WikiWaves - User contributions [en]

Main Page

2006-06-04T05:02:00Z

Funkalot: /* Featured Pages */

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

'''Wikiwaves''' is a water waves [[wikipedia:wiki|wiki]] devoted to the collective creation of technical content for practicing scientists. Please [[Sign up instructions|sign up]], [[browse]] around the site, click on the edit links, and contribute something! The site is nominally centered around water waves, at the moment the focus in on linear water
wave theory, especially as applied to hydroelasticity but we welcome any content. If you are new to the [http://google.com/trends?q=wiki increasingly popular] wiki way, you may want to visit [http://en.wikipedia.org/ Wikipedia] to see a more general effort in action.

__NOTOC__

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===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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===Featured Pages===
*[[Michael Meylan]]: A description of the research of Mike Meylan.
*[[Wave Scattering in the Marginal Ice Zone]]: A description of the geophysical problem in water wave scattring.
*[[Floating Elastic Plate]]: A discussion of this standard model in hydroelasticity.
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===Wikiwaves Announcements===
*This site is just beginning and right now we want people to make contributions.
*If you have any questions, problems etc. contact [[Gareth Hegarty]] or ask on the [[FAQ]]
*'''I have changed the standard format for references.''' Check out the [[FAQ]] for details.
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===About Us===

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

This website was started by [[Michael Meylan]] and is being initially supported by a grant from
the [http://www.nzima.auckland.ac.nz/ New Zealand Institute of Mathematics].

== 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]
* [http://qwiki.caltech.edu/index.php/Converting_LaTex_To_Wiki:A Python Script to Convert Latex to Wiki]
* [[Converting Latex to Wiki]]

Consult the [http://meta.wikipedia.org/wiki/MediaWiki_User%27s_Guide User's Guide] for information on using the wiki software.

Michael Meylan

2006-05-14T10:33:53Z

Funkalot: /* PhD Otago 1991 - 1993 */

Michael Meylan is a senior lecturer at the University of Auckland. He completed his Ph.D. under [[Vernon Squire]]
in 1993 which was concerned with modelling ice floes using linear wave theory.
He has worked on various problem connected with linear water wave theory in the subsequent time.

[[Image:Mikem.jpg|thumb|right|Photo taken in 1999]]

= Research =

== PhD Otago 1991 - 1993==
Mike's PhD thesis concerned a two-dimensional floating elastic plate which was solved
using a Green function method. The motivation for the solution was to model ice floe
and at the time he was ignorant of the engineering applications (e.g. [[VLFS]]).
Mike independently derived the Green function which
was well known in water waves and goes back to [[John_1950a| John 1950]].
The derivation method was copied by [[Squire_Dixon_2000a| Squire and Dixon 2000]]
(based on a close reading of his Phd thesis) for the case, not of a free surface,
but for a free surface covered by a plate
The results
of this research were publised in the ''Journal of Geophysical Research'' were largely
ignored by later researchers. His Phd thesis probably had a much greater influence, through
the researchers who followed at Otago and it is continuing to appear in journal citations.
The solution method using a Green function coupled with a Green function for the plate
(the later Green function does not extend to three dimensions because of the much
more complicated boundary conditions which exist). The solution method has been
superseded by more efficient methods, most notably the [[Wiener-Hopf]] method developed
by [[Tim Williams]] and the [[Eigenfunction Matching Method]] (which applied to
multiple plates) developed by
[[Kohout_Meylan_Sakai_Hanai_Leman_Brossard_2006a | Kohout et. al. 2006]].

== Post-Doc in Otago 1994 - 1996 ==

Mike then extended the two-dimensional solution to a three-dimensional circular elastic plate
([[Meylan_Squire_1996a|Meylan and Squire 1996]]).
This solution again used a Green function method coupled with the eigenfunctions for a circular
plate (which can be computed in exact form, at least up to solving an equation involving
Bessel functions. The solution method has been superseded by [[Peter_Meylan_Chung_2004a | Peter, Meylan and Chung 2004]].
Mike also developed a method to solve for plates of arbitrary geometry, initially using
a variational method ([[Meylan_2001a|Meylan 2001]]) and later using the [[Finite Element Method]]
([[Meylan_2002a|Meylan 2002]]).

