https://wikiwaves.org/wiki/api.php?action=feedcontributions&user=Funkalot&feedformat=atomWikiWaves - User contributions [en]2020-07-06T09:02:06ZUser contributionsMediaWiki 1.33.0https://wikiwaves.org/wiki/index.php?title=Main_Page&diff=2643Main Page2006-06-04T05:02:00Z<p>Funkalot: /* Featured Pages */</p>
<hr />
<div>==Welcome to '''Wikiwaves'''!==<br />
<br />
'''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<br />
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. <br />
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===Getting Started===<br />
*First [[Browse|browse]] around to get a feel for what is here.<br />
*Then follow the [[Sign up instructions|sign up instructions]] to make yourself a profile page.<br />
*After that, learn [[Simple wiki help|how to create and compose pages]] and contribute to the site!<br />
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===Featured Pages===<br />
*[[Michael Meylan]]: A description of the research of Mike Meylan.<br />
*[[Wave Scattering in the Marginal Ice Zone]]: A description of the geophysical problem in water wave scattring. <br />
*[[Floating Elastic Plate]]: A discussion of this standard model in hydroelasticity.<br />
</div><br />
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===Wikiwaves Announcements===<br />
*This site is just beginning and right now we want people to make contributions.<br />
*If you have any questions, problems etc. contact [[Gareth Hegarty]] or ask on the [[FAQ]]<br />
*'''I have changed the standard format for references.''' Check out the [[FAQ]] for details.<br />
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===Site Map===<br />
*[[Browse]]<br />
*[[:Category:People|People]]<br />
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<hr><br />
<br />
===About Us===<br />
<br />
[[Image:NZIMA.jpg|thumb|right]]<br />
<br />
This website was started by [[Michael Meylan]] and is being initially supported by a grant from<br />
the [http://www.nzima.auckland.ac.nz/ New Zealand Institute of Mathematics].<br />
<br />
== Useful Links ==<br />
<br />
* [[FAQ]] (|Frequently asked questions) for the water-waves wiki<br />
* [http://www.mediawiki.org/wiki/Help:FAQ MediaWiki FAQ]<br />
* [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula Using Latex in Wiki]<br />
* [http://qwiki.caltech.edu/index.php/Converting_LaTex_To_Wiki:A Python Script to Convert Latex to Wiki]<br />
* [[Converting Latex to Wiki]]<br />
<br />
Consult the [http://meta.wikipedia.org/wiki/MediaWiki_User%27s_Guide User's Guide] for information on using the wiki software.</div>Funkalothttps://wikiwaves.org/wiki/index.php?title=Michael_Meylan&diff=2028Michael Meylan2006-05-14T10:33:53Z<p>Funkalot: /* PhD Otago 1991 - 1993 */</p>
<hr />
<div>Michael Meylan is a senior lecturer at the University of Auckland. He completed his Ph.D. under [[Vernon Squire]]<br />
in 1993 which was concerned with modelling ice floes using linear wave theory. <br />
He has worked on various problem connected with linear water wave theory in the subsequent time. <br />
<br />
[[Image:Mikem.jpg|thumb|right|Photo taken in 1999]]<br />
<br />
= Research =<br />
<br />
== PhD Otago 1991 - 1993== <br />
Mike's PhD thesis concerned a two-dimensional floating elastic plate which was solved<br />
using a Green function method. The motivation for the solution was to model ice floe<br />
and at the time he was ignorant of the engineering applications (e.g. [[VLFS]]).<br />
Mike independently derived the Green function which <br />
was well known in water waves and goes back to [[John_1950a| John 1950]]. <br />
The derivation method was copied by [[Squire_Dixon_2000a| Squire and Dixon 2000]]<br />
(based on a close reading of his Phd thesis) for the case, not of a free surface,<br />
but for a free surface covered by a plate<br />
The results<br />
of this research were publised in the ''Journal of Geophysical Research'' were largely<br />
ignored by later researchers. His Phd thesis probably had a much greater influence, through<br />
the researchers who followed at Otago and it is continuing to appear in journal citations.<br />
The solution method using a Green function coupled with a Green function for the plate<br />
(the later Green function does not extend to three dimensions because of the much<br />
more complicated boundary conditions which exist). The solution method has been <br />
