth may be approximate as infinite.

-----
This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/C88824A3-CBFC-4857-A3DD-D463461C8B97/0/lecture3.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

[[Category:Linear Water-Wave Theory]]

Main Page

2008-08-19T12:54:18Z

Adipro: /* Welcome to '''Wikiwaves'''! */

==Welcome to '''Wikiwaves'''!==

'''Wikiwaves''' is a water waves [http://en.wikipedia.org/wiki/Wiki wiki]. It is essentially an online book being
written by [[Michael Meylan]] but you are welcome to make a contribution if you would like to.
Check out [[Wiki List]] for a list of wikis in a similar spirit to this one. This wiki was started 12 April 2006.

__NOTOC__

<hr>

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<div style="padding: .4em .9em .9em">
===Getting Started===
*First [[Browse|browse]] around to get a feel for what is here.
*Then follow the [[Sign up instructions|sign up instructions]] to make yourself a profile page.
*After that, learn [[Simple wiki help|how to create and compose pages]] and contribute to the site!
*If you're having any problems, see the [[FAQ]] page.
*We like feedback. If you have suggestions, comments, or additional questions, add them to our [[requests]] page or contact [[About us|us]].
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<div style="padding: .4em .9em .9em">

===Featured Pages===
*[[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]] The eigenfunction matching method.
*[[:Category:Wave Scattering in the Marginal Ice Zone|Wave Scattering in the Marginal Ice Zone]]: A description of the geophysical problem in water wave scattering.
*[[:Category:Interaction Theory|Interaction Theory]]: Presents the theory of multiple body interactions.
</div>

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===Wikiwaves Announcements===
*At the moment we are trying to link this wiki to computer code
*We appreciate all and every edit - even fixing a typing mistake.

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===Site Map===
*[[Browse]]
*[[:Category:People|People]]
*[[:Category:Reference|References]]
*[[test | Test page]]
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<hr>

===About Us===

[[Image:NZIMA.jpg|thumb|right]]

This website is largely the work of [[Michael Meylan]]. It was initially supported by a grant from the
[http://www.nzima.auckland.ac.nz/ New Zealand Institute of Mathematics].
A significant amount of the initial content was derived from the [http://ocw.mit.edu/index.html MIT opencourseware].

== Useful Links ==

* [[FAQ]] (Frequently asked questions) for the water-waves wiki
* [http://www.mediawiki.org/wiki/Help:FAQ MediaWiki FAQ]
* [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula Using Latex in Wiki]
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Consult the [http://meta.wikipedia.org/wiki/MediaWiki_User%27s_Guide User's Guide] for information on using the wiki software.

Wiener-Hopf Elastic Plate Solution

2008-03-10T13:57:38Z

Adipro: /* Solution of the W-H Equation */

= Introduction =

We present here the [[:Category:Wiener-Hopf|Wiener-Hopf]] solution to the problem of a
two semi-infinite [[Two-Dimensional Floating Elastic Plate|Two-Dimensional Floating Elastic Plates]].
The solution method is based on the one presented by [[Chung and Fox 2002]]. This problem
has been well studied and the first solution was by [[Evans and Davies 1968]]
but they did not actually develop the method sufficiently to be able to calculate the solution.
A solution was also developed by [[Balmforth and Craster 1999]] and by [[Tkacheva 2004]].

The theory is described in [[:Category:Wiener-Hopf|Wiener-Hopf]].

= Elastic Plate =

We imagine two semi-infinite [[:category:Floating Elastic Plate|Floating Elastic Plates]]
of (possibly) different properties. The equations are the following
<center><math>
\left( D_{j}\left( \frac{\partial^{2}}{\partial x^{2}}-k^{2}\right)
^{2}+\rho g-m_{j}\omega^{2}\right) \phi_{z}-\rho\omega^{2}\phi
=0,\;j=1,2,\;z=0
</math></center>
<center><math>
\left( \frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
z^{2}}-k^{2}\right) \phi =0,\;-H<z<0,
</math></center>
<center><math>
\phi_{z} =0,\;\;z=-H.
</math></center>
where <math>j=1</math> is to the left and <math>j=2</math> is to the right of
<math>x=0.</math>
We apply the Fourier transform to these equations in
<math>x<0</math> and <math>x>0</math> and obtain algebraic expressions of the Fourier transform of
<math>\phi\left( x,0\right) </math>. The Fourier transforms of <math>\phi\left( x,0\right)
</math> in <math>x<0</math> and <math>x>0</math> are defined as
<center><math>
\Phi^{-}\left( \alpha,z\right) =\int_{-\infty}^{0}\phi\left( x,z\right)
e^{\mathrm{i}\alpha x}dx
</math></center>
and
<center><math>
\Phi^{+}\left( \alpha,z\right)
=\int_{0}^{\infty}\phi\left( x,z\right) e^{\mathrm{i}\alpha
x}dx.
</math></center>
Notice that the superscript `<math>+</math>' and `<math>-</math>' correspond to the integral domain.
The [[Sommerfeld Radiation Condition]]s introduced in section 2.3 restrict the amplitude of
<math>\phi\left( x,z\right) </math> to stay finite as <math>\left| x\right| \rightarrow
\infty</math> because of the absence of dissipation. It follows that <math>\Phi
^{-}\left( \alpha,z\right) </math> and <math>\Phi^{+}\left( \alpha,z\right) </math> are
regular in <math>\operatorname{Im}\alpha<0</math> and <math>\operatorname{Im}\alpha>0</math>, respectively.

It is possible to find the inverse transform of the sum of functions
<math>\Phi=\Phi^{-}+\Phi^{+}</math> using the inverse formula if the two
functions share a strip of their analyticity in which a integral path
<math>-\infty<\varepsilon<\infty</math> can be taken. The Wiener-Hopf technique usually
involves the spliting of complex valued functions into a product of two
regular functions in the lower and upper half planes and then the application
of Liouville's theorem, which states that
''a function that is bounded and analytic in the whole plane is constant everywhere''. A corollary of
Liouville's theorem is that a function which is asymptotically <math>o\left(
\alpha^{n+1}\right) </math> as <math>\left| \alpha\right| \rightarrow\infty</math> must be a
polynomial of <math>n</math>'th order.

We will show two ways of solving the given boundary value problem.
First we figure out the domains of regularity of the
functions of complex variable defined by integrals, thus we are
able to calculate the inverse that has the appropriate asymptotic behaviour.
Secondly we find the asymptotic behaviour of the solution from
the physical conditions, thus we already know the domains in which the Fourier
transforms are regular and are able to calculate the inverse transform.

=Weierstrass's factor theorem =

As mentioned above, we will require splitting a ratio of two functions of a
complex variable in <math>\alpha</math>-plane. We here remind ourselves of Weierstrass's
factor theorem ([[Carrier, Krook and Pearson 1966]] section 2.9) which can be proved using the
Mittag-Leffler theorem.

Let <math>H\left( \alpha\right) </math> denote a function that is analytic in the whole
<math>\alpha</math>-plane (except possibly at infinity) and has zeros of first order at
<math>a_{0}</math>, <math>a_{1}</math>, <math>a_{2}</math>, ..., and no zero is located at the origin. Consider
the Mittag-Leffler expansion of the logarithmic derivative of <math>H\left(
\alpha\right) </math>, i.e.,
<center><math>
\frac{d\log H\left( \alpha\right) }{d\alpha} =\frac{1}{H\left(
\alpha\right) }\frac{dH\left( \alpha\right) }{d\alpha}
=\frac{d\log H\left( 0\right) }{d\alpha}+\sum_{n=0}^{\infty}\left[
\frac{1}{\alpha-a_{n}}+\frac{1}{a_{n}}\right] .
</math></center>
Integrating both sides in <math>\left[ 0,\alpha\right] </math> we have
<center><math>
\log H\left( \alpha\right) =\log H\left( 0\right) +\alpha\frac{d\log
H\left( 0\right) }{d\alpha}+\sum_{n=0}^{\infty}\left[ \log\left(
1-\frac{\alpha}{a_{n}}\right) +\frac{\alpha}{a_{n}}\right] .
</math></center>
Therefore, the expression for <math>H\left( \alpha\right) </math> is
<center><math>
H\left( \alpha\right) =H\left( 0\right) \exp\left[ \alpha\frac{d\log
H\left( 0\right) }{d\alpha}\right] \prod_{n=0}^{\infty}\left(
1-\frac{\alpha}{a_{n}}\right) e^{\alpha/a_{n}}.
</math></center>