== Post Doc Auckland 1996 - 1998 ==
Mike then worked on using the solution for a circular elastic plate to try and construct a model
for wave scattering in the Marginal Ice Zone ([[Meylan_Squire_Fox_1997a| Meylan, Squire and Fox 1997]]).
This model was developed independently of the model of [[Masson_LeBlond_1989a | Masson and LeBlond 1989]]
but shares many similarities with it.

Mike then began to work on a very abstract (and difficult problems) of an eigenfunction
expansion method for the non-selfadjoint operator which arises in the scattering model
of [[Meylan_Squire_Fox_1997a| Meylan, Squire and Fox 1997]]. This work is still
unpublished although a paper has been submitted. It is not a problem in water wave theory.

==Massey University 1999 - 2003 ==

Mike began working on the [[Time-Dependent Linear Water Wave]] problem.
He solved for the time-dependent motion of a [[Floating Elastic Plate]]
on [[Shallow Water]]. The solution was found using a [[Generalised Eigenfunction Expansion]]
and as a sum over [[Scattering Frequencies]] ([[Meylan_2002b|Meylan 2002]]. This lead to a collaboration with
[[Christophe Hazard]] and to a solution of the problem of a [[Floating Elastic Plate]]
on [[Finite Depth Water]].

[[Cynthia Wang]] worked with Mike as a masters (2000) and Phd student (2001-2003). Her master thesis concerned
wave scattering by a [[Floating Elastic Plate]] on water of [[Variable Bottom Topography]]
([[Wang_Meylan_2002a| Wang and Meylan 2002]]). Cynthia's PhD concerned a higher-order
coupled [[Boundary Element Method]] [[Finite Element Method]] for the three-dimensional
[[Floating Elastic Plate]] ([[Wang_Meylan_2004a|Wang and Meylan 2004]]) and applied this
method to the problem of an [[Infinite Array]] of [[Floating Elastic Plate|Floating Elastic Plates]]
([[Wang_Meylan_Porter_2006a|Wang, Meylan and Porter 2006]]).

Mike developed a method to solve for multiple floes using an extension of the method
of [[Meylan_2002a|Meylan 2002]]. This was not published but was used to test the
multiple floe scattering method which was developed with [[Malte Peter]] using [[Kagemoto and Yue Interaction Theory]]
which was developed during his masters in 2002.
Specifically, in [[Peter_Meylan_2004a | Peter and Meylan 2004]] the [[Kagemoto and Yue Interaction Theory]] was extended
to infinite depth and a coherent account of the theory for bodies of arbitrary geometry was given.
This work required the development of very sophisticated wave scattering code for bodies of
arbitrary geometry. As a direct result of this work a new expression for the [[Free-Surface Green Function]] was
developed and this was published separately ([[Peter_Meylan_2004b | Peter and Meylan 2004]]).

Mike also revisited the problem of a floating circular plate and developed a method
based on the [[Eigenfunction Matching Method]] ([[Peter_Meylan_Chung_2004a|Peter, Meylan, and Chung 2004]]).
Rike Grotmaack worked with Mike for an honours project in 2002 on [[Wave Forcing of Small Bodies]]
([[Grotmaack_Meylan_2006a| Grotmaack and Meylan 2006]])

== Auckland 2003 - present ==

[[Malte Peter]] and Mike have continued to work together and have developed an alternative method
for the [[Infinite Array]] based on [[Kagemoto and Yue Interaction Theory]]
([[Peter_Meylan_Linton_2006a|Peter, Meylan and Linton 2006]]). This method has been recently
extended to a [[Semi-Infinite Array]].

Michael Meylan

2006-05-14T10:33:11Z

Funkalot: /* Post-Doc in Otago 1994 - 1996 */

Michael Meylan is a senior lecturer at the University of Auckland. He completed his Ph.D. under [[Vernon Squire]]
in 1993 which was concerned with modelling ice floes using linear wave theory.
He has worked on various problem connected with linear water wave theory in the subsequent time.

[[Image:Mikem.jpg|thumb|right|Photo taken in 1999]]

= Research =

== PhD Otago 1991 - 1993==
Mike's PhD thesis concerned a two-dimensional floating elastic plate which was solved
using a Green function method. The motivation for the solution was to model ice floe
and at the time he was ignorant of the engineering applications (e.g. [[VLFS]]).
Mike independently derived the Green function which
was well known in water waves and goes back to [[John_1950a| John 1950]].
The derivation method was copied by [[Squire_Dixon_2000a| Squire and Dixon 2000]]
(based on a close reading of his Phd thesis) for the case, not of a free surface,
but for a free surface covered by a plate
The results
of this research were publised in the ''Journal of Geophysical Research'' were largely
ignored by later researchers. His Phd thesis probably had a much greater influence, through
the researchers who followed at Otago and it is continuing to appear in journal citations.
The solution method using a Green function coupled with a Green function for the plate
(the later Green function does not extend to three dimensions because of the much
more complicated boundary conditions which exist). The solution method has been
superseeded by more efficient methods, most notably the [[Wiener-Hopf]] method developed
by [[Tim Williams]] and the [[Eigenfunction Matching Method]] (which applied to
multiple plates) developed by
[[Kohout_Meylan_Sakai_Hanai_Leman_Brossard_2006a | Kohout et. al. 2006]].

== Post-Doc in Otago 1994 - 1996 ==

Mike then extended the two-dimensional solution to a three-dimensional circular elastic plate
([[Meylan_Squire_1996a|Meylan and Squire 1996]]).
This solution again used a Green function method coupled with the eigenfunctions for a circular
plate (which can be computed in exact form, at least up to solving an equation involving
Bessel functions. The solution method has been superseded by [[Peter_Meylan_Chung_2004a | Peter, Meylan and Chung 2004]].
Mike also developed a method to solve for plates of arbitrary geometry, initially using
a variational method ([[Meylan_2001a|Meylan 2001]]) and later using the [[Finite Element Method]]
([[Meylan_2002a|Meylan 2002]]).

== Post Doc Auckland 1996 - 1998 ==
Mike then worked on using the solution for a circular elastic plate to try and construct a model
for wave scattering in the Marginal Ice Zone ([[Meylan_Squire_Fox_1997a| Meylan, Squire and Fox 1997]]).
This model was developed independently of the model of [[Masson_LeBlond_1989a | Masson and LeBlond 1989]]
but shares many similarities with it.

Mike then began to work on a very abstract (and difficult problems) of an eigenfunction
expansion method for the non-selfadjoint operator which arises in the scattering model
of [[Meylan_Squire_Fox_1997a| Meylan, Squire and Fox 1997]]. This work is still
unpublished although a paper has been submitted. It is not a problem in water wave theory.

==Massey University 1999 - 2003 ==

Mike began working on the [[Time-Dependent Linear Water Wave]] problem.
He solved for the time-dependent motion of a [[Floating Elastic Plate]]
on [[Shallow Water]]. The solution was found using a [[Generalised Eigenfunction Expansion]]
and as a sum over [[Scattering Frequencies]] ([[Meylan_2002b|Meylan 2002]]. This lead to a collaboration with
[[Christophe Hazard]] and to a solution of the problem of a [[Floating Elastic Plate]]
on [[Finite Depth Water]].

[[Cynthia Wang]] worked with Mike as a masters (2000) and Phd student (2001-2003). Her master thesis concerned
wave scattering by a [[Floating Elastic Plate]] on water of [[Variable Bottom Topography]]
([[Wang_Meylan_2002a| Wang and Meylan 2002]]). Cynthia's PhD concerned a higher-order
coupled [[Boundary Element Method]] [[Finite Element Method]] for the three-dimensional
[[Floating Elastic Plate]] ([[Wang_Meylan_2004a|Wang and Meylan 2004]]) and applied this
method to the problem of an [[Infinite Array]] of [[Floating Elastic Plate|Floating Elastic Plates]]
([[Wang_Meylan_Porter_2006a|Wang, Meylan and Porter 2006]]).

