Waves reflecting off a vertical wall - Revision history

Meylan at 21:53, 3 April 2010

2010-04-03T21:53:51Z

← Older revision		Revision as of 21:53, 3 April 2010
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	~~[[Image:Breakwater.jpg\|thumb\|right\|600px\|Breakwater]]~~
	Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with solid boundaries seen in practice. The other is wavemaker theory.		Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with solid boundaries seen in practice. The other is wavemaker theory.

Meylan at 21:53, 3 April 2010

2010-04-03T21:53:08Z

@@ Line 6: / Line 6: @@
 On <math> x = 0 \, </math>:
 <center><math> \partial_n\phi = \frac{\partial}{\partial x}
-\left( \phi^{\mathrm{I}} + \phi^{mathrm{D}} \right) = 0 </math></center>
+\left( \phi^{\mathrm{I}} + \phi^{\mathrm{D}} \right) = 0 </math></center>
-where <math>\phi^{mathrm{D}}</math> is the diffraction potential
+where <math>\phi^{\mathrm{D}}</math> is the diffraction potential.
 [[Category:Linear Water-Wave Theory]]
 [[Category:Complete Pages]]

Meylan at 21:49, 3 April 2010

2010-04-03T21:49:27Z

← Older revision		Revision as of 21:49, 3 April 2010
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	{{incident plane wave}}		{{incident plane wave}}

			On <math> x = 0 \, </math>:
			<center><math> \partial_n\phi = \frac{\partial}{\partial x}
			\left( \phi^{\mathrm{I}} + \phi^{mathrm{D}} \right) = 0 </math></center>
			where <math>\phi^{mathrm{D}}</math> is the diffraction potential
			or
			<center><math> - \mathrm{i} k A_I + \mathrm{i} k A_R \Longrightarrow A_R = A_I \equiv A </math></center>


	The refl		The refl
Line 11:		Line 19:
	<center><math> \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_R}{\omega} \frac{\cosh k (z+h)}{\cosh k h}		<center><math> \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_R}{\omega} \frac{\cosh k (z+h)}{\cosh k h}
	e^{\mathrm{i} k x +\mathrm{i}\omega t} \right \} </math></center>		e^{\mathrm{i} k x +\mathrm{i}\omega t} \right \} </math></center>
	~~On <math> x = 0 \, </math>:~~
	~~<center><math> \frac{\partial}{\partial x} \left( \Phi_I + \Phi_R \right) = 0 </math></center>~~
	or
	~~<center><math> - \mathrm{i} k A_I + \mathrm{i} k A_R \Longrightarrow A_R = A_I \equiv A </math></center>~~

	The resulting total velocity potential is		The resulting total velocity potential is

Meylan at 21:45, 3 April 2010

2010-04-03T21:45:59Z

← Older revision		Revision as of 21:45, 3 April 2010
Line 1:		Line 1:
	[[Image:Breakwater.jpg\|thumb\|right\|600px\|Breakwater]]		[[Image:Breakwater.jpg\|thumb\|right\|600px\|Breakwater]]
	Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with solid boundaries seen in practice. The other is wavemaker theory.		Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with solid boundaries seen in practice. The other is wavemaker theory.
	~~Let the incident wave elevation be~~
	~~<center><math> \zeta_I = \mathrm~~{~~Re} \left \~~{ ~~A_I e^{-\mathrm{i~~} ~~k x+\mathrm{i}\omega t} \right \~~} ~~</math></center>~~		{{incident plane wave}}
	~~and the reflected is~~
	~~<center><math> \zeta_R = \mathrm{Re} \left \{ A_R e^{\mathrm{i}k x+\mathrm{i}\omega t} \right \} </math></center>~~		The refl
	~~where the sign of the <math> e^{\mathrm{i} k x}\, </math> term has been reversed to denote a wave propagating to the left. <math> A_R\,</math> can be complex in order to allow for phase differences.~~

	The corresponding velocity potentials are		The corresponding velocity potentials are