If <math>H\left( \alpha\right) </math> is even, then <math>dH\left( 0\right) /d\alpha=0</math>
and <math>-a_{n}</math> is a zero if <math>a_{n}</math> is a zero. Then we have the simpler
expression
<center><math>
H\left( \alpha\right) =H\left( 0\right) \prod_{n=0}^{\infty}\left(
1-\frac{\alpha^{2}}{a_{n}^{2}}\right) .
</math></center>

=Derivation of the Wiener-Hopf equation=

We derive algebraic expressions for <math>\Phi^{\pm}\left( \alpha,z\right) </math>
using integral transforms of the equations which gives
<center><math>
\left\{ \frac{\partial^{2}}{\partial z^{2}}-\left( \alpha^{2}+k^{2}\right)
\right\} \Phi^{\pm}\left( \alpha,z\right) =\pm\left\{ \mathrm{i}
\alpha\phi\left( 0,z\right) -\phi_{x}\left( 0,z\right) \right\} .
</math></center>
Hence, the solutions of the above ordinary differential equations with the
Fourier transform of condition ((4-45)),
<center><math>
\Phi_{z}^{\pm}\left( \alpha,-H\right) =0,
</math></center>
can be written as
<center><math>
\Phi^{\pm}\left( \alpha,z\right) =\Phi^{\pm}\left( \alpha,0\right)
\frac{\cosh\gamma\left( z+H\right) }{\cosh\gamma H}\pm g\left(
\alpha,z\right)
</math></center>
where <math>\gamma=\sqrt{\alpha^{2}+k^{2}}</math> and <math>g\left( \alpha,z\right) </math> is a
function determined by <math>\left\{ \mathrm{i}\alpha\phi\left(
0,z\right) -\phi_{x}\left( 0,z\right) \right\} </math>,
<center><math>
g\left( \alpha,z\right) =\frac{h_{z}\left( \alpha,-H\right) }{\gamma
}\left( \tanh\gamma H\cosh\gamma\left( z+H\right) -\sinh\gamma\left(
z+H\right) \right)
</math></center>
<center><math>
+h\left( \alpha,z\right) \left( 1-\frac{\cosh\gamma\left( z+H\right)
}{\cosh\gamma H}\right) ,
</math></center>
<center><math>
h\left( \alpha,z\right) =\int^{z}\frac{\sinh\gamma\left( z-t\right)
}{\gamma}\left\{ \phi_{x}\left( 0,t\right) -\mathrm{i}\alpha
\phi\left( 0,t\right) \right\} dt.
</math></center>
Note that <math>\operatorname{Re}\gamma>0</math> when <math>\operatorname{Re}\alpha>0</math> and
<math>\operatorname{Re}\gamma<0</math> when <math>\operatorname{Re}\alpha<0</math>. We have, by
differentiating both sides with respect to <math>z</math> at <math>z=0</math>
<center><math>
\Phi_{z}^{\pm}\left( \alpha,0\right) =\Phi^{\pm}\left( \alpha,0\right)
\gamma\tanh\gamma H\pm g_{z}\left( \alpha,0\right)
</math></center>
where <math>\Phi_{z}^{\pm}\left( \alpha,0\right) </math> denotes the <math>z</math>-derivative. We
apply the integral transform to the free-surface conditions in <math>x<0</math> and <math>x>0</math>,
<center><math>
\left\{ D_{1}\gamma^{4}-m_{1}\omega^{2}+\rho g\right\} \Phi_{z}^{-}\left(
\alpha,0\right) -\rho\omega^{2}\Phi^{-}\left( \alpha,0\right) +P_{1}\left(
\alpha\right) =0,
</math></center>
<center><math>
\left\{ D_{2}\gamma^{4}-m_{2}\omega^{2}+\rho g\right\} \Phi_{z}^{+}\left(
\alpha,0\right) -\rho\omega^{2}\Phi^{+}\left( \alpha,0\right) -P_{2}\left(
\alpha\right) =0,
</math></center>
where
<center><math>
P_{j}\left( \alpha\right) =D_{j}\left[ c_{3}^{j}-\mathrm{i}c_{2}
^{j}\alpha-\left( \alpha+2k^{2}\right) \left( c_{1}^{j}-\mathrm{i}
c_{0}^{j}\alpha\right) \right] ,\;j=1,2,
</math></center>
<center><math>
c_{\mathrm{i}}^{1}=\left. \left( \frac{\partial}{\partial x}\right)
^{i}\phi_{z}\right| _{x=0-,z=0},\;c_{\mathrm{i}}
^{2}=\left. \left( \frac{\partial}{\partial x}\right) ^{i
}\phi_{z}\right| _{x=0+,z=0},\;i=0,1,2,3.
</math></center>
We therefore have
<center><math>\begin{matrix}
f_{1}\left( \gamma\right) \Phi_{z}^{-}\left( \alpha,0\right) +C_{1}\left(
\alpha\right) & =0 \\
f_{2}\left( \gamma\right) \Phi_{z}^{+}\left( \alpha,0\right) +C_{2}\left(
\alpha\right) & =0
\end{matrix}</math></center>
where
<center><math>
f_{j}\left( \gamma\right) =D_{j}\gamma^{4}-m_{j}\omega^{2}+\rho
g-\frac{\rho\omega^{2}}{\gamma\tanh\gamma H},\;j=1,2,
</math></center>
<center><math>
C_{1}\left( \alpha\right) =-\frac{\rho\omega^{2}g_{z}\left(
\alpha,0\right) }{\gamma\tanh\gamma H}+P_{1}\left( \alpha\right)
,\;C_{2}\left( \alpha\right) =\frac{\rho\omega^{2}g_{z}\left(
\alpha,0\right) }{\gamma\tanh\gamma H}-P_{2}\left( \alpha\right) .
</math></center>

= [[Dispersion Relation for a Floating Elastic Plate]] =

Functions <math>f_{1}</math> and <math>f_{2}</math> are
the [[Dispersion Relation for a Floating Elastic Plate]] and the zeros of these functions are the primary tools in
our method of deriving the solutions.
Functions <math>\Phi_{z}^{-}\left( \alpha,0\right) </math>, and <math>\Phi_{z}^{+}\left(
\alpha,0\right) </math> are defined in <math>\operatorname{Im}\alpha<0</math> and
<math>\operatorname{Im}\alpha>0</math>, respectively. However they can be extended in the
whole plane defined via analytic
continuation. This show that the
singularities of <math>\Phi_{z}^{-}</math> and <math>\Phi_{z}^{+}</math> are determined by the
positions of the zeros of <math>f_{1}</math>\ and <math>f_{2}</math>, since <math>g_{z}\left(
\alpha,0\right) </math> is bounded and zeros of <math>\gamma\tanh\gamma H</math> are not the
singularities of <math>\Phi_{z}^{\pm}</math>. We denote sets of singularities
corresponding to zeros of <math>f_{1}</math> and <math>f_{2}</math> by <math>\mathcal{K}_{1}</math> and
<math>\mathcal{K}_{2}</math> respectively
<center><math>
\mathcal{K}_{j}=\left\{ \alpha\in\mathbb{C}\mid f_{j}\left( \gamma\right)
=0,\;\alpha=\sqrt{\gamma^{2}-k^{2}},\, \operatorname{Im}(\alpha)>0\,\,\,\mathrm{or}\,\,\,
\alpha>0\,\,\,\mathrm{for}\, \alpha\in\mathbb{R}\right\} .
</math></center>
We avoid numbering the roots with this notation, but for numerical purposes this is important
and we order them with increasing size.