superseded by more efficient methods, most notably the [[Wiener-Hopf]] method developed<br />
by [[Tim Williams]] and the [[Eigenfunction Matching Method]] (which applied to <br />
multiple plates) developed by <br />
[[Kohout_Meylan_Sakai_Hanai_Leman_Brossard_2006a | Kohout et. al. 2006]].<br />
<br />
== Post-Doc in Otago 1994 - 1996 ==<br />
<br />
Mike then extended the two-dimensional solution to a three-dimensional circular elastic plate <br />
([[Meylan_Squire_1996a|Meylan and Squire 1996]]).<br />
This solution again used a Green function method coupled with the eigenfunctions for a circular<br />
plate (which can be computed in exact form, at least up to solving an equation involving<br />
Bessel functions. The solution method has been superseded by [[Peter_Meylan_Chung_2004a | Peter, Meylan and Chung 2004]].<br />
Mike also developed a method to solve for plates of arbitrary geometry, initially using<br />
a variational method ([[Meylan_2001a|Meylan 2001]]) and later using the [[Finite Element Method]]<br />
([[Meylan_2002a|Meylan 2002]]).<br />
<br />
== Post Doc Auckland 1996 - 1998 ==<br />
Mike then worked on using the solution for a circular elastic plate to try and construct a model<br />
for wave scattering in the Marginal Ice Zone ([[Meylan_Squire_Fox_1997a| Meylan, Squire and Fox 1997]]).<br />
This model was developed independently of the model of [[Masson_LeBlond_1989a | Masson and LeBlond 1989]]<br />
but shares many similarities with it. <br />
<br />
Mike then began to work on a very abstract (and difficult problems) of an eigenfunction<br />
expansion method for the non-selfadjoint operator which arises in the scattering model<br />
of [[Meylan_Squire_Fox_1997a| Meylan, Squire and Fox 1997]]. This work is still<br />
unpublished although a paper has been submitted. It is not a problem in water wave theory.<br />
<br />
==Massey University 1999 - 2003 ==<br />
<br />
Mike began working on the [[Time-Dependent Linear Water Wave]] problem.<br />
He solved for the time-dependent motion of a [[Floating Elastic Plate]]<br />
on [[Shallow Water]]. The solution was found using a [[Generalised Eigenfunction Expansion]]<br />
and as a sum over [[Scattering Frequencies]] ([[Meylan_2002b|Meylan 2002]]. This lead to a collaboration with<br />
[[Christophe Hazard]] and to a solution of the problem of a [[Floating Elastic Plate]]<br />
on [[Finite Depth Water]]. <br />
<br />
[[Cynthia Wang]] worked with Mike as a masters (2000) and Phd student (2001-2003). Her master thesis concerned <br />
wave scattering by a [[Floating Elastic Plate]] on water of [[Variable Bottom Topography]]<br />
([[Wang_Meylan_2002a| Wang and Meylan 2002]]). Cynthia's PhD concerned a higher-order<br />
coupled [[Boundary Element Method]] [[Finite Element Method]] for the three-dimensional<br />
[[Floating Elastic Plate]] ([[Wang_Meylan_2004a|Wang and Meylan 2004]]) and applied this<br />
method to the problem of an [[Infinite Array]] of [[Floating Elastic Plate|Floating Elastic Plates]]<br />
([[Wang_Meylan_Porter_2006a|Wang, Meylan and Porter 2006]]).<br />
<br />
Mike developed a method to solve for multiple floes using an extension of the method<br />
of [[Meylan_2002a|Meylan 2002]]. This was not published but was used to test the<br />
multiple floe scattering method which was developed with [[Malte Peter]] using [[Kagemoto and Yue Interaction Theory]]<br />
which was developed during his masters in 2002.<br />
Specifically, in [[Peter_Meylan_2004a | Peter and Meylan 2004]] the [[Kagemoto and Yue Interaction Theory]] was extended<br />
to infinite depth and a coherent account of the theory for bodies of arbitrary geometry was given.<br />
This work required the development of very sophisticated wave scattering code for bodies of<br />
arbitrary geometry. As a direct result of this work a new expression for the [[Free-Surface Green Function]] was <br />
developed and this was published separately ([[Peter_Meylan_2004b | Peter and Meylan 2004]]). <br />
<br />
Mike also revisited the problem of a floating circular plate and developed a method<br />
based on the [[Eigenfunction Matching Method]] ([[Peter_Meylan_Chung_2004a|Peter, Meylan, and Chung 2004]]).<br />
Rike Grotmaack worked with Mike for an honours project in 2002 on [[Wave Forcing of Small Bodies]] <br />
