Wavemaker Theory - Revision history

Meylan: /* Introduction */

2023-05-05T01:13:05Z

Introduction

← Older revision		Revision as of 01:13, 5 May 2023
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	We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].		We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].
	A paddle is undergoing small amplitude horizontal oscillations with displacement		A paddle is undergoing small amplitude horizontal oscillations with displacement
	<center><math> \zeta (t) = \mathrm{Re} \left \{\frac{1}{\mathrm{i}\omega} f(z) e^{-i\omega t} \right \} </math></center>		<center><math> \zeta (z,t) = \mathrm{Re} \left \{\frac{1}{-\mathrm{i}\omega} f(z) e^{-i\omega t} \right \} </math></center>
	where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that		where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that
	<math>f(z)\,</math> is real but this is not necessary.		<math>f(z)\,</math> is real but this is not necessary.

Meylan: /* Introduction */

2023-05-05T01:12:28Z

Introduction

← Older revision		Revision as of 01:12, 5 May 2023
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	We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].		We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].
	A paddle is undergoing small amplitude horizontal oscillations with displacement		A paddle is undergoing small amplitude horizontal oscillations with displacement
	<center><math> \zeta (t) = \mathrm{Re} \left \{\frac{1}{\mathrm{i}\omega} f(z) e^{i\omega t} \right \} </math></center>		<center><math> \zeta (t) = \mathrm{Re} \left \{\frac{1}{\mathrm{i}\omega} f(z) e^{-i\omega t} \right \} </math></center>
	where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that		where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that
	<math>f(z)\,</math> is real but this is not necessary.		<math>f(z)\,</math> is real but this is not necessary.

Meylan at 02:16, 4 April 2010

2010-04-04T02:16:14Z

← Older revision		Revision as of 02:16, 4 April 2010
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	{{Ocean Wave Interaction with Ships and Offshore Structures		{{Ocean Wave Interaction with Ships and Offshore Structures
	\| chapter title = Wavemaker Theory		\| chapter title = Wavemaker Theory
	\| next chapter = [[~~Translating Coordinate Systems~~]]		\| next chapter = [[Ship Kelvin Wake]]
	\| previous chapter = [[Wave Momentum Flux]]		\| previous chapter = [[Wave Momentum Flux]]
	}}		}}

Meylan at 22:37, 30 August 2009

2009-08-30T22:37:32Z

← Older revision		Revision as of 22:37, 30 August 2009
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	\| previous chapter = [[Wave Momentum Flux]]		\| previous chapter = [[Wave Momentum Flux]]
	}}		}}

			{{complete pages}}

	== Introduction ==		== Introduction ==

Meylan at 22:35, 30 August 2009

2009-08-30T22:35:37Z

← Older revision		Revision as of 22:35, 30 August 2009
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	== Introduction ==		== Introduction ==

	[[Image:~~Wavemaker~~.~~jpg~~\|600px\|Wavemaker]]		[[Image:Wave_maker.png\|600px\|right\|thumb\|Wavemaker]]

	We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].		We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].

Meylan: /* Expansion of the solution */

2009-08-24T21:40:30Z

Expansion of the solution

← Older revision		Revision as of 21:40, 24 August 2009
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	The equations therefore become		The equations therefore become
	{{standard linear wave scattering equations without body condition}}		{{standard linear wave scattering equations without body condition}}
			The boundary condition at the wavemaker is
	<center>		<center>
	<math>		<math>

Meylan: /* Introduction */

2009-08-24T10:46:07Z

Introduction

← Older revision		Revision as of 10:46, 24 August 2009
Line 11:		Line 11:
	We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].		We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].
	A paddle is undergoing small amplitude horizontal oscillations with displacement		A paddle is undergoing small amplitude horizontal oscillations with displacement
	<center><math> \zeta (t) = \mathrm{Re} \left \{\frac{1}{i\omega} f(z) e^{i\omega t} \right \} </math></center>		<center><math> \zeta (t) = \mathrm{Re} \left \{\frac{1}{\mathrm{i}\omega} f(z) e^{i\omega t} \right \} </math></center>
	where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that		where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that
	<math>f(z)\,</math> is real but this is not necessary.		<math>f(z)\,</math> is real but this is not necessary.

Meylan: /* Introduction */

2009-08-24T10:45:44Z

Introduction

← Older revision		Revision as of 10:45, 24 August 2009
Line 11:		Line 11:
	We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].		We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]].
	A paddle is undergoing small amplitude horizontal oscillations with displacement		A paddle is undergoing small amplitude horizontal oscillations with displacement
	<center><math> \xi (t) = \mathrm{Re} \left \{\frac{1}{i\omega} f(z) e^{i\omega t} \right \} </math></center>		<center><math> \zeta (t) = \mathrm{Re} \left \{\frac{1}{i\omega} f(z) e^{i\omega t} \right \} </math></center>
	where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that		where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that
	<math>f(z)\,</math> is real but this is not necessary.		<math>f(z)\,</math> is real but this is not necessary.