= Solution of the W-H Equation=

Using the Mittag-Leffler theorem ([[Carrier, Krook and Pearson 1966]] section 2.9), functions <math>\Phi_{z}^{\pm}</math> can be expressed by a series of fractional functions that contribute to the solutions. Thus, we have series expansions of <math>\Phi_{z}^{-}</math> and <math>\Phi_{z}^{+}</math>
<center><math>
\Phi_{z}^{-}\left( \alpha,0\right) =\frac{Q_{1}\left( -\lambda\right)
}{\alpha+\lambda}+\sum_{q\in\mathcal{K}_{1}}\frac{Q_{1}\left( q\right)
}{\alpha-q},\;\Phi_{z}^{+}\left( \alpha,0\right) =\sum_{q\in\mathcal{K}_{2}
}\frac{Q_{2}\left( q\right) }{\alpha+q},
</math></center>
where <math>\lambda\ </math>is a positive real singularity of <math>\Phi_{z}^{-}</math> and <math>Q_{1}</math>, <math>Q_{2}</math> are coefficient functions yet to be determined. Note that <math>\Phi _{z}^{-}\left( \alpha,0\right) </math>\ has an additional term corresponding to <math>-\lambda</math>\ because of the incident wave. The solution <math>\phi\left( x,0\right) </math>, <math>x<0</math> is then obtained using the inverse Fourier transform taken over the line shown in Fig.~((roots5)a)
<center><math>
\phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty-\mathrm{i}
\tau}^{\infty-\mathrm{i}\tau}\Phi_{z}^{-}e^{-\mathrm{i}\alpha
x}d\alpha=\mathrm{i}Q_{1}\left( -\lambda\right) e^{\mathrm{i}
\lambda x}+\sum\limits_{q\in\mathcal{K}_{1}}\mathrm{i}Q_{1}\left(
q\right) e^{-\mathrm{i}qx} (4-51)
</math></center>
where <math>\tau</math>\ is an infinitesimally small positive real number. Note that
<math>k=\lambda\sin\theta</math>. Similarly, we obtain <math>\phi\left( x,0\right) </math> for
<math>x>0</math> by taking the integration path shown in Fig.~((roots5)b), then we
have
<center><math>
\phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty+\mathrm{i}
\tau}^{\infty+\mathrm{i}\tau}\Phi_{z}^{+}e^{-\mathrm{i}\alpha
x}d\alpha=-\sum\limits_{q\in\mathcal{K}_{2}}\mathrm{i}Q_{2}\left(
q\right) e^{\mathrm{i}qx}.
</math></center>

The Wiener-Hopf technique enables us to calculate coefficients <math>Q_{1}</math> and <math>Q_{2}</math> without knowing functions <math>C_{1}</math>, <math>C_{2}</math>, or <math>\left\{ \phi _{x}\left( 0,z\right) -\mathrm{i}\alpha\phi\left( 0,z\right) \right\} </math>. It requires the domains of analyticity of Eqn.~((4-46)) and Eqn.~((4-47)) to have a common strip of analyticity which they do not have right now. We create such a strip by shifting a singularity of <math>\Phi_{z}^{-}</math> in Eqn.~((4-46)) to <math>\Phi_{z}^{+}</math> in Eqn.~((4-47)) (we can also create a strip by moving a singularity of <math>\Phi_{z}^{+} </math>, and more than one of the singularities can be moved). Here, we shift <math>-\lambda</math> as shown in Fig.~((roots5)a), so that the common strip of analyticity denoted by <math>\mathcal{D}</math> is created on the real axis, which passes above the two negative real singularities and below the two positive real singularities. We denote the domain above and including <math>\mathcal{D}</math> by <math>\mathcal{D}_{+}</math>\ and below and including <math>\mathcal{D}</math> by <math>\mathcal{D}_{-}</math>. Hence, the zeros of <math>f_{1}</math> and <math>f_{2}</math> belong to either <math>\mathcal{D}_{+}</math> or <math>\mathcal{D}_{-} </math>.

Let <math>\Psi_{z}^{-}</math> be a function created by subtracting a singularity from function <math>\Phi_{z}^{-}</math>. Then\ <math>\Psi_{z}^{-}\left( \alpha,0\right) </math> is regular in <math>\mathcal{D}_{-}</math>.\ Since the removed singularity term makes no contribution to the solution,\ from Eqn.~((4-46)), <math>\Psi_{z}^{-}</math> satisfies
<center><math>
f_{1}\left( \gamma\right) \Psi_{z}^{-}\left( \alpha,0\right) +C_{1}\left(
\alpha\right) =0. (100)
</math></center>
Eqn.~((4-47)) becomes, as a result of modifying function <math>\Phi_{z}^{+} </math> to a function denoted by <math>\Psi_{z}^{+}</math> with an additional singularity term,
<center><math>
f_{2}\left( \gamma\right) \Psi_{z}^{+}\left( \alpha,0\right) -\frac
{f_{2}\left( \lambda^{\prime}\right) Q_{1}\left( -\lambda\right) }
{\alpha+\lambda}+C_{2}\left( \alpha\right) =0. (4-48)
</math></center>
Our aim now is to find a formula for
<center><math>
\Psi_{z}\left( \alpha,0\right) =\Psi_{z}^{-}\left( \alpha,0\right)
+\Psi_{z}^{+}\left( \alpha,0\right)
</math></center>
in <math>\alpha\in\mathcal{D}</math> so that its inverse Fourier transform can be calculated.

Adding both sides of Eqn.~((100)) and Eqn.~((4-48)) gives the Wiener-Hopf equation
<center><math>
f_{1}\left( \gamma\right) \Psi_{z}^{-}\left( \alpha,0\right) +f_{2}\left(
\gamma\right) \Psi_{z}^{+}\left( \alpha,0\right) -\frac{f_{2}\left(
\lambda^{\prime}\right) Q_{1}\left( -\lambda\right) }{\alpha+\lambda
}+C\left( \alpha\right) =0 (4-41)
</math></center>
where <math>C\left( \alpha\right) =C_{1}\left( \alpha\right) -C_{2}\left(
\alpha\right) </math>. This equation can alternatively be written as
<center><math>
\begin{matrix}
[c]{c}
f_{2}\left( \gamma\right) \left[ f\left( \gamma\right) \Psi_{z}
^{+}\left( \alpha,0\right) -\frac{f_{2}\left( \lambda^{\prime}\right)
Q_{1}\left( -\lambda\right) }{\alpha+\lambda}+C\left( \alpha\right)
\right] \\
=-f_{1}\left( \gamma\right) \left[ f\left( \gamma\right) \Psi_{z}
^{-}\left( \alpha,0\right) +\frac{f_{2}\left( \lambda^{\prime}\right)
Q_{1}\left( -\lambda\right) }{\alpha+\lambda}-C\left( \alpha\right)
\right]
\end{matrix}
(eq:WH2)
</math></center>
where <math>f\left( \gamma\right) =f_{2}\left( \gamma\right) -f_{1}\left(
\gamma\right) .</math>

We now modify Eqn.~((eq:WH2)) so that the right and left hand sides of the equation become regular in <math>\mathcal{D}_{-}</math> and <math>\mathcal{D}_{+}</math> respectively. Using Weierstrass's factor theorem given in the previous subsection, the ratio <math>f_{2}/f_{1}</math> can be factorized into infinite products of polynomials <math>\left( 1-\alpha/q\right) </math>, <math>q\in\mathcal{K}_{1}</math> and <math>\mathcal{K}_{2}</math>. Hence, using a regular non-zero function <math>K\left( \alpha\right) </math> in <math>\mathcal{D}_{+}</math>,
<center><math>
K\left( \alpha\right) =\left( \prod\limits_{q\in\mathcal{K}_{1}}
\frac{q^{\prime}}{q+\alpha}\right) \left( \prod\limits_{q\in\mathcal{K}_{2}
}\frac{q+\alpha}{q^{\prime}}\right) (eq:K)
</math></center>
where <math>q^{\prime}=\sqrt{q^{2}+k^{2}}</math>, then we have
<center><math>
\frac{f_{2}}{f_{1}}=K\left( \alpha\right) K\left( -\alpha\right) .
</math></center>
Note that the factorization is done in the <math>\alpha</math>-plane, hence functions
<math>f_{1}</math> and <math>f_{2}</math> are here seen as functions of <math>\alpha</math> and we are actually
factorizing
<center><math>
\frac{f_{2}\left( \gamma\right) \gamma\sinh\gamma H}{f_{1}\left(
\gamma\right) \gamma\sinh\gamma H}
</math></center>
in order to satisfy the conditions given in the previous subsection. Then Eqn.~((eq:WH2)) can be rewritten as
<center><math>
\begin{matrix}
[c]{c}
K\left( \alpha\right) \left[ f\left( \gamma\right) \Psi_{z}^{+}+C\right]
-\left( K\left( \alpha\right) -\frac{1}{K\left( \lambda\right) }\right)
\frac{f_{2}\left( \lambda^{\prime}\right) Q_{1}\left( -\lambda\right)
}{\alpha+\lambda}\\
=-\frac{1}{K\left( -\alpha\right) }\left[ f\left( \gamma\right) \Psi
_{z}^{-}-C\right] -\left( \frac{1}{K\left( -\alpha\right) }-\frac
{1}{K\left( \lambda\right) }\right) \frac{f_{2}\left( \lambda^{\prime
}\right) Q_{1}\left( -\lambda\right) }{\alpha+\lambda}.
\end{matrix}
(4-26)
</math></center>
Note that the infinite products in Eqn.~((eq:K)) converge in the order of <math>q^{-5}</math> as <math>\left| q\right| </math> becomes large, thus numerical computation of <math>K\left( \alpha\right) </math> does not pose any difficulties.