([[Grotmaack_Meylan_2006a| Grotmaack and Meylan 2006]])<br />
<br />
== Auckland 2003 - present ==<br />
<br />
[[Malte Peter]] and Mike have continued to work together and have developed an alternative method<br />
for the [[Infinite Array]] based on [[Kagemoto and Yue Interaction Theory]] <br />
([[Peter_Meylan_Linton_2006a|Peter, Meylan and Linton 2006]]). This method has been recently<br />
extended to a [[Semi-Infinite Array]].</div>Funkalothttps://wikiwaves.org/wiki/index.php?title=Michael_Meylan&diff=2027Michael Meylan2006-05-14T10:33:11Z<p>Funkalot: /* Post-Doc in Otago 1994 - 1996 */</p>
<hr />
<div>Michael Meylan is a senior lecturer at the University of Auckland. He completed his Ph.D. under [[Vernon Squire]]<br />
in 1993 which was concerned with modelling ice floes using linear wave theory. <br />
He has worked on various problem connected with linear water wave theory in the subsequent time. <br />
<br />
[[Image:Mikem.jpg|thumb|right|Photo taken in 1999]]<br />
<br />
= Research =<br />
<br />
== PhD Otago 1991 - 1993== <br />
Mike's PhD thesis concerned a two-dimensional floating elastic plate which was solved<br />
using a Green function method. The motivation for the solution was to model ice floe<br />
and at the time he was ignorant of the engineering applications (e.g. [[VLFS]]).<br />
Mike independently derived the Green function which <br />
was well known in water waves and goes back to [[John_1950a| John 1950]]. <br />
The derivation method was copied by [[Squire_Dixon_2000a| Squire and Dixon 2000]]<br />
(based on a close reading of his Phd thesis) for the case, not of a free surface,<br />
but for a free surface covered by a plate<br />
The results<br />
of this research were publised in the ''Journal of Geophysical Research'' were largely<br />
ignored by later researchers. His Phd thesis probably had a much greater influence, through<br />
the researchers who followed at Otago and it is continuing to appear in journal citations.<br />
The solution method using a Green function coupled with a Green function for the plate<br />
(the later Green function does not extend to three dimensions because of the much<br />
more complicated boundary conditions which exist). The solution method has been <br />
superseeded by more efficient methods, most notably the [[Wiener-Hopf]] method developed<br />
by [[Tim Williams]] and the [[Eigenfunction Matching Method]] (which applied to <br />
multiple plates) developed by <br />
[[Kohout_Meylan_Sakai_Hanai_Leman_Brossard_2006a | Kohout et. al. 2006]].<br />
<br />
== Post-Doc in Otago 1994 - 1996 ==<br />
<br />
Mike then extended the two-dimensional solution to a three-dimensional circular elastic plate <br />
([[Meylan_Squire_1996a|Meylan and Squire 1996]]).<br />
This solution again used a Green function method coupled with the eigenfunctions for a circular<br />
plate (which can be computed in exact form, at least up to solving an equation involving<br />
Bessel functions. The solution method has been superseded by [[Peter_Meylan_Chung_2004a | Peter, Meylan and Chung 2004]].<br />
Mike also developed a method to solve for plates of arbitrary geometry, initially using<br />
a variational method ([[Meylan_2001a|Meylan 2001]]) and later using the [[Finite Element Method]]<br />
([[Meylan_2002a|Meylan 2002]]).<br />
<br />
== Post Doc Auckland 1996 - 1998 ==<br />
Mike then worked on using the solution for a circular elastic plate to try and construct a model<br />
for wave scattering in the Marginal Ice Zone ([[Meylan_Squire_Fox_1997a| Meylan, Squire and Fox 1997]]).<br />
This model was developed independently of the model of [[Masson_LeBlond_1989a | Masson and LeBlond 1989]]<br />
but shares many similarities with it. <br />
<br />
Mike then began to work on a very abstract (and difficult problems) of an eigenfunction<br />
expansion method for the non-selfadjoint operator which arises in the scattering model<br />
of [[Meylan_Squire_Fox_1997a| Meylan, Squire and Fox 1997]]. This work is still<br />
unpublished although a paper has been submitted. It is not a problem in water wave theory.<br />
<br />
==Massey University 1999 - 2003 ==<br />
<br />
Mike began working on the [[Time-Dependent Linear Water Wave]] problem.<br />
He solved for the time-dependent motion of a [[Floating Elastic Plate]]<br />
on [[Shallow Water]]. The solution was found using a [[Generalised Eigenfunction Expansion]]<br />
and as a sum over [[Scattering Frequencies]] ([[Meylan_2002b|Meylan 2002]]. This lead to a collaboration with<br />
[[Christophe Hazard]] and to a solution of the problem of a [[Floating Elastic Plate]]<br />
on [[Finite Depth Water]]. <br />
<br />
[[Cynthia Wang]] worked with Mike as a masters (2000) and Phd student (2001-2003). Her master thesis concerned <br />
wave scattering by a [[Floating Elastic Plate]] on water of [[Variable Bottom Topography]]<br />
([[Wang_Meylan_2002a| Wang and Meylan 2002]]). Cynthia's PhD concerned a higher-order<br />
coupled [[Boundary Element Method]] [[Finite Element Method]] for the three-dimensional<br />
[[Floating Elastic Plate]] ([[Wang_Meylan_2004a|Wang and Meylan 2004]]) and applied this<br />
method to the problem of an [[Infinite Array]] of [[Floating Elastic Plate|Floating Elastic Plates]]<br />
([[Wang_Meylan_Porter_2006a|Wang, Meylan and Porter 2006]]).<br />
<br />
Mike developed a method to solve for multiple floes using an extension of the method<br />
of [[Meylan_2002a|Meylan 2002]]. This was not published but was used to test the<br />
multiple floe scattering method which was developed with [[Malte Peter]] using [[Kagemoto and Yue Interaction Theory]]<br />
which was developed during his masters in 2002.<br />
Specifically, in [[Peter_Meylan_2004a | Peter and Meylan 2004]] the [[Kagemoto and Yue Interaction Theory]] was extended<br />
to infinite depth and a coherent account of the theory for bodies of arbitrary geometry was given.<br />
This work required the development of very sophisticated wave scattering code for bodies of<br />
arbitrary geometry. As a direct result of this work a new expression for the [[Free-Surface Green Function]] was <br />
developed and this was published separately ([[Peter_Meylan_2004b | Peter and Meylan 2004]]). <br />
<br />
Mike also revisited the problem of a floating circular plate and developed a method<br />
based on the [[Eigenfunction Matching Method]] ([[Peter_Meylan_Chung_2004a|Peter, Meylan, and Chung 2004]]).<br />
Rike Grotmaack worked with Mike for an honours project in 2002 on [[Wave Forcing of Small Bodies]] <br />
([[Grotmaack_Meylan_2006a| Grotmaack and Meylan 2006]])<br />
<br />
== Auckland 2003 - present ==<br />
<br />
[[Malte Peter]] and Mike have continued to work together and have developed an alternative method<br />
for the [[Infinite Array]] based on [[Kagemoto and Yue Interaction Theory]] <br />
([[Peter_Meylan_Linton_2006a|Peter, Meylan and Linton 2006]]). This method has been recently<br />
extended to a [[Semi-Infinite Array]].</div>Funkalothttps://wikiwaves.org/wiki/index.php?title=Main_Page&diff=1599Main Page2006-05-03T12:07:34Z<p>Funkalot: Added link to Latex/Wiki help</p>
<hr />
<div>= Water Waves Website =<br />
<br />
[[contents | contents page]] <br />
<br />
[[index | index page ]]<br />
<br />
[[test | test page]]<br />
<br />
[[FAQ]]<br />
<br />
----<br />
<br />
Welcome to the water waves website. This site is just beginning and right now we want people to make<br />
contributions. We are still not sure exactly what is the best format and we are aware that there will<br />
be lots of questions, changes etc. that will have to be made as we go along.<br />
<br />
The first thing to do is to create an account so that you can login and create or edit pages. <br />
If you are unfamiliar with wiki sites then start with the [[test | test page]] <br />
which (hopefully) will show you how to get started.<br />
Then start to <br />
enter the kind of information that you think will be useful, this should be the best guide to what<br />
other will find useful. You should feel empowered to make any changes you like (including changing this<br />
page or changing the structure of the site).<br />
<br />
We need content! Please write anything you want. The first<br />
page was on [[Scattering Frequencies]] which you might like to check out (and add to, fix errors<br />
in etc.). While we need pages describing the basic theory, there is no need to start here.<br />
We also strongly encourage you to include a page about yourself and create links to this. <br />
<br />
We are starting with three kinds of page. The first is a topic page which describes a topic in the<br />
water-waves. We<br />
suggest that every topic page has a link in the [[contents | contents page]] and the [[index | index page ]]. <br />
The second kind of page is for an individual.<br />
The third type of page is for each article which is cited. This should contain at least the citation<br />
information and ideally will have a brief synopsis of the article. There is a standard format for<br />
citations described on the [[FAQ]].<br />
<br />
The aim of this site is to be as '''useful''' as possible. This is different from being accurate <br />
(but of course accuracy is useful). Basically, we prefer content with errors (small hopefully)<br />
to no content at all. A page which describes the theory with a few errors in the equations<br />
will still be useful. Furthermore we can expect that someone else will spot these errors<br />
and fix them.<br />
<br />
If you have any questions, problems etc. (and we are expecting these at the moment as we have only<br />
just begun to experiment with this website) please contact [[Gareth Hegarty]] or alternatively<br />
you can asked them on the [[FAQ]] page (which is in the process of evolving).<br />
<br />
[[Image:NZIMA.jpg|thumb|right|NZIMA logo.]]<br />
This website was started by [[Michael Meylan]] and is being initially supported by a grant from<br />
the [http://www.nzima.auckland.ac.nz/ New Zealand institute of mathematics]. <br />
<br />
== Getting started ==<br />
<br />
[[FAQ]] (|Frequently asked questions) for the water-waves wiki<br />
<br />
* [http://www.mediawiki.org/wiki/Help:Configuration_settings Configuration settings list]<br />
* [http://www.mediawiki.org/wiki/Help:FAQ MediaWiki FAQ]<br />
* [http://mail.wikipedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula Using Latex in Wiki]<br />
<br />
Consult the [http://meta.wikipedia.org/wiki/MediaWiki_User%27s_Guide User's Guide] for information on using the wiki software.</div>Funkalothttps://wikiwaves.org/wiki/index.php?title=Cylindrical_Eigenfunction_Expansion&diff=1598Cylindrical Eigenfunction Expansion2006-05-02T13:11:38Z<p>Funkalot: </p>
<hr />
<div>= Introduction =<br />
<br />
There are any situations where we want to expand the three-dimensional linear water wave<br />
solution in cylindrical co-ordinates. For example, scattering from a <br />
[[Bottom Mounted Cylinder]] or scattering from a [[Circular Elastic Plate]]. In these cases it is easy to find<br />
the solution by an expansion in the cylindrical eigenfunctions. If the depth dependence can be<br />
removed the solution reduces to a two dimensional problem (see [[Removing The Depth Dependence]]). While<br />
the theory here does apply in this two dimensional situtation, the theory is presented here<br />
for the fully three dimensional (depth dependent) case. We begin by assuming the [[Frequency Domain Problem]].<br />
<br />
= Outine of the theory = <br />
<br />
<br />
The problem for the complex water velocity potential in suitable non-dimensionalised<br />
cylindrical coordinates, <math>\phi (r,\theta,z)</math>, is given by<br />
<br />
<math> \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial<br />
\phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2<br />
\phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0,<br />
\quad (r,\theta,z) \in \mathbb{R}_{>0} \, \times \ ]- \pi, \pi] <br />
\times \mathbb{R}_{<0}, </math><br />
<br />
<math>\frac{\partial \phi}{\partial z} - \alpha \phi = 0, \quad<br />
(r,\theta,z) \in \mathbb{R}_{>0}\,<br />
\times \, ]\!- \pi, \pi] \times \{ 0 \},</math><br />
<br />
as well as<br />
<br />
<math><br />
\frac{\partial \phi}{\partial z} = 0, \quad (r,\theta,z) \in<br />
\mathbb{R}_{>0}\, \times \,]\!- \pi, \pi] \times \{ -d \},<br />
</math><br />
<br />
in the case of constant finite water depth <math>d</math> and<br />
<br />
<math><br />
\sup \big\{ \, |\phi| \ \big| \ (r,\theta,z) \in \mathbb{R}_{>0}\,<br />
\times \, ]\!- \pi, \pi] \times \mathbb{R}_{<0} \,\big\} < \infty<br />
</math><br />
<br />
in the case of infinite water depth. Moreover, the radiation condition<br />
<br />
<math><br />
\lim_{r \rightarrow \infty} \sqrt{r} \, \Big(<br />
\frac{\partial}{\partial r} - \mathrm{i} k \Big) \phi = 0<br />
</math><br />
<br />
with the wavenumber <math>k</math> also applies.<br />
<br />
== The case of water of finite depth ==<br />
<br />
The solution of the problem for the potential in finite water depth<br />
