Variable Depth Shallow Water Wave Equation - Revision history

Meylan at 03:04, 18 July 2009

2009-07-18T03:04:45Z

← Older revision		Revision as of 03:04, 18 July 2009
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	We consider here the problem of waves reflected by a region of variable depth in a finite region or in		We consider here the problem of waves reflected by a region of variable depth in a finite region or in
	an otherwise uniform depth region assuming the equations of [[:Category:Shallow Depth\|Shallow Depth]].		an otherwise uniform depth region assuming the equations of [[:Category:Shallow Depth\|Shallow Depth]] (assuming the problem is linear).
	We consider slightly more general equations of motion so that the same method could be used for a variable		We consider slightly more general equations of motion so that the same method could be used for a variable
	density string.		density string.

Adi Kurniawan: /* Generalised Eigenfunction Expansion */

2009-04-07T10:06:14Z

Generalised Eigenfunction Expansion

← Older revision		Revision as of 10:06, 7 April 2009
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	\implies \zeta(x,t) = \int_{0}^{\infty} \left\{ \cos(\omega t)\left(c_1(\omega)\zeta_+(x,\omega)+d_1(\omega)\zeta_-(x,\omega)\right)		\implies \zeta(x,t) = \int_{0}^{\infty} \left\{ \cos(\omega t)\left(c_1(\omega)\zeta_+(x,\omega)+d_1(\omega)\zeta_-(x,\omega)\right)
	+ \frac{\sin(\omega t)}{\omega}\left(c_2(\omega)\zeta_+(x,\omega)+d_2(\omega)\zeta_-(x,\omega)\right)		+ \frac{\sin(\omega t)}{\omega}\left(c_2(\omega)\zeta_+(x,\omega)+d_2(\omega)\zeta_-(x,\omega)\right)
	\right\} d \omega \qquad (10)		\right\} \mathrm{d} \omega \qquad (10)
	</math></center>		</math></center>

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	Giving the initial conditions:		Giving the initial conditions:
	<center><math>		<center><math>
	F(x) = \zeta(x,0) = \int_{0}^{\infty} \left\{ c_1(\omega)\zeta_+(x,\omega)+d_1(\omega)\zeta_-(x,\omega) \right\} d \omega		F(x) = \zeta(x,0) = \int_{0}^{\infty} \left\{ c_1(\omega)\zeta_+(x,\omega)+d_1(\omega)\zeta_-(x,\omega) \right\} \mathrm{d} \omega
	</math></center>		</math></center>
	<center><math>		<center><math>
	G(x) = \partial_t \zeta(x,0) = \int_{0}^{\infty} \left\{ c_2(\omega)\zeta_+(x,\omega)+d_2(\omega)\zeta_-(x,\omega) \right\} d \omega		G(x) = \partial_t \zeta(x,0) = \int_{0}^{\infty} \left\{ c_2(\omega)\zeta_+(x,\omega)+d_2(\omega)\zeta_-(x,\omega) \right\} \mathrm{d} \omega
	</math></center>		</math></center>

Adi Kurniawan: /* Generalised Eigenfunction Expansion */

2009-04-07T10:05:21Z

Generalised Eigenfunction Expansion

← Older revision		Revision as of 10:05, 7 April 2009
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	Now we have the generalised eigenfunctions <math>\zeta_+(x,\omega)</math> and <math>\zeta_-(x,\omega)</math>, which have the orthogonality relationships:		Now we have the generalised eigenfunctions <math>\zeta_+(x,\omega)</math> and <math>\zeta_-(x,\omega)</math>, which have the orthogonality relationships:
	<center><math>		<center><math>
	\int_{-\infty}^{\infty}\zeta_+(x,\omega)\zeta_-(x,\omega')dx= 0 \qquad (7)		\int_{-\infty}^{\infty}\zeta_+(x,\omega)\zeta_-(x,\omega')\mathrm{d}x= 0 \qquad (7)
	</math></center>		</math></center>
	<center><math>		<center><math>
	\int_{-\infty}^{\infty}\zeta_+(x,\omega)\zeta_+(x,\omega')dx= 2 \pi \delta(\omega-\omega') \qquad (8)		\int_{-\infty}^{\infty}\zeta_+(x,\omega)\zeta_+(x,\omega')\mathrm{d}x= 2 \pi \delta(\omega-\omega') \qquad (8)
	</math></center>		</math></center>
	<center><math>		<center><math>
	\int_{-\infty}^{\infty}\zeta_-(x,\omega)\zeta_-(x,\omega')dx= 2 \pi \delta(\omega-\omega') \qquad (9)		\int_{-\infty}^{\infty}\zeta_-(x,\omega)\zeta_-(x,\omega')\mathrm{d}x= 2 \pi \delta(\omega-\omega') \qquad (9)
	</math></center>		</math></center>

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	The general solution to the PDE is therefore:		The general solution to the PDE is therefore:
	<center><math>		<center><math>
	\zeta(x,t) = \int_{0}^{\infty} \zeta(x,t,\omega) d \omega		\zeta(x,t) = \int_{0}^{\infty} \zeta(x,t,\omega) \mathrm{d} \omega
	</math></center>		</math></center>
	<center><math>		<center><math>