Mike developed a method to solve for multiple floes using an extension of the method
of [[Meylan_2002a|Meylan 2002]]. This was not published but was used to test the
multiple floe scattering method which was developed with [[Malte Peter]] using [[Kagemoto and Yue Interaction Theory]]
which was developed during his masters in 2002.
Specifically, in [[Peter_Meylan_2004a | Peter and Meylan 2004]] the [[Kagemoto and Yue Interaction Theory]] was extended
to infinite depth and a coherent account of the theory for bodies of arbitrary geometry was given.
This work required the development of very sophisticated wave scattering code for bodies of
arbitrary geometry. As a direct result of this work a new expression for the [[Free-Surface Green Function]] was
developed and this was published separately ([[Peter_Meylan_2004b | Peter and Meylan 2004]]).

Mike also revisited the problem of a floating circular plate and developed a method
based on the [[Eigenfunction Matching Method]] ([[Peter_Meylan_Chung_2004a|Peter, Meylan, and Chung 2004]]).
Rike Grotmaack worked with Mike for an honours project in 2002 on [[Wave Forcing of Small Bodies]]
([[Grotmaack_Meylan_2006a| Grotmaack and Meylan 2006]])

== Auckland 2003 - present ==

[[Malte Peter]] and Mike have continued to work together and have developed an alternative method
for the [[Infinite Array]] based on [[Kagemoto and Yue Interaction Theory]]
([[Peter_Meylan_Linton_2006a|Peter, Meylan and Linton 2006]]). This method has been recently
extended to a [[Semi-Infinite Array]].

Main Page

2006-05-03T12:07:34Z

Funkalot: Added link to Latex/Wiki help

= Water Waves Website =

[[contents | contents page]]

[[index | index page ]]

[[test | test page]]

[[FAQ]]

----

Welcome to the water waves website. This site is just beginning and right now we want people to make
contributions. We are still not sure exactly what is the best format and we are aware that there will
be lots of questions, changes etc. that will have to be made as we go along.

The first thing to do is to create an account so that you can login and create or edit pages.
If you are unfamiliar with wiki sites then start with the [[test | test page]]
which (hopefully) will show you how to get started.
Then start to
enter the kind of information that you think will be useful, this should be the best guide to what
other will find useful. You should feel empowered to make any changes you like (including changing this
page or changing the structure of the site).

We need content! Please write anything you want. The first
page was on [[Scattering Frequencies]] which you might like to check out (and add to, fix errors
in etc.). While we need pages describing the basic theory, there is no need to start here.
We also strongly encourage you to include a page about yourself and create links to this.

We are starting with three kinds of page. The first is a topic page which describes a topic in the
water-waves. We
suggest that every topic page has a link in the [[contents | contents page]] and the [[index | index page ]].
The second kind of page is for an individual.
The third type of page is for each article which is cited. This should contain at least the citation
information and ideally will have a brief synopsis of the article. There is a standard format for
citations described on the [[FAQ]].

The aim of this site is to be as '''useful''' as possible. This is different from being accurate
(but of course accuracy is useful). Basically, we prefer content with errors (small hopefully)
to no content at all. A page which describes the theory with a few errors in the equations
will still be useful. Furthermore we can expect that someone else will spot these errors
and fix them.

If you have any questions, problems etc. (and we are expecting these at the moment as we have only
just begun to experiment with this website) please contact [[Gareth Hegarty]] or alternatively
you can asked them on the [[FAQ]] page (which is in the process of evolving).

[[Image:NZIMA.jpg|thumb|right|NZIMA logo.]]
This website was started by [[Michael Meylan]] and is being initially supported by a grant from
the [http://www.nzima.auckland.ac.nz/ New Zealand institute of mathematics].

== Getting started ==

[[FAQ]] (|Frequently asked questions) for the water-waves wiki

* [http://www.mediawiki.org/wiki/Help:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Help:FAQ MediaWiki FAQ]
* [http://mail.wikipedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]
* [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula Using Latex in Wiki]

Consult the [http://meta.wikipedia.org/wiki/MediaWiki_User%27s_Guide User's Guide] for information on using the wiki software.

Cylindrical Eigenfunction Expansion

2006-05-02T13:11:38Z

Funkalot:

= Introduction =

There are any situations where we want to expand the three-dimensional linear water wave
solution in cylindrical co-ordinates. For example, scattering from a
[[Bottom Mounted Cylinder]] or scattering from a [[Circular Elastic Plate]]. In these cases it is easy to find
the solution by an expansion in the cylindrical eigenfunctions. If the depth dependence can be
removed the solution reduces to a two dimensional problem (see [[Removing The Depth Dependence]]). While
the theory here does apply in this two dimensional situtation, the theory is presented here
for the fully three dimensional (depth dependent) case. We begin by assuming the [[Frequency Domain Problem]].

= Outine of the theory =

The problem for the complex water velocity potential in suitable non-dimensionalised
cylindrical coordinates, <math>\phi (r,\theta,z)</math>, is given by

<math> \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial
\phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2
\phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0,
\quad (r,\theta,z) \in \mathbb{R}_{>0} \, \times \ ]- \pi, \pi]
\times \mathbb{R}_{<0}, </math>

<math>\frac{\partial \phi}{\partial z} - \alpha \phi = 0, \quad
(r,\theta,z) \in \mathbb{R}_{>0}\,
\times \, ]\!- \pi, \pi] \times \{ 0 \},</math>

as well as

<math>
\frac{\partial \phi}{\partial z} = 0, \quad (r,\theta,z) \in
\mathbb{R}_{>0}\, \times \,]\!- \pi, \pi] \times \{ -d \},
</math>

in the case of constant finite water depth <math>d</math> and

<math>
\sup \big\{ \, |\phi| \ \big| \ (r,\theta,z) \in \mathbb{R}_{>0}\,
\times \, ]\!- \pi, \pi] \times \mathbb{R}_{<0} \,\big\} < \infty
</math>

in the case of infinite water depth. Moreover, the radiation condition

<math>
\lim_{r \rightarrow \infty} \sqrt{r} \, \Big(
\frac{\partial}{\partial r} - \mathrm{i} k \Big) \phi = 0
</math>

with the wavenumber <math>k</math> also applies.

== The case of water of finite depth ==

The solution of the problem for the potential in finite water depth
can be found by a separation ansatz,

<math>
\phi (r,\theta,z) =: Y(r,\theta) Z(z).\,
</math>

Substituting this into the equation for <math>\pi</math> yields

<math>
\frac{1}{Y(r,\theta)} \left[ \frac{1}{r} \frac{\partial}{\partial
r} \left( r \frac{\partial Y}{\partial r} \right) + \frac{1}{r^2}
\frac{\partial^2 Y}{\partial \theta^2} \right] = - \frac{1}{Z(z)}
\frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} = \eta^2.
</math>

The possible separation constants <math>\eta</math> will be determined by the
free surface condition and the bed condition.

In the setting of water of finite depth, the general solution
<math>Z(z)</math> can be written as

<math>
Z(z) = F \cos \big( \eta (z+d) \big) + G \sin \big( \eta (z+d) \big),
\quad \eta \in \mathbb{C} \backslash \{ 0 \},
</math>

since <math>\eta = 0</math> is not an eigenvalue.
To satisfy the bed condition, <math>G</math> must be <math>0</math>.
<math>Z(z)</math> satisfies the free surface condition, provided the separation
constants <math>\eta</math> are roots of the equation

<math>
- F \eta \sin \big( \eta (z+d) \big) - \alpha F \cos \big( \eta (z+d)
\big) = 0, \quad z=0,
</math>

or, equivalently, if they satisfy

<math>
\alpha + \eta \tan \eta d = 0.
</math>

This equation, also called dispersion relation, has an
infinite number of real roots, denoted by <math>k_m</math> and <math>-k_m</math> (<math>m \geq
1</math>), but the negative roots produce the same eigenfunctions as the
positive ones and will therefore not be considered. It also has a pair of purely imaginary roots which
will be denoted by <math>k_0</math>. Writing <math>k_0 = - \mathrm{i} k</math>, <math>k</math> is the
(positive) root of the dispersion relation

<math>
\alpha = k \tanh k d,
</math>

again it suffices to consider only the positive root. The solutions can
therefore be written as

<math>
Z_m(z) = F_m \cos \big( k_m (z+d) \big), \quad m \geq 0.
</math>

It follows that <math>k</math> is the previously introduced wavenumber and the dispersion relation gives the required relation to the radian frequency.