Meylan at 21:37, 3 April 2010

2010-04-03T21:37:37Z

← Older revision		Revision as of 21:37, 3 April 2010
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	<center><math> \zeta_I = \mathrm{Re} \left \{ A_I e^{-\mathrm{i} k x+\mathrm{i}\omega t} \right \} </math></center>		<center><math> \zeta_I = \mathrm{Re} \left \{ A_I e^{-\mathrm{i} k x+\mathrm{i}\omega t} \right \} </math></center>
	and the reflected is		and the reflected is
	<center><math> \zeta_R = \mathrm{Re} \left \{ A_R e^{+\mathrm{i}k x+\mathrm{i}\omega t} \right \} </math></center>		<center><math> \zeta_R = \mathrm{Re} \left \{ A_R e^{\mathrm{i}k x+\mathrm{i}\omega t} \right \} </math></center>
	where the sign of the <math> e^{~~iKX~~}\, </math> term has been reversed to denote a wave propagating to the left. <math> A_R\,</math> can be complex in order to allow for phase differences.		where the sign of the <math> e^{\mathrm{i} k x}\, </math> term has been reversed to denote a wave propagating to the left. <math> A_R\,</math> can be complex in order to allow for phase differences.

	The corresponding velocity potentials are		The corresponding velocity potentials are
	<center><math> \Phi_I = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_I}{\omega} \frac{\cosh k k h (z+h)}{\cosh k h}		<center><math> \Phi_I = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_I}{\omega} \frac{\cosh k k h (z+h)}{\cosh k h}
	e^{-\mathrm{i}+i\omega t} \right \} </math></center>		e^{-\mathrm{i} k x + \mathrm{i}\omega t} \right \} </math></center>
	<center><math> \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_R}{\omega} \frac{\cosh k (z+h)}{\cosh k h}		<center><math> \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_R}{\omega} \frac{\cosh k (z+h)}{\cosh k h}
	e^{\mathrm{i} k x +\mathrm{i}\omega t} \right \} </math></center>		e^{\mathrm{i} k x +\mathrm{i}\omega t} \right \} </math></center>
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	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]
			[[Category:Complete Pages]]

Meylan: Created page with 'Breakwater Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with…'

2010-04-03T21:34:30Z

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New page

[[Image:Breakwater.jpg|thumb|right|600px|Breakwater]]
Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with solid boundaries seen in practice. The other is wavemaker theory.
Let the incident wave elevation be
<center><math> \zeta_I = \mathrm{Re} \left \{ A_I e^{-\mathrm{i} k x+\mathrm{i}\omega t} \right \} </math></center>
and the reflected is
<center><math> \zeta_R = \mathrm{Re} \left \{ A_R e^{+\mathrm{i}k x+\mathrm{i}\omega t} \right \} </math></center>
where the sign of the <math> e^{iKX}\, </math> term has been reversed to denote a wave propagating to the left. <math> A_R\,</math> can be complex in order to allow for phase differences.

The corresponding velocity potentials are
<center><math> \Phi_I = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_I}{\omega} \frac{\cosh k k h (z+h)}{\cosh k h}
e^{-\mathrm{i}+i\omega t} \right \} </math></center>
<center><math> \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_R}{\omega} \frac{\cosh k (z+h)}{\cosh k h}
e^{\mathrm{i} k x +\mathrm{i}\omega t} \right \} </math></center>
On <math> x = 0 \, </math>:
<center><math> \frac{\partial}{\partial x} \left( \Phi_I + \Phi_R \right) = 0 </math></center>
or
<center><math> - \mathrm{i} k A_I + \mathrm{i} k A_R \Longrightarrow A_R = A_I \equiv A </math></center>

The resulting total velocity potential is
<center><math> \Phi = \Phi_I + \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i}gA_R}{\omega}
\frac{\cosh k (z+h}{\cosh kh} e^{\mathrm{i}\omega t} \left( e^{\mathrm{i}kx} + e^{-\mathrm{i}kx} \right) \right \} </math></center>
<center><math> = 2 A \mathrm{Re} \left \{ \frac{\mathrm{i}g}{\omega}
\frac{\cosh k (z+h)}{\cosh kh} \cos kx e^{\mathrm{i}\omega t} \right \} </math></center>

The resulting standing-wave elevation is:
<center><math> \zeta = \zeta_I + \zeta_R = 2 A \cos k x \cos \omega t \, </math></center>

[[Category:Linear Water-Wave Theory]]