Meylan: /* Expansion of the solution */

2009-08-24T10:44:52Z

Expansion of the solution

← Older revision		Revision as of 10:44, 24 August 2009
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	== Expansion of the solution ==		== Expansion of the solution ==

	~~Let the total velocity potential be~~		{{frequency definition}}
	~~<center><math> \Phi(x,z,t) = \mathrm~~{Re} ~~\left \~~{ ~~\phi(x,z) e^~~{~~i\omega t~~} ~~\right \~~} ~~</math></center>.~~
	~~This gives us a [[Frequency Domain Problem]].~~
	The ~~water is assumed to have~~		{{velocity potential in frequency domain}}
	~~constant finite depth <math>h \,</math> and the <math>z \,</math>-direction points vertically~~
	~~upward with the water surface at <math>z=0 \,</math> and the sea floor at <math>z=-h \,</math>. The~~		The equations therefore become
	~~boundary value problem can~~ therefore ~~be expressed as~~		{{standard linear wave scattering equations without body condition}}
	~~<center>~~
	~~<math>~~
	~~\Delta\phi=0, \,\, -h<z<0,\,x>0,~~
	~~</math>~~
	~~</center>~~
	~~<center>~~
	~~<math>~~
	~~\phi_~~{z}~~=0, \,\, z=-h,~~
	~~</math>~~
	~~</center>~~
	~~<center>~~
	~~<math>~~
	~~\partial_z\phi=\alpha\phi, \,\, z=0,\,x>0,~~
	~~</math>~~
	~~</center>~~
	<center>		<center>
	<math>		<math>
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	</math>		</math>
	</center>		</center>
	We		We must also apply the [[Sommerfeld Radiation Condition]]
	must also apply the [[Sommerfeld Radiation Condition]]
	as <math>x\rightarrow\infty</math>. This essentially implies		as <math>x\rightarrow\infty</math>. This essentially implies
	that the only wave at infinity is propagating away.		that the only wave at infinity is propagating away.

Adi Kurniawan at 05:21, 18 May 2009

2009-05-18T05:21:17Z

← Older revision		Revision as of 05:21, 18 May 2009
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	}}		}}

	= Introduction =		== Introduction ==

	[[Image:Wavemaker.jpg\|600px\|Wavemaker]]		[[Image:Wavemaker.jpg\|600px\|Wavemaker]]
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	as a function of <math> \omega, f(z) \, </math> and <math> h \, </math>. Time-dependent wavemaker theories can also be developed.		as a function of <math> \omega, f(z) \, </math> and <math> h \, </math>. Time-dependent wavemaker theories can also be developed.

	= Expansion of the solution =		== Expansion of the solution ==

	Let the total velocity potential be		Let the total velocity potential be
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	{{separation of variables for a free surface}}		{{separation of variables for a free surface}}

	= Expansion in Eigenfunctions =		== Expansion in Eigenfunctions ==

	The wavemaker velocity potential <math> \phi \,</math> can be expressed simply in terms of eigenfunctions		The wavemaker velocity potential <math> \phi \,</math> can be expressed simply in terms of eigenfunctions
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	<center><math> a_n = -\frac{1}{k_n A_n} \int_{-h}^0 \phi_n(z) f(z)\mathrm{d}z </math></center>		<center><math> a_n = -\frac{1}{k_n A_n} \int_{-h}^0 \phi_n(z) f(z)\mathrm{d}z </math></center>

	== Far Field Wave ==		=== Far Field Wave ===
	One of the primary objecives of wavemaker theory is to determine the far-field wave amplitude <math> A \, </math> in terms of <math> f(z) \, </math>.		One of the primary objecives of wavemaker theory is to determine the far-field wave amplitude <math> A \, </math> in terms of <math> f(z) \, </math>.
	The far-field wave component representing progagating waves is given by:		The far-field wave component representing progagating waves is given by:
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	Unfortunately this is not a "practical" displacement since <math> \phi_0 (z) \, </math> depends on <math> \omega\, </math>, so one would need to build a flexible paddle.		Unfortunately this is not a "practical" displacement since <math> \phi_0 (z) \, </math> depends on <math> \omega\, </math>, so one would need to build a flexible paddle.

	= Matlab Code =		== Matlab Code ==

	A program to calculate the coefficients for the wave maker problems can be found here		A program to calculate the coefficients for the wave maker problems can be found here
	[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/wavemaker.m wavemaker.m]		[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/wavemaker.m wavemaker.m]

	== Additional code ==		=== Additional code ===

	This program requires		This program requires