The left hand side of Eqn.~((4-26)) is regular in <math>\mathcal{D}_{+}</math> and the right hand side is regular in <math>\mathcal{D}_{-}</math>. Notice that a function is added to both sides of the equation to make the right hand side of the equation regular in <math>\mathcal{D}_{-}</math>. The left hand side of Eqn.~((4-26)) is <math>o\left( \alpha^{4}\right) </math> as <math>\left| \alpha\right| \rightarrow \infty</math> in <math>\mathcal{D}_{+}</math>, since <math>\Psi_{z}^{+}\rightarrow0</math> and <math>K\left( \alpha\right) =O\left( 1\right) </math>\ as <math>\left| \alpha\right| \rightarrow\infty</math> in <math>\mathcal{D}_{+}</math>. The right hand side of Eqn.~((4-26)) has the equivalent analytic properties in <math>\mathcal{D}_{-}</math>. Liouville's theorem (Carrier, Krook and Pearson [[carrier]] section 2.4) tells us that there exists a function, which we denote <math>J\left( \alpha\right) </math>, uniquely defined by Eqn.~((4-26)), and function <math>J\left( \alpha\right) </math> is a polynomial of degree three in the whole plane. Hence
<center><math>
J\left( \alpha\right) =d_{0}+d_{1}\alpha+d_{2}\alpha^{2}+d_{3}\alpha^{3}.
</math></center>

Equating Eqn.~((4-26)) for <math>\Psi_{z}</math> gives
<center><math>
\Psi_{z}\left( \alpha,0\right) =\frac{-F\left( \alpha\right) }{K\left(
\alpha\right) f_{1}\left( \gamma\right) }\;=or= \;-\frac{K\left(
-\alpha\right) F\left( \alpha\right) }{f_{2}\left( \gamma\right)
} (4-50)
</math></center>
where
<center><math>
F\left( \alpha\right) =J\left( \alpha\right) -\frac{Q_{1}\left(
-\lambda\right) f_{2}\left( \lambda^{\prime}\right) }{\left(
\alpha+\lambda\right) K\left( \lambda\right) }.
</math></center>
Notice that procedure from Eqn.~((eq:WH2)) to Eqn.~((4-26)) eliminates
the need for calculating constant <math>C</math> in Eqn.~((4-26)).

For <math>x<0</math> we close the integral contour in <math>\mathcal{D}_{+}</math>, and put the
incident wave back, then we have
<center><math>
\phi_{z}\left( x,0\right) =\mathrm{i}Q_{1}\left( -\lambda\right)
e^{\mathrm{i}\lambda x}-\sum\limits_{q\in\mathcal{K}_{1}}
\frac{\mathrm{i}F\left( q\right) q^{\prime}R_{1}\left( q^{\prime
}\right) }{qK\left( q\right) }e^{-\mathrm{i}qx}, (eq:solution1)
</math></center>
where <math>R_{1}\left( q^{\prime}\right) </math> is a residue of <math>\left[ f_{1}\left(
\gamma\right) \right] ^{-1}<math> at </math>\gamma=q^{\prime}</math>
<center><math>\begin{matrix}
R_{1}\left( q^{\prime}\right) & =\left( \left. \frac{df_{1}\left(
\gamma\right) }{d\gamma}\right| _{\gamma=q^{\prime}}\right) ^{-1}
\\
& =\left\{ 5D_{1}q^{\prime3}+\frac{b_{1}}{q^{\prime}}+\frac{H}{q^{\prime}
}\left( \frac{\left( D_{1}q^{\prime5}+b_{1}q^{\prime}\right) ^{2}-\left(
\rho\omega^{2}\right) ^{2}}{\rho\omega^{2}}\right) \right\} ^{-1}. (R)
\end{matrix}</math></center>
We used <math>b_{1}=-m_{1}\omega^{2}+\rho g</math> and <math>f_{1}\left( q^{\prime}\right)
=0</math> to simplify the formula. Displacement <math>w\left( x\right) </math> can be
obtained by multiplying Eqn.~((eq:solution1)) by <math>-\mathrm{i}
/\omega</math>. Notice that the formula for the residue is again expressed by a
polynomial using the dispersion equation as shown in section (sec:3),
which gives us a stable numerical computation of the solutions.

The velocity potential <math>\phi\left( x,z\right) </math> can be obtained using
Eqn.~((4-44)) and Eqn.~((eq:4)),
<center><math>
\phi\left( x,z\right) =\frac{\mathrm{i}Q_{1}\left( -\lambda\right)
\cosh\lambda^{\prime}\left( z+H\right) }{\lambda^{\prime}\sinh
\lambda^{\prime}H}e^{\mathrm{i}\lambda x}-\sum\limits_{q\in
\mathcal{K}_{1}}\frac{\mathrm{i}F\left( q\right) R_{1}\left(
q^{\prime}\right) \cosh q^{\prime}\left( z+H\right) }{qK\left( q\right)
\sinh q^{\prime}H}e^{-\mathrm{i}qx}
</math></center>
where <math>\lambda^{\prime}=\sqrt{\lambda^{2}+k^{2}}</math>.

For <math>x>0</math>, the functions <math>\phi_{z}\left( x,0\right) </math> and <math>\phi\left(
x,z\right) </math> are obtained by closing the integral contour in <math>\mathcal{D}
_{-}</math>,
<center><math>\begin{matrix}
\phi_{z}\left( x,0\right) & =-\sum\limits_{q\in\mathcal{K}_{2}}
\frac{\mathrm{i}K\left( q\right) F\left( -q\right) q^{\prime}
R_{2}\left( q^{\prime}\right) }{q}e^{\mathrm{i}qx}, (4-28)\\
\phi\left( x,z\right) & =-\sum\limits_{q\in\mathcal{K}_{2}}\frac
{\mathrm{i}K\left( q\right) F\left( -q\right) R_{2}\left(
q^{\prime}\right) \cosh q^{\prime}\left( z+H\right) }{q\sinh q^{\prime}
H}e^{\mathrm{i}qx},
\end{matrix}</math></center>
where <math>R_{2}</math> is a residue of <math>\left[ f_{2}\left( \gamma\right) \right]
^{-1}</math> and its formula can be obtained by replacing the subscript <math>1</math> with
<math>2</math> in Eqn.~((R)). Notice that since <math>R_{j}\sim O\left( q^{-9}\right) </math>,
<math>j=1,2</math>, the coefficients of <math>\phi_{z}</math> of Eqn.~((4-28)) decay as
<math>O\left( q^{-6}\right) </math> as <math>\left| q\right| </math> becomes large, so the
displacement is bounded up to the fourth <math>x</math>-derivatives. In a physical sense,
the biharmonic term of the plate equation for the vertical displacement is
associated with the strain energy due to bending of the plate as explained in
chapter 2. Hence, up to fourth derivative of the displacement function should
be bounded, as has been confirmed. The coefficients of <math>\phi</math>, have an extra
<math>1/q^{\prime}\tanh q^{\prime}H</math> term which is <math>O\left( q^{4}\right) </math>, hence
the coefficients decay as <math>O\left( q^{-2}\right) </math> as <math>\left| q\right| </math>
becomes large. Therefore, <math>\phi</math> is bounded everywhere including at <math>x=0</math>.