For the solution of

<math>
\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial
Y}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 Y}{\partial
\theta^2} = k_m^2 Y(r,\theta),
</math>

another separation will be used,

<math>
Y(r,\theta) =: R(r) \Theta(\theta).
</math>

Substituting this into Laplace's equation yields

<math>
\frac{r^2}{R(r)} \left[ \frac{1}{r} \frac{\mathrm{d}}{\mathrm{d}r} \left( r
\frac{\mathrm{d} R}{\mathrm{d}r} \right) - k_m^2 R(r) \right] = -
\frac{1}{\Theta (\theta)} \frac{\mathrm{d}^2 \Theta}{\mathrm{d}
\theta^2} = \eta^2,
</math>

where the separation constant <math>\eta</math> must be an integer, say <math>\nu</math>,
in order for the potential to be continuous. <math>\Theta
(\theta)</math> can therefore be expressed as

<math>
\Theta (\theta) = C \, \mathrm{e}^{\mathrm{i} \nu \theta}, \quad \nu \in \mathbb{Z}.
</math>

Equation (\ref{pot_cyl_rt2}) also yields

<math>
r \frac{\mathrm{d}}{\mathrm{d}r} \left( r \frac{\mathrm{d}
R}{\mathrm{d} r} \right) - (\nu^2 + k_m^2 r^2) R(r) = 0, \quad \nu \in
\mathbb{Z}.
</math>

Substituting <math>\tilde{r}:=k_m r</math> and writing <math>\tilde{R} (\tilde{r}) :=
R(\tilde{r}/k_m) = R(r)</math>, this can be rewritten as

<math>
\tilde{r}^2 \frac{\mathrm{d}^2 \tilde{R}}{\mathrm{d} \tilde{r}^2}
+ \tilde{r} \frac{\mathrm{d} \tilde{R}}{\mathrm{d} \tilde{r}}
- (\nu^2 + \tilde{r}^2)\, \tilde{R} = 0, \quad \nu \in \mathbb{Z},
</math>

which is the modified version of Bessel's equation. Substituting back,
the general solution is given by

<math>
R(r) = D \, I_\nu(k_m r) + E \, K_\nu(k_m r), \quad m \in
\mathbb{N},\ \nu \in \mathbb{Z},
</math>

where <math>I_\nu</math> and <math>K_\nu</math> are the modified Bessel functions of the first
and second kind, respectively, of order <math>\nu</math>.

The potential <math>\phi</math> can thus be expressed in local cylindrical
coordinates as

<math>
\phi (r,\theta,z) = \sum_{m = 0}^{\infty} Z_m(z) \sum_{\nu = -
\infty}^{\infty} \left[ D_{m\nu} I_\nu (k_m r) + E_{m\nu} K_\nu (k_m
r) \right] \mathrm{e}^{\mathrm{i} \nu \theta},
</math>

where <math>Z_m(z)</math> is given by equation \eqref{sol_Z_fin}. Substituting <math>Z_m</math>
back as well as noting that <math>k_0=-\mathrm{i} k</math> yields

<math> \phi (r,\theta,z)
= F_0 \cos(-\mathrm{i} k (z+d)) \sum_{\nu = - \infty}^{\infty}
\left[ D_{0\nu} I_\nu (-\mathrm{i} k r) + E_{0\nu} K_\nu (-\mathrm{i} k r)\right]
\mathrm{e}^{\mathrm{i} \nu \theta}
+ \sum_{m = 1}^{\infty} F_m \cos(k_m(z+d)) \sum_{\nu = -
\infty}^{\infty} \left[ D_{m\nu} I_\nu (k_m r) + E_{m\nu} K_\nu (k_m
r) \right] \mathrm{e}^{\mathrm{i} \nu \theta}.
</math>

Noting that <math>\cos \mathrm{i} x = \cosh x</math> is an even function and the
relations <math>I_\nu(-\mathrm{i} x) = (-\mathrm{i})^{\nu} J_\nu(x)</math> where <math>J_\nu</math> is the Bessel
function of the first kind of order <math>\nu</math> and <math>K_\nu (-\mathrm{i} x) = \pi / 2\,\,
\mathrm{i}^{\nu+1} H_\nu^{(1)}(x)</math> with <math>H_\nu^{(1)}</math> denoting
the Hankel function of the first kind of order <math>\nu</math>, it follows that

<math>
\phi (r,\theta,z)
= \cosh(k (z+d)) \sum_{\nu = - \infty}^{\infty}
\left[ D_{0\nu}' J_\nu (k r) + E_{0\nu}' H_\nu^{(1)} (k r)\right]
\mathrm{e}^{\mathrm{i} \nu \theta} + \sum_{m = 1}^{\infty} F_m \cos(k_m(z+d)) \sum_{\nu = -
\infty}^{\infty} \left[ D_{m\nu}' I_\nu (k_m r) + E_{m\nu}' K_\nu (k_m
r) \right] \mathrm{e}^{\mathrm{i} \nu \theta}.
</math>

However, <math>J_\nu</math> does not satisfy the radiation
condition \eqref{water_rad} and neither does <math>I_\nu</math>
since it becomes unbounded for increasing real argument. These
two solutions represent incoming waves which will also be
required later.

Therefore, the solution of the problem requires <math>D_{m\nu}'=0</math>
for all <math>m,\nu</math>. Therefore, the
eigenfunction expansion of the water velocity potential in
cylindrical outgoing waves with coefficients <math>A_{m\nu}</math> is given by

<math>
\phi (r,\theta,z) = \frac{\cosh(k (z+d))}{\cosh kd} \sum_{\nu = -
\infty}^{\infty} A_{0\nu} H_\nu^{(1)} (k r) \mathrm{e}^{\mathrm{i} \nu \theta} + \sum_{m = 1}^{\infty} \frac{\cos(k_m(z+d))}{\cos k_m d}
\sum_{\nu = - \infty}^{\infty} A_{m\nu} K_\nu (k_m r) \mathrm{e}^{\mathrm{i} \nu \theta}.
</math>

The two terms describe the propagating and the decaying wavefields
respectively.

== The case of infinitely deep water ==

A solution will be developed for the same setting as before but under the
assumption of water of infinite depth. As in the previous section,
Laplace's equation must be solved in cylindrical coordinates
satisfying the free surface and the radiation condition. However,
instead of the bed condition, the water velocity potential is also required to
satisfy the depth condition. Therefore, <math>Z(z)</math> must be solved for satisfying the depth condition. It will turn out that in the case of
infinitely deep water an uncountable amount of separation constants
<math>\eta</math> is valid.

As above, the general solution can be represented as

<math>
Z(z) = F \mathrm{e}^{\mathrm{i} \eta z} + G \mathrm{e}^{- \mathrm{i} \eta z}, \quad \eta \in \mathbb{C}
\backslash \{0\}.
</math>

Assuming <math>\eta</math> has got a positive
imaginary part, in order to satisfy the depth condition, <math>F<math> must be
zero. <math>Z(z)</math> then satisfies the free surface condition if <math>\eta</math> is a root of

<math>
-G \mathrm{i} \eta \mathrm{e}^{-\mathrm{i} \eta z} - \alpha G \mathrm{e}^{-\mathrm{i} \eta z} = 0, \quad z=0,
</math>

which yields the dispersion relation

<math>
\eta = - \mathrm{i} \alpha.
</math>

Therefore, <math>\eta</math> must even be purely imaginary. For <math>\Im \eta < 0</math>,
this is also obtained, but with a minus sign in front of
<math>\eta</math>. However, this yields the same solution. One solution can
therefore be written as

<math>
Z(z) = G \mathrm{e}^{\alpha z}.
</math>

Now, <math>\eta</math> is assumed real. In this case, it is convenient to write
the general solution in terms of cosine and sine,

<math>
Z(z) = F \cos(\eta z) + G \sin(\eta z), \quad \eta \in \mathbb{R}
\backslash \{0\}.
</math>

This solution satisfies the depth condition automatically.
Making use of the free surface condition, it follows that

