Wave Energy Density and Flux - Revision history

MuTau: the context indicates that energy density should be lower case epsilon

2011-02-23T17:44:05Z

the context indicates that energy density should be lower case epsilon

← Older revision		Revision as of 17:44, 23 February 2011
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	[[Image:volume_normal.png\|thumb\|right\|600px\|Moving Volume]]		[[Image:volume_normal.png\|thumb\|right\|600px\|Moving Volume]]

	''Energy flux'' <math>\mathcal{P}(t)</math> is the rate of change of energy density <math> \~~mathcal{e}~~(t) </math>. It is the flux of energy which is critical to ocean waves. While the individual fluid particles do not move the waves carry energy. We begin by deriving the energy flux in general conditions.		''Energy flux'' <math>\mathcal{P}(t)</math> is the rate of change of energy density <math> \epsilon(t) </math>. It is the flux of energy which is critical to ocean waves. While the individual fluid particles do not move the waves carry energy. We begin by deriving the energy flux in general conditions.

	<center><math> \mathcal{P}(t) \equiv \frac{\mathrm{d}\mathcal{E}(t)}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \iiint_{\Omega(t)} \epsilon(t) \mathrm{d}V		<center><math> \mathcal{P}(t) \equiv \frac{\mathrm{d}\mathcal{E}(t)}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \iiint_{\Omega(t)} \epsilon(t) \mathrm{d}V

MuTau: the context indicates that energy density should be lower case epsilon

2011-02-23T17:42:20Z

the context indicates that energy density should be lower case epsilon

← Older revision		Revision as of 17:42, 23 February 2011
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	[[Image:volume_normal.png\|thumb\|right\|600px\|Moving Volume]]		[[Image:volume_normal.png\|thumb\|right\|600px\|Moving Volume]]

	''Energy flux'' <math>\mathcal{P}(t)</math> is the rate of change of energy density <math> \mathcal{E}(t) </math>. It is the flux of energy which is critical to ocean waves. While the individual fluid particles do not move the waves carry energy. We begin by deriving the energy flux in general conditions.		''Energy flux'' <math>\mathcal{P}(t)</math> is the rate of change of energy density <math> \mathcal{e}(t) </math>. It is the flux of energy which is critical to ocean waves. While the individual fluid particles do not move the waves carry energy. We begin by deriving the energy flux in general conditions.

	<center><math> \mathcal{P}(t) \equiv \frac{\mathrm{d}\mathcal{E}(t)}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \iiint_{\Omega(t)} \epsilon(t) \mathrm{d}V		<center><math> \mathcal{P}(t) \equiv \frac{\mathrm{d}\mathcal{E}(t)}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \iiint_{\Omega(t)} \epsilon(t) \mathrm{d}V

Adi Kurniawan: /* Rayleigh's proof of the group velocity formula */

2010-11-06T11:29:18Z

Rayleigh's proof of the group velocity formula

← Older revision		Revision as of 11:29, 6 November 2010
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	<center><math> \zeta(x,t) = A \cos ( \omega_1 t - k_1 x) + A \cos ( \omega_2 t - k_2 x) \, </math></center>		<center><math> \zeta(x,t) = A \cos ( \omega_1 t - k_1 x) + A \cos ( \omega_2 t - k_2 x) \, </math></center>

	where the amplitude is assumed to be common and:		where the amplitude is assumed to be common and

	<center><math> \omega_2 = \omega_1 + \Delta \omega, \quad \| \Delta\omega \| \ll \omega_1 , \omega_2 ~~</math></center>~~		<center><math>
			\begin{align}
	~~<center><math>~~ k_2 = k_1 + \Delta k, \quad \| \Delta k \| \ll k_1 , k_2 </math></center>		\omega_2 &= \omega_1 + \Delta \omega, \quad \| \Delta\omega \| \ll \omega_1 , \omega_2 \\
			k_2 &= k_1 + \Delta k, \quad \| \Delta k \| \ll k_1 , k_2
			\end{align}
			</math></center>

	Converting into complex notation:		Converting into complex notation:

	<center><math> \zeta(x,t) = A \mathrm{Re} \{ e^{i\omega_1 t - i k_1 x} + e^{i\omega_2 t - i k_2 x} \}		<center><math>
	= A \mathrm{Re} \{ e^{i\omega_1 t - i k_1 x} + e^{i\omega_1 t - i k_1 x + i \Delta\omega t - i \Delta k x} \} ~~</math></center> <br>~~		\begin{align}
	~~<center><math>~~ = A \mathrm{Re} \{ e^{i\omega_1 t - i k_1 x} \left( 1 + e^{i\Delta\omega t - i \Delta k x} \right) \} </math></center>		\zeta(x,t) &= A \mathrm{Re} \{ e^{i\omega_1 t - i k_1 x} + e^{i\omega_2 t - i k_2 x} \} \\
			&= A \mathrm{Re} \{ e^{i\omega_1 t - i k_1 x} + e^{i\omega_1 t - i k_1 x + i \Delta\omega t - i \Delta k x} \} \\
			&= A \mathrm{Re} \{ e^{i\omega_1 t - i k_1 x} \left( 1 + e^{i\Delta\omega t - i \Delta k x} \right) \}
			\end{align}
			</math></center>

	The combined wave elevation <math> \zeta \,</math> vanishes identically where <math> \left( 1 + e^{i\Delta\omega t - i \Delta k x} \right) = 0 \, </math>,		The combined wave elevation <math> \zeta \,</math> vanishes identically where <math> \left( 1 + e^{i\Delta\omega t - i \Delta k x} \right) = 0 \, </math>,

Adi Kurniawan: /* Energy flux across a vertical fluid boundary fixed in space */

2010-11-06T11:27:14Z

Energy flux across a vertical fluid boundary fixed in space

← Older revision		Revision as of 11:27, 6 November 2010
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	=== Energy flux across a vertical fluid boundary fixed in space ===		=== Energy flux across a vertical fluid boundary fixed in space ===

	<center><math> \mathcal{P}(t) = - \rho \int_{-\infty}^{\zeta(t)} \frac{\partial\Phi}{\partial t} \Phi_n \mathrm{d}z =		<center><math>
	- \rho \left( \int_{-\infty}^0 + \int_0^\zeta \right) \frac{\partial\Phi}{\partial t} \Phi_n \mathrm{d}z		\begin{align}
	= - \rho \int_{-\infty}^0 \frac{\partial\Phi}{\partial t} \frac{\partial\Phi}{\partial n} \mathrm{d}z + O(A^3) </math></center>		\mathcal{P}(t) &= - \rho \int_{-\infty}^{\zeta(t)} \frac{\partial\Phi}{\partial t} \Phi_n \mathrm{d}z \\
			&= - \rho \left( \int_{-\infty}^0 + \int_0^\zeta \right) \frac{\partial\Phi}{\partial t} \Phi_n \mathrm{d}z \\
			&= - \rho \int_{-\infty}^0 \frac{\partial\Phi}{\partial t} \frac{\partial\Phi}{\partial n} \mathrm{d}z + O(A^3)
			\end{align}
			</math></center>

	Mean energy flux for a [[Linear Plane Progressive Regular Waves]] follows upon substitution of the regular wave velocity potential and taking mean values:		Mean energy flux for a [[Linear Plane Progressive Regular Waves]] follows upon substitution of the regular wave velocity potential and taking mean values:
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	Where terms of <math> O(A^3) </math> have been neglected. Note that within linear theory, energy density and energy flux are quantities of <math>O(A^2)</math>. If higher-order terms are kept then we need to consider the treatment of second-order surface wave theory, at least.		Where terms of <math> O(A^3) </math> have been neglected. Note that within linear theory, energy density and energy flux are quantities of <math>O(A^2)</math>. If higher-order terms are kept then we need to consider the treatment of second-order surface wave theory, at least.
	Solving the above equation for <math> U </math> we obtain:		Solving the above equation for <math> U </math> we obtain:
	<center><math> U = \~~frac~~{\rho \ {\overline{\int_{-\infty}^0 \~~frac~~{\partial\Phi}{\partial t} \~~frac~~{\partial\Phi}{\partial x} \mathrm{d}z}}}		<center><math> U = \dfrac{\rho \ {\overline{\int_{-\infty}^0 \dfrac{\partial\Phi}{\partial t} \dfrac{\partial\Phi}{\partial x} \mathrm{d}z}}}
	{{\overline{\int_{-\infty}^0 \left( \~~frac~~{P}{\rho} + \~~frac~~{\partial\Phi}{\partial t} \right) \mathrm{d}z}}} </math></center>		{{\overline{\int_{-\infty}^0 \left( \dfrac{P}{\rho} + \dfrac{\partial\Phi}{\partial t} \right) \mathrm{d}z}}} </math></center>

	Upon substitution of the plane progressive wave velocity potential and definition of pressure from Bernoulli's equation we obtain:		Upon substitution of the plane progressive wave velocity potential and definition of pressure from Bernoulli's equation we obtain:

Adi Kurniawan: /* Energy Flux */

2010-11-06T11:22:55Z

Energy Flux

← Older revision		Revision as of 11:22, 6 November 2010
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	(the last result following from the transport theorem) where <math> U_n</math> is the normal velocity of surface <math> \partial\Omega(t) </math> outwards of the enclosed volume <math> \Omega(t)</math>.		(the last result following from the transport theorem) where <math> U_n</math> is the normal velocity of surface <math> \partial\Omega(t) </math> outwards of the enclosed volume <math> \Omega(t)</math>.
	We know that		We know that
	<center><math> \frac{\partial \epsilon}{\partial t} = \frac{\partial}{\partial t} \{ \frac{1}{2} \rho \|\mathbf{v}\|^2 + \rho g z \}		<center><math>
	= \frac{1}{2} \rho \frac{\partial}{\partial t} ( \nabla\Phi \cdot \nabla\Phi) ~~</math></center>~~		\begin{align}
	~~<center><math>~~ = \rho \nabla \cdot \left( \frac{\partial\Phi}{\partial t} \nabla\Phi \right) - \rho \frac{\partial\Phi}{\partial t} \nabla^2 \Phi </math></center>		\frac{\partial \epsilon}{\partial t} &= \frac{\partial}{\partial t} \{ \frac{1}{2} \rho \|\mathbf{v}\|^2 + \rho g z \} \\
			&= \frac{1}{2} \rho \frac{\partial}{\partial t} ( \nabla\Phi \cdot \nabla\Phi) \\
			&= \rho \nabla \cdot \left( \frac{\partial\Phi}{\partial t} \nabla\Phi \right) - \rho \frac{\partial\Phi}{\partial t} \nabla^2 \Phi
			\end{align}
			</math></center>
	so that		so that
	<center><math> \mathcal{P}(t) = \rho \iiint_{\Omega(t)} \nabla \cdot \left( \frac{\partial \Phi}{\partial t} \nabla \Phi \right) \mathrm{d}V		<center><math> \mathcal{P}(t) = \rho \iiint_{\Omega(t)} \nabla \cdot \left( \frac{\partial \Phi}{\partial t} \nabla \Phi \right) \mathrm{d}V

Adi Kurniawan: /* Energy in Linear Plane Progressive Regular Waves */

2010-11-06T11:19:26Z

Energy in Linear Plane Progressive Regular Waves

← Older revision		Revision as of 11:19, 6 November 2010
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	<center><math> \mathrm{Re} \{ B e^{i\omega t} \} = B(t) \,\!</math></center>		<center><math> \mathrm{Re} \{ B e^{i\omega t} \} = B(t) \,\!</math></center>
	then it follows that		then it follows that
	<center><math> \overline{A(t)B(t)} = \frac{1}{2} \mathrm{Re} \{ A B^* \} </math></center>.		<center><math> \overline{A(t)B(t)} = \frac{1}{2} \mathrm{Re} \{ A B^* \}. </math></center>

	This allows us to write the following expression		This allows us to write the following expression