can be found by a separation ansatz,<br />
<br />
<math><br />
\phi (r,\theta,z) =: Y(r,\theta) Z(z).\,<br />
</math><br />
<br />
Substituting this into the equation for <math>\pi</math> yields<br />
<br />
<math><br />
\frac{1}{Y(r,\theta)} \left[ \frac{1}{r} \frac{\partial}{\partial<br />
r} \left( r \frac{\partial Y}{\partial r} \right) + \frac{1}{r^2}<br />
\frac{\partial^2 Y}{\partial \theta^2} \right] = - \frac{1}{Z(z)}<br />
\frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} = \eta^2.<br />
</math><br />
<br />
The possible separation constants <math>\eta</math> will be determined by the <br />
free surface condition and the bed condition.<br />
<br />
In the setting of water of finite depth, the general solution <br />
<math>Z(z)</math> can be written as<br />
<br />
<math><br />
Z(z) = F \cos \big( \eta (z+d) \big) + G \sin \big( \eta (z+d) \big),<br />
\quad \eta \in \mathbb{C} \backslash \{ 0 \},<br />
</math><br />
<br />
since <math>\eta = 0</math> is not an eigenvalue.<br />
To satisfy the bed condition, <math>G</math> must be <math>0</math>. <br />
<math>Z(z)</math> satisfies the free surface condition, provided the separation <br />
constants <math>\eta</math> are roots of the equation<br />
<br />
<math><br />
- F \eta \sin \big( \eta (z+d) \big) - \alpha F \cos \big( \eta (z+d)<br />
\big) = 0, \quad z=0,<br />
</math><br />
<br />
or, equivalently, if they satisfy <br />
<br />
<math><br />
\alpha + \eta \tan \eta d = 0.<br />
</math><br />
<br />
This equation, also called dispersion relation, has an<br />
infinite number of real roots, denoted by <math>k_m</math> and <math>-k_m</math> (<math>m \geq<br />
1</math>), but the negative roots produce the same eigenfunctions as the<br />
positive ones and will therefore not be considered. It also has a pair of purely imaginary roots which<br />
will be denoted by <math>k_0</math>. Writing <math>k_0 = - \mathrm{i} k</math>, <math>k</math> is the<br />
(positive) root of the dispersion relation<br />
<br />
<math><br />
\alpha = k \tanh k d,<br />
</math><br />
<br />
again it suffices to consider only the positive root. The solutions can<br />
therefore be written as<br />
<br />
<math><br />
Z_m(z) = F_m \cos \big( k_m (z+d) \big), \quad m \geq 0.<br />
</math><br />
<br />
It follows that <math>k</math> is the previously introduced wavenumber and the dispersion relation gives the required relation to the radian frequency. <br />
<br />
For the solution of <br />
<br />
<math><br />
\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial<br />
Y}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 Y}{\partial<br />
\theta^2} = k_m^2 Y(r,\theta),<br />
</math><br />
<br />
another separation will be used,<br />
<br />
<math><br />
Y(r,\theta) =: R(r) \Theta(\theta).<br />
</math><br />
<br />
Substituting this into Laplace's equation yields<br />
<br />
<math><br />
\frac{r^2}{R(r)} \left[ \frac{1}{r} \frac{\mathrm{d}}{\mathrm{d}r} \left( r<br />
\frac{\mathrm{d} R}{\mathrm{d}r} \right) - k_m^2 R(r) \right] = -<br />
\frac{1}{\Theta (\theta)} \frac{\mathrm{d}^2 \Theta}{\mathrm{d}<br />
\theta^2} = \eta^2, <br />
</math><br />
<br />
<br />
where the separation constant <math>\eta</math> must be an integer, say <math>\nu</math>, <br />
in order for the potential to be continuous. <math>\Theta<br />
(\theta)</math> can therefore be expressed as <br />
<br />
<math><br />
\Theta (\theta) = C \, \mathrm{e}^{\mathrm{i} \nu \theta}, \quad \nu \in \mathbb{Z}.<br />
</math><br />
<br />
Equation (\ref{pot_cyl_rt2}) also yields<br />
<br />
<math><br />
r \frac{\mathrm{d}}{\mathrm{d}r} \left( r \frac{\mathrm{d}<br />
R}{\mathrm{d} r} \right) - (\nu^2 + k_m^2 r^2) R(r) = 0, \quad \nu \in<br />
\mathbb{Z}.<br />
</math><br />
<br />
Substituting <math>\tilde{r}:=k_m r</math> and writing <math>\tilde{R} (\tilde{r}) :=<br />
R(\tilde{r}/k_m) = R(r)</math>, this can be rewritten as<br />
<br />
<math><br />
\tilde{r}^2 \frac{\mathrm{d}^2 \tilde{R}}{\mathrm{d} \tilde{r}^2}<br />
+ \tilde{r} \frac{\mathrm{d} \tilde{R}}{\mathrm{d} \tilde{r}}<br />
- (\nu^2 + \tilde{r}^2)\, \tilde{R} = 0, \quad \nu \in \mathbb{Z},<br />
</math><br />
<br />
which is the modified version of Bessel's equation. Substituting back,<br />
the general solution is given by<br />
<br />
<math><br />
R(r) = D \, I_\nu(k_m r) + E \, K_\nu(k_m r), \quad m \in<br />
\mathbb{N},\ \nu \in \mathbb{Z},<br />
</math><br />
<br />
where <math>I_\nu</math> and <math>K_\nu</math> are the modified Bessel functions of the first<br />
and second kind, respectively, of order <math>\nu</math>.<br />
<br />
The potential <math>\phi</math> can thus be expressed in local cylindrical<br />
coordinates as<br />
<br />
<math><br />
\phi (r,\theta,z) = \sum_{m = 0}^{\infty} Z_m(z) \sum_{\nu = -<br />
\infty}^{\infty} \left[ D_{m\nu} I_\nu (k_m r) + E_{m\nu} K_\nu (k_m<br />
r) \right] \mathrm{e}^{\mathrm{i} \nu \theta}, <br />
</math><br />
<br />
where <math>Z_m(z)</math> is given by equation \eqref{sol_Z_fin}. Substituting <math>Z_m</math><br />
back as well as noting that <math>k_0=-\mathrm{i} k</math> yields <br />
<br />
<math> \phi (r,\theta,z) <br />
= F_0 \cos(-\mathrm{i} k (z+d)) \sum_{\nu = - \infty}^{\infty}<br />
\left[ D_{0\nu} I_\nu (-\mathrm{i} k r) + E_{0\nu} K_\nu (-\mathrm{i} k r)\right]<br />
\mathrm{e}^{\mathrm{i} \nu \theta}<br />
+ \sum_{m = 1}^{\infty} F_m \cos(k_m(z+d)) \sum_{\nu = -<br />
\infty}^{\infty} \left[ D_{m\nu} I_\nu (k_m r) + E_{m\nu} K_\nu (k_m<br />
r) \right] \mathrm{e}^{\mathrm{i} \nu \theta}.<br />
</math><br />
<br />
Noting that <math>\cos \mathrm{i} x = \cosh x</math> is an even function and the<br />
relations <math>I_\nu(-\mathrm{i} x) = (-\mathrm{i})^{\nu} J_\nu(x)</math> where <math>J_\nu</math> is the Bessel<br />
function of the first kind of order <math>\nu</math> and <math>K_\nu (-\mathrm{i} x) = \pi / 2\,\,<br />
\mathrm{i}^{\nu+1} H_\nu^{(1)}(x)</math> with <math>H_\nu^{(1)}</math> denoting<br />
the Hankel function of the first kind of order <math>\nu</math>, it follows that<br />
<br />
<math><br />
\phi (r,\theta,z) <br />
= \cosh(k (z+d)) \sum_{\nu = - \infty}^{\infty}<br />
\left[ D_{0\nu}' J_\nu (k r) + E_{0\nu}' H_\nu^{(1)} (k r)\right]<br />
\mathrm{e}^{\mathrm{i} \nu \theta} + \sum_{m = 1}^{\infty} F_m \cos(k_m(z+d)) \sum_{\nu = -<br />
\infty}^{\infty} \left[ D_{m\nu}' I_\nu (k_m r) + E_{m\nu}' K_\nu (k_m<br />
r) \right] \mathrm{e}^{\mathrm{i} \nu \theta}.<br />
</math><br />
<br />
However, <math>J_\nu</math> does not satisfy the radiation<br />
condition \eqref{water_rad} and neither does <math>I_\nu</math><br />
since it becomes unbounded for increasing real argument. These<br />
two solutions represent incoming waves which will also be<br />
required later.<br />
<br />
Therefore, the solution of the problem requires <math>D_{m\nu}'=0</math><br />
for all <math>m,\nu</math>. Therefore, the<br />
eigenfunction expansion of the water velocity potential in<br />
cylindrical outgoing waves with coefficients <math>A_{m\nu}</math> is given by <br />
<br />
<math><br />
\phi (r,\theta,z) = \frac{\cosh(k (z+d))}{\cosh kd} \sum_{\nu = -<br />
\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}<br />
\sum_{\nu = - \infty}^{\infty} A_{m\nu} K_\nu (k_m r) \mathrm{e}^{\mathrm{i} \nu \theta}.<br />
</math><br />
<br />
The two terms describe the propagating and the decaying wavefields<br />
respectively.<br />
<br />
<br />
<br />
<br />
== The case of infinitely deep water == <br />
<br />
A solution will be developed for the same setting as before but under the<br />
assumption of water of infinite depth. As in the previous section,<br />
Laplace's equation must be solved in cylindrical coordinates<br />
satisfying the free surface and the radiation condition. However,<br />
instead of the bed condition, the water velocity potential is also required to<br />
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<br />
infinitely deep water an uncountable amount of separation constants<br />
<math>\eta</math> is valid. <br />
<br />
As above, the general solution can be represented as<br />
<br />
<math><br />
Z(z) = F \mathrm{e}^{\mathrm{i} \eta z} + G \mathrm{e}^{- \mathrm{i} \eta z}, \quad \eta \in \mathbb{C}<br />
\backslash \{0\}.<br />
</math><br />
<br />
Assuming <math>\eta</math> has got a positive<br />
imaginary part, in order to satisfy the depth condition, <math>F<math> must be<br />
zero. <math>Z(z)</math> then satisfies the free surface condition if <math>\eta</math> is a root of<br />
<br />
<math><br />
-G \mathrm{i} \eta \mathrm{e}^{-\mathrm{i} \eta z} - \alpha G \mathrm{e}^{-\mathrm{i} \eta z} = 0, \quad z=0,<br />
</math><br />
<br />
which yields the dispersion relation <br />
<br />
<math><br />