Shifting a singularity of one function to the other is equivalent to
subtracting an incident wave from both functions then solving the boundary
value problem for the scattered field as in [[Balmforth and Craster 1999]]. As mentioned,
any one of the singularities can be shifted as long as it creates a common
strip of analyticity for the newly created functions. We chose <math>-\lambda</math>
because of the convenience of the symmetry in locations of the singularities.
The method of subtracting either incoming or transmitting wave requires the
Fourier transform be performed twice, first to express the solution with a
series expansion, and second to solve the system of equations for the newly
created functions. Thus, we find the method of shifting a singularity shown
here is advantageous to other methods since it needs the Fourier transform
only once to obtain the Wiener-Hopf equation.

The polynomial <math>J\left( \alpha\right) </math> is yet to be determined. In the
following section the coefficients of <math>J\left( \alpha\right) </math> will be
determined from conditions at <math>x=0\pm</math>, <math>-\infty<y<\infty</math>, <math>z=0</math>.

[[Category:Floating Elastic Plate]]
[[Category:Wiener-Hopf]]

Wiener-Hopf Elastic Plate Solution

2008-03-10T13:53:49Z

Adipro: /* Solution of the W-H Equation */

= Introduction =

We present here the [[:Category:Wiener-Hopf|Wiener-Hopf]] solution to the problem of a
two semi-infinite [[Two-Dimensional Floating Elastic Plate|Two-Dimensional Floating Elastic Plates]].
The solution method is based on the one presented by [[Chung and Fox 2002]]. This problem
has been well studied and the first solution was by [[Evans and Davies 1968]]
but they did not actually develop the method sufficiently to be able to calculate the solution.
A solution was also developed by [[Balmforth and Craster 1999]] and by [[Tkacheva 2004]].

The theory is described in [[:Category:Wiener-Hopf|Wiener-Hopf]].

= Elastic Plate =

We imagine two semi-infinite [[:category:Floating Elastic Plate|Floating Elastic Plates]]
of (possibly) different properties. The equations are the following
<center><math>
\left( D_{j}\left( \frac{\partial^{2}}{\partial x^{2}}-k^{2}\right)
^{2}+\rho g-m_{j}\omega^{2}\right) \phi_{z}-\rho\omega^{2}\phi
=0,\;j=1,2,\;z=0
</math></center>
<center><math>
\left( \frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
z^{2}}-k^{2}\right) \phi =0,\;-H<z<0,
</math></center>
<center><math>
\phi_{z} =0,\;\;z=-H.
</math></center>
where <math>j=1</math> is to the left and <math>j=2</math> is to the right of
<math>x=0.</math>
We apply the Fourier transform to these equations in
<math>x<0</math> and <math>x>0</math> and obtain algebraic expressions of the Fourier transform of
<math>\phi\left( x,0\right) </math>. The Fourier transforms of <math>\phi\left( x,0\right)
</math> in <math>x<0</math> and <math>x>0</math> are defined as
<center><math>
\Phi^{-}\left( \alpha,z\right) =\int_{-\infty}^{0}\phi\left( x,z\right)
e^{\mathrm{i}\alpha x}dx
</math></center>
and
<center><math>
\Phi^{+}\left( \alpha,z\right)
=\int_{0}^{\infty}\phi\left( x,z\right) e^{\mathrm{i}\alpha
x}dx.
</math></center>
Notice that the superscript `<math>+</math>' and `<math>-</math>' correspond to the integral domain.
The [[Sommerfeld Radiation Condition]]s introduced in section 2.3 restrict the amplitude of
<math>\phi\left( x,z\right) </math> to stay finite as <math>\left| x\right| \rightarrow
\infty</math> because of the absence of dissipation. It follows that <math>\Phi
^{-}\left( \alpha,z\right) </math> and <math>\Phi^{+}\left( \alpha,z\right) </math> are
regular in <math>\operatorname{Im}\alpha<0</math> and <math>\operatorname{Im}\alpha>0</math>, respectively.

It is possible to find the inverse transform of the sum of functions
<math>\Phi=\Phi^{-}+\Phi^{+}</math> using the inverse formula if the two
functions share a strip of their analyticity in which a integral path
<math>-\infty<\varepsilon<\infty</math> can be taken. The Wiener-Hopf technique usually
involves the spliting of complex valued functions into a product of two
regular functions in the lower and upper half planes and then the application
of Liouville's theorem, which states that
''a function that is bounded and analytic in the whole plane is constant everywhere''. A corollary of
Liouville's theorem is that a function which is asymptotically <math>o\left(
\alpha^{n+1}\right) </math> as <math>\left| \alpha\right| \rightarrow\infty</math> must be a
polynomial of <math>n</math>'th order.

We will show two ways of solving the given boundary value problem.
First we figure out the domains of regularity of the
functions of complex variable defined by integrals, thus we are
able to calculate the inverse that has the appropriate asymptotic behaviour.
Secondly we find the asymptotic behaviour of the solution from
the physical conditions, thus we already know the domains in which the Fourier
transforms are regular and are able to calculate the inverse transform.

=Weierstrass's factor theorem =

As mentioned above, we will require splitting a ratio of two functions of a
complex variable in <math>\alpha</math>-plane. We here remind ourselves of Weierstrass's
factor theorem ([[Carrier, Krook and Pearson 1966]] section 2.9) which can be proved using the
Mittag-Leffler theorem.

Let <math>H\left( \alpha\right) </math> denote a function that is analytic in the whole
<math>\alpha</math>-plane (except possibly at infinity) and has zeros of first order at
<math>a_{0}</math>, <math>a_{1}</math>, <math>a_{2}</math>, ..., and no zero is located at the origin. Consider
the Mittag-Leffler expansion of the logarithmic derivative of <math>H\left(
\alpha\right) </math>, i.e.,
<center><math>
\frac{d\log H\left( \alpha\right) }{d\alpha} =\frac{1}{H\left(
\alpha\right) }\frac{dH\left( \alpha\right) }{d\alpha}
=\frac{d\log H\left( 0\right) }{d\alpha}+\sum_{n=0}^{\infty}\left[
\frac{1}{\alpha-a_{n}}+\frac{1}{a_{n}}\right] .
</math></center>
Integrating both sides in <math>\left[ 0,\alpha\right] </math> we have
<center><math>
\log H\left( \alpha\right) =\log H\left( 0\right) +\alpha\frac{d\log
H\left( 0\right) }{d\alpha}+\sum_{n=0}^{\infty}\left[ \log\left(
1-\frac{\alpha}{a_{n}}\right) +\frac{\alpha}{a_{n}}\right] .
</math></center>
Therefore, the expression for <math>H\left( \alpha\right) </math> is
<center><math>
H\left( \alpha\right) =H\left( 0\right) \exp\left[ \alpha\frac{d\log
H\left( 0\right) }{d\alpha}\right] \prod_{n=0}^{\infty}\left(
1-\frac{\alpha}{a_{n}}\right) e^{\alpha/a_{n}}.
</math></center>

If <math>H\left( \alpha\right) </math> is even, then <math>dH\left( 0\right) /d\alpha=0</math>
and <math>-a_{n}</math> is a zero if <math>a_{n}</math> is a zero. Then we have the simpler
expression
<center><math>
H\left( \alpha\right) =H\left( 0\right) \prod_{n=0}^{\infty}\left(
1-\frac{\alpha^{2}}{a_{n}^{2}}\right) .
</math></center>

=Derivation of the Wiener-Hopf equation=

We derive algebraic expressions for <math>\Phi^{\pm}\left( \alpha,z\right) </math>
using integral transforms of the equations which gives
<center><math>
\left\{ \frac{\partial^{2}}{\partial z^{2}}-\left( \alpha^{2}+k^{2}\right)
\right\} \Phi^{\pm}\left( \alpha,z\right) =\pm\left\{ \mathrm{i}
\alpha\phi\left( 0,z\right) -\phi_{x}\left( 0,z\right) \right\} .
</math></center>
Hence, the solutions of the above ordinary differential equations with the
Fourier transform of condition ((4-45)),
<center><math>
\Phi_{z}^{\pm}\left( \alpha,-H\right) =0,
</math></center>
can be written as
<center><math>
\Phi^{\pm}\left( \alpha,z\right) =\Phi^{\pm}\left( \alpha,0\right)
\frac{\cosh\gamma\left( z+H\right) }{\cosh\gamma H}\pm g\left(
\alpha,z\right)
</math></center>
where <math>\gamma=\sqrt{\alpha^{2}+k^{2}}</math> and <math>g\left( \alpha,z\right) </math> is a
function determined by <math>\left\{ \mathrm{i}\alpha\phi\left(
0,z\right) -\phi_{x}\left( 0,z\right) \right\} </math>,
<center><math>
g\left( \alpha,z\right) =\frac{h_{z}\left( \alpha,-H\right) }{\gamma
}\left( \tanh\gamma H\cosh\gamma\left( z+H\right) -\sinh\gamma\left(
z+H\right) \right)
</math></center>
<center><math>
+h\left( \alpha,z\right) \left( 1-\frac{\cosh\gamma\left( z+H\right)
}{\cosh\gamma H}\right) ,
</math></center>
<center><math>
h\left( \alpha,z\right) =\int^{z}\frac{\sinh\gamma\left( z-t\right)
}{\gamma}\left\{ \phi_{x}\left( 0,t\right) -\mathrm{i}\alpha
\phi\left( 0,t\right) \right\} dt.
</math></center>
Note that <math>\operatorname{Re}\gamma>0</math> when <math>\operatorname{Re}\alpha>0</math> and
<math>\operatorname{Re}\gamma<0</math> when <math>\operatorname{Re}\alpha<0</math>. We have, by
differentiating both sides with respect to <math>z</math> at <math>z=0</math>
<center><math>
\Phi_{z}^{\pm}\left( \alpha,0\right) =\Phi^{\pm}\left( \alpha,0\right)
\gamma\tanh\gamma H\pm g_{z}\left( \alpha,0\right)
</math></center>
where <math>\Phi_{z}^{\pm}\left( \alpha,0\right) </math> denotes the <math>z</math>-derivative. We
apply the integral transform to the free-surface conditions in <math>x<0</math> and <math>x>0</math>,
<center><math>
\left\{ D_{1}\gamma^{4}-m_{1}\omega^{2}+\rho g\right\} \Phi_{z}^{-}\left(
\alpha,0\right) -\rho\omega^{2}\Phi^{-}\left( \alpha,0\right) +P_{1}\left(
\alpha\right) =0,
</math></center>
<center><math>
\left\{ D_{2}\gamma^{4}-m_{2}\omega^{2}+\rho g\right\} \Phi_{z}^{+}\left(
\alpha,0\right) -\rho\omega^{2}\Phi^{+}\left( \alpha,0\right) -P_{2}\left(
\alpha\right) =0,
</math></center>
where
<center><math>
P_{j}\left( \alpha\right) =D_{j}\left[ c_{3}^{j}-\mathrm{i}c_{2}
^{j}\alpha-\left( \alpha+2k^{2}\right) \left( c_{1}^{j}-\mathrm{i}
c_{0}^{j}\alpha\right) \right] ,\;j=1,2,
</math></center>
<center><math>
c_{\mathrm{i}}^{1}=\left. \left( \frac{\partial}{\partial x}\right)
^{i}\phi_{z}\right| _{x=0-,z=0},\;c_{\mathrm{i}}
^{2}=\left. \left( \frac{\partial}{\partial x}\right) ^{i
}\phi_{z}\right| _{x=0+,z=0},\;i=0,1,2,3.
</math></center>
We therefore have
<center><math>\begin{matrix}
f_{1}\left( \gamma\right) \Phi_{z}^{-}\left( \alpha,0\right) +C_{1}\left(
\alpha\right) & =0 \\
f_{2}\left( \gamma\right) \Phi_{z}^{+}\left( \alpha,0\right) +C_{2}\left(
\alpha\right) & =0
\end{matrix}</math></center>
where
<center><math>
f_{j}\left( \gamma\right) =D_{j}\gamma^{4}-m_{j}\omega^{2}+\rho
g-\frac{\rho\omega^{2}}{\gamma\tanh\gamma H},\;j=1,2,
</math></center>
<center><math>
C_{1}\left( \alpha\right) =-\frac{\rho\omega^{2}g_{z}\left(
\alpha,0\right) }{\gamma\tanh\gamma H}+P_{1}\left( \alpha\right)
,\;C_{2}\left( \alpha\right) =\frac{\rho\omega^{2}g_{z}\left(
\alpha,0\right) }{\gamma\tanh\gamma H}-P_{2}\left( \alpha\right) .
</math></center>

= [[Dispersion Relation for a Floating Elastic Plate]] =

Functions <math>f_{1}</math> and <math>f_{2}</math> are
the [[Dispersion Relation for a Floating Elastic Plate]] and the zeros of these functions are the primary tools in
our method of deriving the solutions.
Functions <math>\Phi_{z}^{-}\left( \alpha,0\right) </math>, and <math>\Phi_{z}^{+}\left(
\alpha,0\right) </math> are defined in <math>\operatorname{Im}\alpha<0</math> and
<math>\operatorname{Im}\alpha>0</math>, respectively. However they can be extended in the
whole plane defined via analytic
continuation. This show that the
singularities of <math>\Phi_{z}^{-}</math> and <math>\Phi_{z}^{+}</math> are determined by the
positions of the zeros of <math>f_{1}</math>\ and <math>f_{2}</math>, since <math>g_{z}\left(
\alpha,0\right) </math> is bounded and zeros of <math>\gamma\tanh\gamma H</math> are not the
singularities of <math>\Phi_{z}^{\pm}</math>. We denote sets of singularities
corresponding to zeros of <math>f_{1}</math> and <math>f_{2}</math> by <math>\mathcal{K}_{1}</math> and
<math>\mathcal{K}_{2}</math> respectively
<center><math>
\mathcal{K}_{j}=\left\{ \alpha\in\mathbb{C}\mid f_{j}\left( \gamma\right)
=0,\;\alpha=\sqrt{\gamma^{2}-k^{2}},\, \operatorname{Im}(\alpha)>0\,\,\,\mathrm{or}\,\,\,
\alpha>0\,\,\,\mathrm{for}\, \alpha\in\mathbb{R}\right\} .
</math></center>
We avoid numbering the roots with this notation, but for numerical purposes this is important
and we order them with increasing size.

= Solution of the W-H Equation=

Using the Mittag-Leffler theorem ([[Carrier, Krook and Pearson 1966]] section 2.9), functions <math>\Phi_{z}^{\pm}</math> can be expressed by a series of fractional functions that contribute to the solutions. Thus, we have series expansions of <math>\Phi_{z}^{-}</math> and <math>\Phi_{z}^{+}</math>
<center><math>
\Phi_{z}^{-}\left( \alpha,0\right) =\frac{Q_{1}\left( -\lambda\right)
}{\alpha+\lambda}+\sum_{q\in\mathcal{K}_{1}}\frac{Q_{1}\left( q\right)
}{\alpha-q},\;\Phi_{z}^{+}\left( \alpha,0\right) =\sum_{q\in\mathcal{K}_{2}
}\frac{Q_{2}\left( q\right) }{\alpha+q},
</math></center>
where <math>\lambda\ </math>is a positive real singularity of <math>\Phi_{z}^{-}</math> and <math>Q_{1}</math>, <math>Q_{2}</math> are coefficient functions yet to be determined. Note that <math>\Phi _{z}^{-}\left( \alpha,0\right) </math>\ has an additional term corresponding to <math>-\lambda</math>\ because of the incident wave. The solution <math>\phi\left( x,0\right) </math>, <math>x<0</math> is then obtained using the inverse Fourier transform taken over the line shown in Fig.~((roots5)a)
<center><math>
\phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty-\mathrm{i}
\tau}^{\infty-\mathrm{i}\tau}\Phi_{z}^{-}e^{-\mathrm{i}\alpha
x}d\alpha=\mathrm{i}Q_{1}\left( -\lambda\right) e^{\mathrm{i}
\lambda x}+\sum\limits_{q\in\mathcal{K}_{1}}\mathrm{i}Q_{1}\left(
q\right) e^{-\mathrm{i}qx} (4-51)
</math></center>
where <math>\tau</math>\ is an infinitesimally small positive real number. Note that
<math>k=\lambda\sin\theta</math>. Similarly, we obtain <math>\phi\left( x,0\right) </math> for
<math>x>0</math> by taking the integration path shown in Fig.~((roots5)b), then we
have
<center><math>
\phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty+\mathrm{i}
\tau}^{\infty+\mathrm{i}\tau}\Phi_{z}^{+}e^{-\mathrm{i}\alpha
x}d\alpha=-\sum\limits_{q\in\mathcal{K}_{2}}\mathrm{i}Q_{2}\left(
q\right) e^{\mathrm{i}qx}.
</math></center>

The Wiener-Hopf technique enables us to calculate coefficients <math>Q_{1}</math> and <math>Q_{2}</math> without knowing functions <math>C_{1}</math>, <math>C_{2}</math>, or <math>\left\{ \phi _{x}\left( 0,z\right) -\mathrm{i}\alpha\phi\left( 0,z\right) \right\} </math>. It requires the domains of analyticity of Eqn.~((4-46)) and Eqn.~((4-47)) to have a common strip of analyticity which they do not have right now. We create such a strip by shifting a singularity of <math>\Phi_{z}^{-}</math> in Eqn.~((4-46)) to <math>\Phi_{z}^{+}</math> in Eqn.~((4-47)) (we can also create a strip by moving a singularity of <math>\Phi_{z}^{+} </math>, and more than one of the singularities can be moved). Here, we shift <math>-\lambda</math> as shown in Fig.~((roots5)a), so that the common strip of analyticity denoted by <math>\mathcal{D}</math> is created on the real axis, which passes above the two negative real singularities and below the two positive real singularities. We denote the domain above and including <math>\mathcal{D}</math> by <math>\mathcal{D}_{+}</math>\ and below and including <math>\mathcal{D}</math> by <math>\mathcal{D}_{-}</math>. Hence, the zeros of <math>f_{1}</math> and <math>f_{2}</math> belong to either <math>\mathcal{D}_{+}</math> or <math>\mathcal{D}_{-} </math>.

Let <math>\Psi_{z}^{-}</math> be a function created by subtracting a singularity from function <math>\Phi_{z}^{-}</math>. Then\ <math>\Psi_{z}^{-}\left( \alpha,0\right) </math> is regular in <math>\mathcal{D}_{-}</math>.\ Since the removed singularity term makes no contribution to the solution,\ from Eqn.~((4-46)), <math>\Psi_{z}^{-}</math> satisfies
<center><math>
f_{1}\left( \gamma\right) \Psi_{z}^{-}\left( \alpha,0\right) +C_{1}\left(
\alpha\right) =0. (100)
</math></center>
Eqn.~((4-47)) becomes, as a result of modifying function <math>\Phi_{z}^{+} </math> to a function denoted by <math>\Psi_{z}^{+}</math> with an additional singularity term,
<center><math>
f_{2}\left( \gamma\right) \Psi_{z}^{+}\left( \alpha,0\right) -\frac
{f_{2}\left( \lambda^{\prime}\right) Q_{1}\left( -\lambda\right) }
{\alpha+\lambda}+C_{2}\left( \alpha\right) =0. (4-48)
</math></center>
Our aim now is to find a formula for
<center><math>
\Psi_{z}\left( \alpha,0\right) =\Psi_{z}^{-}\left( \alpha,0\right)
+\Psi_{z}^{+}\left( \alpha,0\right)
</math></center>
in <math>\alpha\in\mathcal{D}</math> so that its inverse Fourier transform can be calculated.

Adding both sides of Eqn.~((100)) and Eqn.~((4-48)) gives the Wiener-Hopf equation
<center><math>
f_{1}\left( \gamma\right) \Psi_{z}^{-}\left( \alpha,0\right) +f_{2}\left(
\gamma\right) \Psi_{z}^{+}\left( \alpha,0\right) -\frac{f_{2}\left(
\lambda^{\prime}\right) Q_{1}\left( -\lambda\right) }{\alpha+\lambda
}+C\left( \alpha\right) =0 (4-41)
</math></center>
where <math>C\left( \alpha\right) =C_{1}\left( \alpha\right) -C_{2}\left(
\alpha\right) </math>. This equation can alternatively be written as
<center><math>
\begin{matrix}
[c]{c}
f_{2}\left( \gamma\right) \left[ f\left( \gamma\right) \Psi_{z}
^{+}\left( \alpha,0\right) -\frac{f_{2}\left( \lambda^{\prime}\right)
Q_{1}\left( -\lambda\right) }{\alpha+\lambda}+C\left( \alpha\right)
\right] \\
=-f_{1}\left( \gamma\right) \left[ f\left( \gamma\right) \Psi_{z}
^{-}\left( \alpha,0\right) +\frac{f_{2}\left( \lambda^{\prime}\right)
Q_{1}\left( -\lambda\right) }{\alpha+\lambda}-C\left( \alpha\right)
\right]
\end{matrix}
(eq:WH2)
</math></center>
where <math>f\left( \gamma\right) =f_{2}\left( \gamma\right) -f_{1}\left(
\gamma\right) .</math>

We now modify Eqn.~((eq:WH2)) so that the right and left hand sides of the equation become regular in <math>\mathcal{D}_{-}</math> and <math>\mathcal{D}_{+}</math> respectively. Using Weierstrass's factor theorem given in the previous subsection, the ratio <math>f_{2}/f_{1}</math> can be factorized into infinite products of polynomials <math>\left( 1-\alpha/q\right) </math>, <math>q\in\mathcal{K}_{1}</math> and <math>\mathcal{K}_{2}</math>. Hence, using a regular non-zero function <math>K\left( \alpha\right) </math> in <math>\mathcal{D}_{+}</math>,
<center><math>
K\left( \alpha\right) =\left( \prod\limits_{q\in\mathcal{K}_{1}}
\frac{q^{\prime}}{q+\alpha}\right) \left( \prod\limits_{q\in\mathcal{K}_{2}
}\frac{q+\alpha}{q^{\prime}}\right) (eq:K)
</math></center>
where <math>q^{\prime}=\sqrt{q^{2}+k^{2}}</math>, then we have
<center><math>
\frac{f_{2}}{f_{1}}=K\left( \alpha\right) K\left( -\alpha\right) .
</math></center>
Note that the factorization is done in the <math>\alpha</math>-plane, hence functions
<math>f_{1}</math> and <math>f_{2}</math> are here seen as functions of <math>\alpha</math> and we are actually
factorizing
<center><math>
\frac{f_{2}\left( \gamma\right) \gamma\sinh\gamma H}{f_{1}\left(
\gamma\right) \gamma\sinh\gamma H}
</math></center>
in order to satisfy the conditions given in the previous subsection. Then Eqn.~((eq:WH2)) can be rewritten as
<center><math>
\begin{matrix}
[c]{c}
K\left( \alpha\right) \left[ f\left( \gamma\right) \Psi_{z}^{+}+C\right]
-\left( K\left( \alpha\right) -\frac{1}{K\left( \lambda\right) }\right)
\frac{f_{2}\left( \lambda^{\prime}\right) Q_{1}\left( -\lambda\right)
}{\alpha+\lambda}\\
=-\frac{1}{K\left( -\alpha\right) }\left[ f\left( \gamma\right) \Psi
_{z}^{-}-C\right] -\left( \frac{1}{K\left( -\alpha\right) }-\frac
{1}{K\left( \lambda\right) }\right) \frac{f_{2}\left( \lambda^{\prime
}\right) Q_{1}\left( -\lambda\right) }{\alpha+\lambda}.
\end{matrix}
(4-26)
</math></center>
Note that the infinite products in Eqn.~((eq:K)) converge in the order of <math>q^{-5}</math> as <math>\left| q\right| </math> becomes large, thus numerical computation of <math>K\left( \alpha\right) </math> does not pose any difficulties.

