Dispersion Relation for a Free Surface - Revision history

Meylan at 09:18, 20 October 2009

2009-10-20T09:18:46Z

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			{{complete pages}}

	== Introduction ==		== Introduction ==

Meylan at 21:36, 24 August 2009

2009-08-24T21:36:51Z

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 The dispersion equation for a free surface is one of the most important equations in linear water wave theory. It arises when [http://en.wikipedia.org/wiki/Separation_of_Variables separating variables] subject to the boundary conditions for a free surface.
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 {{separation of variables for a free surface}}
 == See Also ==

Meylan at 21:35, 24 August 2009

2009-08-24T21:35:42Z

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	The same equation arises when separating variables in two or three dimensions and we present here the two-dimensional version. We denote the vertical coordinate by <math>z\,,</math> which points vertically upwards, and the free surface is at <math>z=0\,.</math>		The same equation arises when separating variables in two or three dimensions and we present here the two-dimensional version. We denote the vertical coordinate by <math>z\,,</math> which points vertically upwards, and the free surface is at <math>z=0\,.</math>

	~~The equations for the [[Frequency Domain Problem]] with radial frequency <math>\,\omega</math> in terms of the potential alone which are~~		{{frequency definition}}
	~~<center><math>\nabla^2\phi =0\,, </math></center>~~
	~~<center><math>\frac~~{~~\partial \phi}~~{~~\partial z} = \alpha \phi, \quad z=0\,,</math></center>~~
	~~<center><math>\frac{\partial \phi~~}~~{\partial z~~} ~~= 0, \quad z=-h\,,</math></center>~~
	~~where <math>\alpha\,</math> is the wavenumber in [[Infinite Depth]] which is given by~~
	~~<math>\alpha=\omega^2/g\,,</math> where <math>g\,</math> is gravity. We generally non-dimensionalize~~
	~~so that gravitational constant is unity.~~

	~~We then find a separation of variables solution to [[Laplace's Equation]] and~~
	~~apply the boundary condition at <math>z=-h\,,</math> obtaining the~~
	~~following expression for the velocity potential:~~
	~~<center>~~
	~~<math>\phi(x,z) = e^{kx} \cos k(z+h) \,.</math>~~
	~~</center>~~
	~~If we then apply the condition at <math>z=0\,,</math> we see that the constant <math>\,k</math>~~
	~~(which corresponds to the wavenumber) is given by~~
	~~<center><math> -k \sin(kh) = \alpha \cos(kh)\,, </math></center>~~
	or
	~~<center>~~
	~~<math> k \tan(kh) = -\alpha\,.\quad(1)</math>~~
	~~</center>~~
	~~This is the dispersion equation for a free surface.~~

	~~We can also write the separation of variables as~~		{{velocity potential in frequency domain}}
	~~<center>~~
	~~<math>\phi(x,z) = e^~~{~~ikx~~} ~~\cosh k(z+h) \,<~~/~~math>~~		The equations therefore become
	</~~center>~~		{{standard linear wave scattering equations without body condition}}
	in ~~which case the dispersion equation becomes~~
	~~<center>~~		We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables]
	~~<math> k \tanh(kh) = \alpha\,\,\,(2)</math>~~		{{separation of variables in two dimensions}}
	~~</center>~~
			{{separation of variables for a free surface}}


	== Solution of the dispersion equation ==		== Solution of the dispersion equation ==

Adi Kurniawan: /* Matlab Code */

2008-09-23T02:29:51Z

Matlab Code

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	* [http://en.wikipedia.org/wiki/Dispersion_(water_waves) Dispersion (water waves)]		* [http://en.wikipedia.org/wiki/Dispersion_(water_waves) Dispersion (water waves)]

	= Matlab Code =		== Matlab Code ==

	A program to calculate solutions to the dispersion relation for a free surface		A program to calculate solutions to the dispersion relation for a free surface

Adi Kurniawan: /* See Also */

2008-09-23T02:29:18Z

Adi Kurniawan: /* Solution of the dispersion equation */

2008-09-23T02:29:03Z

Solution of the dispersion equation

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	</center>		</center>

	= Solution of the dispersion equation =		== Solution of the dispersion equation ==

	Equation (1) has two purely imaginary solutions plus a countable number		Equation (1) has two purely imaginary solutions plus a countable number

Adi Kurniawan: /* See Also */

2008-09-23T02:28:14Z

Adi Kurniawan: add link

2008-09-23T02:27:53Z

add link

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	normal derivative at <math>z=h</math> which is why we have the		normal derivative at <math>z=h</math> which is why we have the
	cosine series).		cosine series).

			= See Also =
			* [[http://en.wikipedia.org/wiki/Dispersion_(water_waves) Dispersion (water waves)]]

	= Matlab Code =		= Matlab Code =

Adi Kurniawan: /* Solution of the dispersion equation */

2008-08-20T08:47:24Z

Solution of the dispersion equation

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	The dispersion equation is a classical Sturm-Liouville equation.		The dispersion equation is a classical Sturm-Liouville equation.
	The vertical eigenfunctions <math>\cos k_n (z-H)</math>		The vertical eigenfunctions <math>\cos k_n (z+h)</math>
	form complete set for <math>L_2[-h,0]\,</math> and they are orthogonal.		form complete set for <math>L_2[-h,0]\,</math> and they are orthogonal.
	Also, as <math>n\to\infty</math> <math>k_n \to \ n\pi/H</math>		Also, as <math>n\to\infty</math> <math>k_n \to \ n\pi/h</math>
	so that in the limit the vertical eigenfunctions become the same as		so that in the limit the vertical eigenfunctions become the same as
	the Fourier cosine series for <math>L_2[-h,0]\,</math> (remembering		the Fourier cosine series for <math>L_2[-h,0]\,</math> (remembering

Adi Kurniawan: /* Solution of the dispersion equation */

2008-08-20T08:45:01Z

Solution of the dispersion equation

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	of positive and negative real solutions. Note that the equation is		of positive and negative real solutions. Note that the equation is
	even in <math>k</math> so that for every solution the negative is also a solution.		even in <math>k</math> so that for every solution the negative is also a solution.
	The solutions of equation(2) are just <math>i</math> times the solutions		The solutions of equation (2) are just <math>i</math> times the solutions
	to equation (1). Sometimes (especially in older works) both equations		to equation (1). Sometimes (especially in older works) both equations
	are used so that each equation needs only to be solved for real solutions.		are used so that each equation needs only to be solved for real solutions.
	We denote the solutions to (1) by <math>k_n</math>		We denote the solutions to (1) by <math>k_n</math>

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	cosine series).		cosine series).

	= See Also =		== See Also ==
	* [http://en.wikipedia.org/wiki/Dispersion_(water_waves) Dispersion (water waves)]		* [http://en.wikipedia.org/wiki/Dispersion_(water_waves) Dispersion (water waves)]

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	= See Also =		= See Also =
	* [[http://en.wikipedia.org/wiki/Dispersion_(water_waves) Dispersion (water waves)]]		* [http://en.wikipedia.org/wiki/Dispersion_(water_waves) Dispersion (water waves)]

	= Matlab Code =		= Matlab Code =