\eta = - \mathrm{i} \alpha.<br />
</math><br />
<br />
Therefore, <math>\eta</math> must even be purely imaginary. For <math>\Im \eta < 0</math>, <br />
this is also obtained, but with a minus sign in front of<br />
<math>\eta</math>. However, this yields the same solution. One solution can<br />
therefore be written as<br />
<br />
<math><br />
Z(z) = G \mathrm{e}^{\alpha z}.<br />
</math><br />
<br />
Now, <math>\eta</math> is assumed real. In this case, it is convenient to write<br />
the general solution in terms of cosine and sine,<br />
<br />
<math><br />
Z(z) = F \cos(\eta z) + G \sin(\eta z), \quad \eta \in \mathbb{R}<br />
\backslash \{0\}.<br />
</math><br />
<br />
This solution satisfies the depth condition automatically.<br />
Making use of the free surface condition, it follows that<br />
<br />
<math><br />
(-\eta F - \alpha G) \sin (\eta z) + (\eta G - \alpha F) \cos(\eta z)<br />
= 0, \quad z=0,<br />
</math><br />
<br />
which can be solved for <math>G</math>,<br />
<br />
<math><br />
G = \frac{\alpha}{\eta} F.<br />
</math><br />
<br />
Substituting this back gives<br />
<br />
<math><br />
Z(z) = F \big( \cos(\eta z) + \frac{\alpha}{\eta} \sin(\eta z)<br />
\big) , \quad \eta \in \mathbb{R} \backslash \{0\}. <br />
</math><br />
<br />
Obviously, a negative value of <math>\eta</math> produces the same<br />
eigenfunction as the positive one. Therefore, only positive ones are<br />
considered, leading to the definition<br />
<br />
<math><br />
\psi(z,\eta) := \cos(\eta z) + \frac{\alpha}{\eta} \sin(\eta z), \quad<br />
(z,\eta) \in \mathbb{R}_{\leq0} \times \mathbb{R}_{>0}.<br />
</math><br />
<br />
This gives the vertical eigenfunctions in infinite depth.<br />
<br />
For the radial and angular coordinate the same separation can be used<br />
as in the finite depth case so that the general solution of problem<br />
can be written as<br />
<br />
<math><br />
\phi (r,\theta,z) = \mathrm{e}^{\alpha z} \sum_{\nu = -<br />
\infty}^{\infty} \left[ E_\nu (-\mathrm{i} \alpha) I_\nu (-\mathrm{i} \alpha r) + <br />
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 = -<br />
\infty}^{\infty} \left[ E_\nu I_\nu (\eta r) + F_{\nu} (\eta) K_\nu<br />
(\eta r) \right] \mathrm{e}^{\mathrm{i} \nu \theta} \mathrm{d}\eta. <br />
</math><br />
<br />
Making use of the radiation condition as<br />
well as the relations of the Bessel functions in the same way as in<br />
the finite depth case, this can be rewritten as the eigenfunction <br />
expansion of the water velocity potential into cylindrical outgoing<br />
waves in water of infinite depth, <br />
<br />
<math><br />
\phi (r,\theta,z) = \mathrm{e}^{\alpha z} \sum_{\nu = -<br />
\infty}^{\infty} A_{\nu} (\mathrm{i} \alpha) H_\nu^{(1)} (\alpha r) \mathrm{e}^{\mathrm{i} \nu<br />
\theta} + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\nu = -<br />
\infty}^{\infty} A_{\nu} (\eta) K_\nu (\eta r) \mathrm{e}^{\mathrm{i} \nu<br />
\theta} \mathrm{d}\eta.<br />
</math></div>Funkalothttps://wikiwaves.org/wiki/index.php?title=WikiWaves:Test&diff=1282WikiWaves:Test2006-04-09T07:09:35Z<p>Funkalot: </p>
<hr />
<div>Here is a link [[Main Page]].<br />
<br />
Here is some <math><br />
\begin{equation}<br />
\left\{<br />
\begin{array}{l}<br />
\ddot r - \gamma\ddot r_{ss} + r_{ssss}=(\tau r_s)_s,\ \ r_s.r_s=1,\\<br />
r(0,t)=(0,0), \;\; r_{s}(0,t)=(1,0),\\<br />
(\gamma\ddot r_s+\tau r_{s}-r_{sss})|_{(1,t)}=-\alpha\dot{r}(1,t),\<br />
r_{ss}(1,t)=-\beta\dot r_s(1,t),\\<br />
r(s,0)=r_0(s),\ \dot r(s,0)=r_1(s).<br />
\end{array}\right.<br />
\end{equation}</math><br />
<br />
and a picture:<br />
\begin{figure}[ht]<br />
\begin{center}<br />
\begin{pspicture}(0,-1.5)(10,1.5)<br />
\psplot[linecolor=lightgray,linewidth=10pt]{0}{10}{x .07 mul 2 exp x .07 mul .5<br />
sub mul x .07 mul .8 sub mul 600 mul x .07 mul 5 add div}<br />
\psline[linecolor=lightgray,linewidth=10pt](-.3,0)(0,0)<br />
\psframe[linestyle=dashed,dash=5pt 8pt](-.3,-.22)(10,.22)<br />
\psline[linestyle=dashed]{|<->|}(4,0)(4,1)<br />
\rput(4.9,1){$y(x,t)$}<br />
\psline[arrowsize=6.5pt,arrowlength=1.5]{->}(0,0)(11,0)<br />
\psline[arrowsize=6.5pt,arrowlength=1.5]{<->}(0,-1.5)(0,1.5)<br />
\rput(-.3,1){$y$}<br />
\rput(10.5,-.2){$x$}<br />
\rput(9.9,.6){$x=1$}<br />
\end{pspicture}<br />
\caption{\small The linear beam}<br />
\label{f1}\end{center}<br />
\end{figure}</div>Funkalot