Meylan at 04:07, 6 April 2009

2009-04-06T04:07:12Z

← Older revision		Revision as of 04:07, 6 April 2009
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	== Waves in a finite basin ==		== Waves in a finite basin ==

			[[Image:finite_string.gif\|right\|A finite shallow basin with variable depth]]

	We consider the problem of waves in a finite basin <math>-L<x<L</math>. At the edge of the basin the boundary conditions are		We consider the problem of waves in a finite basin <math>-L<x<L</math>. At the edge of the basin the boundary conditions are

Meylan: /* Generalised Eigenfunction Expansion */

2009-04-03T03:11:20Z

Generalised Eigenfunction Expansion

← Older revision		Revision as of 03:11, 3 April 2009
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\implies \zeta(x,t) = \int_{0}^{\infty} \left\{ cos(\omega t)\left(c_1(\omega)\zeta_+(x,\omega)+d_1(\omega)\zeta_-(x,\omega)\right) + \frac{sin(\omega t)}{\omega}\left(c_2(\omega)\zeta_+(x,\omega)+d_2(\omega)\zeta_-(x,\omega)\right)		\implies \zeta(x,t) = \int_{0}^{\infty} \left\{ \cos(\omega t)\left(c_1(\omega)\zeta_+(x,\omega)+d_1(\omega)\zeta_-(x,\omega)\right)
			+ \frac{\sin(\omega t)}{\omega}\left(c_2(\omega)\zeta_+(x,\omega)+d_2(\omega)\zeta_-(x,\omega)\right)
	\right\} d \omega \qquad (10)		\right\} d \omega \qquad (10)
	</math></center>		</math></center>

Meylan: /* Matlab Code */

2009-04-03T03:07:43Z

Matlab Code

← Older revision		Revision as of 03:07, 3 April 2009
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	[[Category:Shallow Depth]]		[[Category:Shallow Depth]]
	[[Category:Simple Linear Waves]]		[[Category:Simple Linear Waves]]
	[[Category:~~Compete~~ Pages]]		[[Category:Complete Pages]]

Meylan at 02:35, 3 April 2009

2009-04-03T02:35:37Z

← Older revision		Revision as of 02:35, 3 April 2009
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	[[Category:Shallow Depth]]		[[Category:Shallow Depth]]
	[[Category:Simple Linear Waves]]		[[Category:Simple Linear Waves]]
			[[Category:Compete Pages]]

Meylan at 02:34, 3 April 2009

2009-04-03T02:34:30Z

← Older revision		Revision as of 02:34, 3 April 2009
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	~~== Introduction ==~~		We consider here the problem of waves reflected by a region of variable depth in a finite region or in

	We consider here the problem of waves reflected by a region of variable depth in
	an otherwise uniform depth region assuming the equations of [[:Category:Shallow Depth\|Shallow Depth]].		an otherwise uniform depth region assuming the equations of [[:Category:Shallow Depth\|Shallow Depth]].
			We consider slightly more general equations of motion so that the same method could be used for a variable
			density string.

	== Equations ==		== Equations ==

Meylan: /* Matlab Code */

2009-04-03T02:27:57Z

Matlab Code

← Older revision		Revision as of 02:27, 3 April 2009
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	Which we can use in combination with (10) to solve the IVP.		Which we can use in combination with (10) to solve the IVP.

	== Matlab Code ==		=== Matlab Code ===

	~~A program~~ to calculate a ~~movie of the finite problem~~ can be found here		Code to calculate the solution in a infinite basin can be found here {{one dimensional shallow wave infinite}}
	~~[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_plate.m semiinfinite_plate.m]~~

	[[Category:Shallow Depth]]		[[Category:Shallow Depth]]
	[[Category:Simple Linear Waves]]		[[Category:Simple Linear Waves]]

Meylan: /* Generalised Eigenfunction Expansion */

2009-04-03T02:27:06Z

Generalised Eigenfunction Expansion

← Older revision		Revision as of 02:27, 3 April 2009
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	Note a_+, b_+, a_- and b_- are found frm their corresponding R and T values.		Note a_+, b_+, a_- and b_- are found frm their corresponding R and T values.

	== Generalised Eigenfunction Expansion ==		=== Generalised Eigenfunction Expansion ===


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	For any particular <math>\omega </math> the general solution to the differential equation can be written as:		For any particular <math>\omega </math> the general solution to the differential equation can be written as:
	<center><math>		<center><math>
	\zeta(x,t,\omega) = cos(\omega t)\left(c_1(\omega)\zeta_+(x,\omega)+d_1(\omega)\zeta_-(x,\omega)\right) + \frac{sin(\omega t)}{\omega}\left(c_2(\omega)\zeta_+(x,\omega)+d_2(\omega)\zeta_-(x,\omega)\right)		\zeta(x,t,\omega) = \cos(\omega t)\left(c_1(\omega)\zeta_+(x,\omega)+d_1(\omega)\zeta_-(x,\omega)\right) + \frac{\sin(\omega t)}{\omega}\left(c_2(\omega)\zeta_+(x,\omega)+d_2(\omega)\zeta_-(x,\omega)\right)
	</math></center>		</math></center>
	The general solution to the PDE is therefore:		The general solution to the PDE is therefore: