Linear and Second-Order Wave Theory - Revision history

Xjwiki: /* Dynamic condition */

2019-07-17T09:03:11Z

Dynamic condition

← Older revision		Revision as of 09:03, 17 July 2019
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	<center><math>		<center><math>
	\left . \begin{matrix}		\left . \begin{matrix}
	\zeta = \dfrac{1}{g} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0}\\		\zeta =-\dfrac{1}{g} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0}\\
	\dfrac{1}{g} \zeta \dfrac{\partial}{\partial z} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0} + \cdots		-\dfrac{1}{g} \zeta \dfrac{\partial}{\partial z} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0} + \cdots
	\end{matrix} \right \}		\end{matrix} \right \}
	\begin{matrix}		\begin{matrix}

Meylan: /* Second-order problem */

2011-04-26T12:15:49Z

Second-order problem

← Older revision		Revision as of 12:15, 26 April 2011
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	This article is based on the MIT open course notes and the original article can be found		This article is based on the MIT open course notes and the original article can be found
	[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/0B7683D3-9B31-453E-B98F-9F71A3C36C58/0/lecture2.pdf here]		[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/0B7683D3-9B31-453E-B98F-9F71A3C36C58/0/lecture2.pdf here]

	~~[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]~~

	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]
	[[Category:Nonlinear Water-Wave Theory]]		[[Category:Nonlinear Water-Wave Theory]]

Adi Kurniawan: /* Kinematic condition */

2010-11-06T11:11:18Z

Kinematic condition

← Older revision		Revision as of 11:11, 6 November 2010
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	<center><math> \left ( \frac{\partial \zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right )_{z=\zeta} = \left ( \frac{\partial \Phi}{\partial z} \right )_{z=\zeta} </math></center>		<center><math> \left ( \frac{\partial \zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right )_{z=\zeta} = \left ( \frac{\partial \Phi}{\partial z} \right )_{z=\zeta} </math></center>
	We expand this equation about <math>\zeta = 0</math>, which we can do because we have assumed that the slope is small. In fact the slope is our parameter <math>\epsilon</math>. It is obvious at this point that the theory does not apply to very steep waves. This gives us the following equation		We expand this equation about <math>\zeta = 0</math>, which we can do because we have assumed that the slope is small. In fact the slope is our parameter <math>\epsilon</math>. It is obvious at this point that the theory does not apply to very steep waves. This gives us the following equation
	<center><math> \left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{z=0} + \zeta \frac{\partial}{\partial z} \left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{z=0} + \cdots ~~</math></center><br>~~		<center><math>
	~~<center><math>~~ = \left( \frac{\partial\Phi}{\partial z} \right)_{z=0} + \zeta \left( \frac{\partial^2 \Phi}{\partial z^2} \right)_{z=0} + \cdots </math></center>		\left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{z=0} + \zeta \frac{\partial}{\partial z} \left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{z=0} + \;\cdots = \left( \frac{\partial\Phi}{\partial z} \right)_{z=0} + \zeta \left( \frac{\partial^2 \Phi}{\partial z^2} \right)_{z=0} + \;\cdots
			</math></center>
	where we have only taken the first order expansion. We then substitute our expressions		where we have only taken the first order expansion. We then substitute our expressions
	<center> <math>		<center> <math>
	\begin{~~matrix~~}		\begin{align}
	\zeta = \zeta_1 + \zeta_2 + \cdots \\		\zeta &= \zeta_1 + \zeta_2 + \cdots \\
	\Phi = \Phi_1 + \Phi_2 + \cdots		\Phi &= \Phi_1 + \Phi_2 + \cdots
	\end{~~matrix~~}		\end{align}
	</math></center>		</math></center>
	and keep terms of <math>\ O(\varepsilon), \ O(\varepsilon^2)</math>, remembering that <math>\zeta_1\Phi_1</math> is <math>O(\varepsilon^2)</math> etc.		and keep terms of <math>\ O(\varepsilon), \ O(\varepsilon^2)</math>, remembering that <math>\zeta_1\Phi_1</math> is <math>O(\varepsilon^2)</math> etc.

Adi Kurniawan: /* Linearization of Free-surface Conditions */

2010-11-06T11:07:15Z

Linearization of Free-surface Conditions

← Older revision		Revision as of 11:07, 6 November 2010
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	We use [http://en.wikipedia.org/wiki/Perturbation_theory perturbation theory] to expand the solution as follows		We use [http://en.wikipedia.org/wiki/Perturbation_theory perturbation theory] to expand the solution as follows
	<center><math>\zeta = \zeta_1 + \zeta_2 + \zeta_3 + \cdots ~~</math></center>~~		<center><math>
	~~<center><math>~~ \Phi = \Phi_1 + \Phi_2 + \Phi_3 + \cdots </math></center>		\begin{align}
			\zeta &= \zeta_1 + \zeta_2 + \zeta_3 + \cdots \\
			\Phi &= \Phi_1 + \Phi_2 + \Phi_3 + \cdots
			\end{align}
			</math></center>
	where we are assuming that there exists a small parameter (the wave slope) and that with respect to this the		where we are assuming that there exists a small parameter (the wave slope) and that with respect to this the
	<math>\Phi_i</math> is proportional to <math>\epsilon^i</math> or <math>O(\epsilon^i)</math>. We then derive the boundary value problem for <math> \zeta_i,\Phi_i </math>. Rarely we need to go beyond <math> i = 3 </math> (in fact it is unlikely that the terms beyond		<math>\Phi_i</math> is proportional to <math>\epsilon^i</math> or <math>O(\epsilon^i)</math>. We then derive the boundary value problem for <math> \zeta_i,\Phi_i </math>. Rarely we need to go beyond <math> i = 3 </math> (in fact it is unlikely that the terms beyond