The left hand side of Eqn.~((4-26)) is regular in <math>\mathcal{D}_{+}</math> and the right hand side is regular in <math>\mathcal{D}_{-}</math>. Notice that a function is added to both sides of the equation to make the right hand side of the equation regular in <math>\mathcal{D}_{-}</math>. The left hand side of Eqn.~((4-26)) is <math>o\left( \alpha^{4}\right) </math> as <math>\left| \alpha\right| \rightarrow \infty</math> in <math>\mathcal{D}_{+}</math>, since <math>\Psi_{z}^{+}\rightarrow0</math> and <math>K\left( \alpha\right) =O\left( 1\right) </math>\ as <math>\left| \alpha\right| \rightarrow\infty</math> in <math>\mathcal{D}_{+}</math>. The right hand side of Eqn.~((4-26)) has the equivalent analytic properties in <math>\mathcal{D}_{-}</math>. Liouville's theorem (Carrier, Krook and Pearson [[carrier]] section 2.4) tells us that there exists a function, which we denote <math>J\left( \alpha\right) </math>, uniquely defined by Eqn.~((4-26)), and function <math>J\left( \alpha\right) </math> is a polynomial of degree three in the whole plane. Hence
<center><math>
J\left( \alpha\right) =d_{0}+d_{1}\alpha+d_{2}\alpha^{2}+d_{3}\alpha^{3}.
</math></center>

Equating Eqn.~((4-26)) for <math>\Psi_{z}</math> gives
<center><math>
\Psi_{z}\left( \alpha,0\right) =\frac{-F\left( \alpha\right) }{K\left(
\alpha\right) f_{1}\left( \gamma\right) }\;=or= \;-\frac{K\left(
-\alpha\right) F\left( \alpha\right) }{f_{2}\left( \gamma\right)
} (4-50)
</math></center>
where
<center><math>
F\left( \alpha\right) =J\left( \alpha\right) -\frac{Q_{1}\left(
-\lambda\right) f_{2}\left( \lambda^{\prime}\right) }{\left(
\alpha+\lambda\right) K\left( \lambda\right) }.
</math></center>
Notice that procedure from Eqn.~((eq:WH2)) to Eqn.~((4-26)) eliminates
the need for calculating constant <math>C</math> in Eqn.~((4-26)).

For <math>x<0</math> we close the integral contour in <math>\mathcal{D}_{+}</math>, and put the
incident wave back, then we have
<center><math>
\phi_{z}\left( x,0\right) =\mathrm{i}Q_{1}\left( -\lambda\right)
e^{\mathrm{i}\lambda x}-\sum\limits_{q\in\mathcal{K}_{1}}
\frac{\mathrm{i}F\left( q\right) q^{\prime}R_{1}\left( q^{\prime
}\right) }{qK\left( q\right) }e^{-\mathrm{i}qx}, (eq:solution1)
</math></center>
where <math>R_{1}\left( q^{\prime}\right) </math> is a residue of <math>\left[ f_{1}\left(
\gamma\right) \right] ^{-1}<math> at </math>\gamma=q^{\prime}</math>
<center><math>\begin{matrix}
R_{1}\left( q^{\prime}\right) & =\left( \left. \frac{df_{1}\left(
\gamma\right) }{d\gamma}\right| _{\gamma=q^{\prime}}\right) ^{-1}
\\
& =\left\{ 5D_{1}q^{\prime3}+\frac{b_{1}}{q^{\prime}}+\frac{H}{q^{\prime}
}\left( \frac{\left( D_{1}q^{\prime5}+b_{1}q^{\prime}\right) ^{2}-\left(
\rho\omega^{2}\right) ^{2}}{\rho\omega^{2}}\right) \right\} ^{-1}. (R)
\end{matrix}</math></center>
We used <math>b_{1}=-m_{1}\omega^{2}+\rho g</math> and <math>f_{1}\left( q^{\prime}\right)
=0</math> to simplify the formula. Displacement <math>w\left( x\right) </math> can be
obtained by multiplying Eqn.~((eq:solution1)) by <math>-\mathrm{i}
/\omega</math>. Notice that the formula for the residue is again expressed by a
polynomial using the dispersion equation as shown in section (sec:3),
which gives us a stable numerical computation of the solutions.

The velocity potential <math>\phi\left( x,z\right) </math> can be obtained using
Eqn.~((4-44)) and Eqn.~((eq:4)),
<center><math>
\phi\left( x,z\right) =\frac{\mathrm{i}Q_{1}\left( -\lambda\right)
\cosh\lambda^{\prime}\left( z+H\right) }{\lambda^{\prime}\sinh
\lambda^{\prime}H}e^{\mathrm{i}\lambda x}-\sum\limits_{q\in
\mathcal{K}_{1}}\frac{\mathrm{i}F\left( q\right) R_{1}\left(
q^{\prime}\right) \cosh q^{\prime}\left( z+H\right) }{qK\left( q\right)
\sinh q^{\prime}H}e^{-\mathrm{i}qx}
</math></center>
where <math>\lambda^{\prime}=\sqrt{\lambda^{2}+k^{2}}</math>.

For <math>x>0</math>, the functions <math>\phi_{z}\left( x,0\right) </math> and <math>\phi\left(
x,z\right) </math>\ are obtained by closing the integral contour in <math>\mathcal{D}
_{-}</math>,
<center><math>\begin{matrix}
\phi_{z}\left( x,0\right) & =-\sum\limits_{q\in\mathcal{K}_{2}}
\frac{\mathrm{i}K\left( q\right) F\left( -q\right) q^{\prime}
R_{2}\left( q^{\prime}\right) }{q}e^{\mathrm{i}qx}, (4-28)\\
\phi\left( x,z\right) & =-\sum\limits_{q\in\mathcal{K}_{2}}\frac
{\mathrm{i}K\left( q\right) F\left( -q\right) R_{2}\left(
q^{\prime}\right) \cosh q^{\prime}\left( z+H\right) }{q\sinh q^{\prime}
H}e^{\mathrm{i}qx},
\end{matrix}</math></center>
where <math>R_{2}</math> is a residue of <math>\left[ f_{2}\left( \gamma\right) \right]
^{-1}</math>\ and its formula can be obtained by replacing the subscript <math>1</math> with
<math>2</math> in Eqn.~((R)). Notice that since <math>R_{j}\sim O\left( q^{-9}\right) </math>,
<math>j=1,2</math>, the coefficients of <math>\phi_{z}</math> of Eqn.~((4-28)) decay as
<math>O\left( q^{-6}\right) </math> as <math>\left| q\right| </math> becomes large, so the
displacement is bounded up to the fourth <math>x</math>-derivatives. In a physical sense,
the biharmonic term of the plate equation for the vertical displacement is
associated with the strain energy due to bending of the plate as explained in
chapter 2. Hence, up to fourth derivative of the displacement function should
be bounded, as has been confirmed. The coefficients of <math>\phi</math>, have an extra
<math>1/q^{\prime}\tanh q^{\prime}H</math> term which is <math>O\left( q^{4}\right) </math>, hence
the coefficients decay as <math>O\left( q^{-2}\right) </math> as <math>\left| q\right| </math>
becomes large. Therefore, <math>\phi</math> is bounded everywhere including at <math>x=0</math>.

Shifting a singularity of one function to the other is equivalent to
subtracting an incident wave from both functions then solving the boundary
value problem for the scattered field as in [[Balmforth and Craster 1999]]. As mentioned,
any one of the singularities can be shifted as long as it creates a common
strip of analyticity for the newly created functions. We chose <math>-\lambda</math>
because of the convenience of the symmetry in locations of the singularities.
The method of subtracting either incoming or transmitting wave requires the
Fourier transform be performed twice, first to express the solution with a
series expansion, and second to solve the system of equations for the newly
created functions. Thus, we find the method of shifting a singularity shown
here is advantageous to other methods since it needs the Fourier transform
only once to obtain the Wiener-Hopf equation.

The polynomial <math>J\left( \alpha\right) </math> is yet to be determined. In the
following section the coefficients of <math>J\left( \alpha\right) </math> will be
determined from conditions at <math>x=0\pm</math>, <math>-\infty<y<\infty</math>, <math>z=0</math>.

[[Category:Floating Elastic Plate]]
[[Category:Wiener-Hopf]]