<math>
(-\eta F - \alpha G) \sin (\eta z) + (\eta G - \alpha F) \cos(\eta z)
= 0, \quad z=0,
</math>

which can be solved for <math>G</math>,

<math>
G = \frac{\alpha}{\eta} F.
</math>

Substituting this back gives

<math>
Z(z) = F \big( \cos(\eta z) + \frac{\alpha}{\eta} \sin(\eta z)
\big) , \quad \eta \in \mathbb{R} \backslash \{0\}.
</math>

Obviously, a negative value of <math>\eta</math> produces the same
eigenfunction as the positive one. Therefore, only positive ones are
considered, leading to the definition

<math>
\psi(z,\eta) := \cos(\eta z) + \frac{\alpha}{\eta} \sin(\eta z), \quad
(z,\eta) \in \mathbb{R}_{\leq0} \times \mathbb{R}_{>0}.
</math>

This gives the vertical eigenfunctions in infinite depth.

For the radial and angular coordinate the same separation can be used
as in the finite depth case so that the general solution of problem
can be written as

<math>
\phi (r,\theta,z) = \mathrm{e}^{\alpha z} \sum_{\nu = -
\infty}^{\infty} \left[ E_\nu (-\mathrm{i} \alpha) I_\nu (-\mathrm{i} \alpha r) +
F_{\nu} (-\mathrm{i} \alpha) K_\nu (-\mathrm{i} \alpha r) \right] \mathrm{e}^{\mathrm{i} \nu \theta} + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\nu = -
\infty}^{\infty} \left[ E_\nu I_\nu (\eta r) + F_{\nu} (\eta) K_\nu
(\eta r) \right] \mathrm{e}^{\mathrm{i} \nu \theta} \mathrm{d}\eta.
</math>

Making use of the radiation condition as
well as the relations of the Bessel functions in the same way as in
the finite depth case, this can be rewritten as the eigenfunction
expansion of the water velocity potential into cylindrical outgoing
waves in water of infinite depth,

<math>
\phi (r,\theta,z) = \mathrm{e}^{\alpha z} \sum_{\nu = -
\infty}^{\infty} A_{\nu} (\mathrm{i} \alpha) H_\nu^{(1)} (\alpha r) \mathrm{e}^{\mathrm{i} \nu
\theta} + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\nu = -
\infty}^{\infty} A_{\nu} (\eta) K_\nu (\eta r) \mathrm{e}^{\mathrm{i} \nu
\theta} \mathrm{d}\eta.
</math>

WikiWaves:Test

2006-04-09T07:09:35Z

Funkalot:

Here is a link [[Main Page]].

Here is some <math>
\begin{equation}
\left\{
\begin{array}{l}
\ddot r - \gamma\ddot r_{ss} + r_{ssss}=(\tau r_s)_s,\ \ r_s.r_s=1,\\
r(0,t)=(0,0), \;\; r_{s}(0,t)=(1,0),\\
(\gamma\ddot r_s+\tau r_{s}-r_{sss})|_{(1,t)}=-\alpha\dot{r}(1,t),\
r_{ss}(1,t)=-\beta\dot r_s(1,t),\\
r(s,0)=r_0(s),\ \dot r(s,0)=r_1(s).
\end{array}\right.
\end{equation}</math>

and a picture:
\begin{figure}[ht]
\begin{center}
\begin{pspicture}(0,-1.5)(10,1.5)
\psplot[linecolor=lightgray,linewidth=10pt]{0}{10}{x .07 mul 2 exp x .07 mul .5
sub mul x .07 mul .8 sub mul 600 mul x .07 mul 5 add div}
\psline[linecolor=lightgray,linewidth=10pt](-.3,0)(0,0)
\psframe[linestyle=dashed,dash=5pt 8pt](-.3,-.22)(10,.22)
\psline[linestyle=dashed]{|<->|}(4,0)(4,1)
\rput(4.9,1){$y(x,t)$}
\psline[arrowsize=6.5pt,arrowlength=1.5]{->}(0,0)(11,0)
\psline[arrowsize=6.5pt,arrowlength=1.5]{<->}(0,-1.5)(0,1.5)
\rput(-.3,1){$y$}
\rput(10.5,-.2){$x$}
\rput(9.9,.6){$x=1$}
\end{pspicture}
\caption{\small The linear beam}
\label{f1}\end{center}
\end{figure}