Adi Kurniawan: /* Classical linear free surface condition */

2009-10-17T20:25:27Z

Classical linear free surface condition

← Older revision		Revision as of 20:25, 17 October 2009
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	If we eliminate <math> \zeta_1 </math> from the kinematic and dynamic free surface conditions, we obtain the classical linear free surface condition:		If we eliminate <math> \zeta_1 </math> from the kinematic and dynamic free surface conditions, we obtain the classical linear free surface condition:
	<center><math> \begin{cases}		<center><math> \begin{cases}
	\~~frac~~{\partial^2\Phi_1}{\partial t^2} + g \~~frac~~{\partial\Phi_1}{\partial z} = 0, \qquad z=0\\		\dfrac{\partial^2\Phi_1}{\partial t^2} + g \dfrac{\partial\Phi_1}{\partial z} = 0, \qquad z=0\\
	\zeta_1 = - \~~frac~~{1}{g} \~~frac~~{\partial\Phi_1}{\partial t}, \qquad z=0		\zeta_1 = - \dfrac{1}{g} \dfrac{\partial\Phi_1}{\partial t}, \qquad z=0
	\end{cases} </math></center>		\end{cases} </math></center>

Adi Kurniawan: /* Dynamic condition */

2009-10-17T20:24:36Z

Dynamic condition

← Older revision		Revision as of 20:24, 17 October 2009
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	<center><math>		<center><math>
	\left . \begin{matrix}		\left . \begin{matrix}
	\zeta = \~~frac~~{1}{g} \left( \~~frac~~{\partial\Phi}{\partial t} + \~~frac~~{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0}\\		\zeta = \dfrac{1}{g} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0}\\
	\~~frac~~{1}{g} \zeta \~~frac~~{\partial}{\partial z} \left( \~~frac~~{\partial\Phi}{\partial t} + \~~frac~~{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0} + \cdots		\dfrac{1}{g} \zeta \dfrac{\partial}{\partial z} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0} + \cdots
	\end{matrix} \right \}		\end{matrix} \right \}
	\begin{matrix}		\begin{matrix}

Meylan at 10:36, 30 August 2009

2009-08-30T10:36:15Z

← Older revision		Revision as of 10:36, 30 August 2009
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	\| previous chapter = [[Conservation Laws and Boundary Conditions]]		\| previous chapter = [[Conservation Laws and Boundary Conditions]]
	}}		}}

			{{complete pages}}

	We saw in [[Conservation Laws and Boundary Conditions]] that the potential flow model for wave propagation is given Laplace's equation plus the free-surface conditions. In this section we present the linear and second order theory for these equations. The linear theory is valid for small wave heights and the second order theory is an improvement on this. However, neither of these theories work for very steep waves and of course the potential theory breaks down once the wave begins to break and completely different methods are required in this situation.		We saw in [[Conservation Laws and Boundary Conditions]] that the potential flow model for wave propagation is given Laplace's equation plus the free-surface conditions. In this section we present the linear and second order theory for these equations. The linear theory is valid for small wave heights and the second order theory is an improvement on this. However, neither of these theories work for very steep waves and of course the potential theory breaks down once the wave begins to break and completely different methods are required in this situation.

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	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]
	[[Category:Nonlinear Water-Wave Theory]]		[[Category:Nonlinear Water-Wave Theory]]
	~~[[Category:Complete Pages]]~~

Meylan at 00:02, 27 July 2009

2009-07-27T00:02:17Z

← Older revision		Revision as of 00:02, 27 July 2009
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	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]
	[[Category:~~Non-linear~~ Water-Wave Theory]]		[[Category:Nonlinear Water-Wave Theory]]
	[[Category:Complete Pages]]		[[Category:Complete Pages]]

Adi Kurniawan at 03:54, 2 March 2009

2009-03-02T03:54:36Z

← Older revision		Revision as of 03:54, 2 March 2009
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	\| previous chapter = [[Conservation Laws and Boundary Conditions]]		\| previous chapter = [[Conservation Laws and Boundary Conditions]]
	}}		}}

	We saw in [[Conservation Laws and Boundary Conditions]] that the potential flow model for wave propagation is given Laplace's equation plus the free-surface conditions. In this section we present the linear and second order theory for these equations. The linear theory is valid for small wave heights and the second order theory is an improvement on this. However, neither of these theories work for very steep waves and of course the potential theory breaks down once the wave begins to break and completely different methods are required in this situation.		We saw in [[Conservation Laws and Boundary Conditions]] that the potential flow model for wave propagation is given Laplace's equation plus the free-surface conditions. In this section we present the linear and second order theory for these equations. The linear theory is valid for small wave heights and the second order theory is an improvement on this. However, neither of these theories work for very steep waves and of course the potential theory breaks down once the wave begins to break and completely different methods are required in this situation.

Meylan: /* Second-order problem */

2009-02-28T20:27:23Z

Second-order problem

← Older revision		Revision as of 20:27, 28 February 2009
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	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]
	[[Category:Non-linear Water-Wave Theory]]		[[Category:Non-linear Water-Wave Theory]]
			[[Category:Complete Pages]]