Adi Kurniawan: /* Energy in Linear Plane Progressive Regular Waves */

2010-11-06T11:19:00Z

Energy in Linear Plane Progressive Regular Waves

← Older revision		Revision as of 11:19, 6 November 2010
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	This allows us to write the following expression		This allows us to write the following expression
	<center><math>		<center><math>
	\overline{\mathcal{E}_{kin}} = \frac{1}{2} \rho \overline{ ( \int_{-\infty}^0 + \int_0^\zeta )		\begin{align}
	\left( (\partial_x\Phi)^2 + (\partial_z\Phi)^2 \right) } \mathrm{d}z		\overline{\mathcal{E}_{kin}} &= \frac{1}{2} \rho \overline{ ( \int_{-\infty}^0 + \int_0^\zeta )
			\left( (\partial_x\Phi)^2 + (\partial_z\Phi)^2 \right) } \mathrm{d}z \\
			&= \frac{1}{2} \rho \int_{-\infty}^0 \overline{ \left( (\partial_x\Phi)^2 + (\partial_z\Phi)^2 \right) } \mathrm{d}z + O (A^3) \\
			&= \rho \frac{\omega^2 A^2}{4k} = \frac{1}{4} \rho g A^2 , \qquad \mbox{for} \ k=\omega^2/g \\
			\overline{\mathcal{E}_{pot}} &= \frac{1}{2} \rho g {\overline{\zeta(t)}}^2 = \frac{1}{4} \rho g A^2 .
			\end{align}
	</math></center>		</math></center>

	~~<center><math> = \frac{1}{2} \rho \int_{-\infty}^0 \overline{ \left( (\partial_x\Phi)^2 + (\partial_z\Phi)^2 \right) } \mathrm{d}z + O (A^3) </math></center>~~

	~~<center><math> = \rho \frac{\omega^2 A^2}{4k} = \frac{1}{4} \rho g A^2 , \qquad \mbox{for} \ k=\omega^2/g </math></center>~~
	~~<center>~~
	~~<math> \overline{\mathcal{E}_{pot}} = \frac{1}{2} \rho g {\overline{\zeta(t)}}^2 = \frac{1}{4} \rho g A^2 </math>~~
	~~</center>~~
	Note that it is a standard feature of linear oscillations that the average potential and kinetic energies are equal. Hence		Note that it is a standard feature of linear oscillations that the average potential and kinetic energies are equal. Hence
	<center><math> \overline{\mathcal{E}} = \overline{\mathcal{E}_{kin}} + \overline{\mathcal{E}_{pot}} = \frac{1}{2} \rho g A^2 </math></center>		<center><math> \overline{\mathcal{E}} = \overline{\mathcal{E}_{kin}} + \overline{\mathcal{E}_{pot}} = \frac{1}{2} \rho g A^2 </math></center>

Adi Kurniawan: /* Summary */

2010-08-06T15:36:00Z

Summary

← Older revision		Revision as of 15:36, 6 August 2010
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	The formulae for the energy flux derived above are very general and for potential flow nonlinear surface waves that are not breaking constitute the energy conservation principle.		The formulae for the energy flux derived above are very general and for potential flow nonlinear surface waves that are not breaking constitute the energy conservation principle.
	Energy flux (power) input into the fluid domain by any mechanism, wavemaker wind (in a conservative manner), a ship or any floating body must be "~~retreived~~" at some distance away. Deriving expressions of the energy flux ~~retreived~~ at "infinity" is a powerful method for estimating the wave resistance of ships (more on this later), the wave damping of floating bodies etc.		Energy flux (power) input into the fluid domain by any mechanism, wavemaker, wind (in a conservative manner), a ship or any floating body must be "retrieved" at some distance away. Deriving expressions of the energy flux retrieved at "infinity" is a powerful method for estimating the wave resistance of ships (more on this later), the wave damping of floating bodies, etc.
	Yet, the only general way of evaluating wave forces on floating bodies (moving or not) or solid boundaries is by applying the [[Wave Momentum Flux\| Wave Momentum]].		Yet, the only general way of evaluating wave forces on floating bodies (moving or not) or solid boundaries is by applying the [[Wave Momentum Flux\| Wave Momentum]].

	-----		-----
	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/C2653597~~-~~0443~~-~~4705~~-~~A4E5~~-~~07A93AFCA3F5~~/0/lecture4.pdf here]		[http://ocw.mit.edu/courses/mechanical-engineering/2-24-ocean-wave-interaction-with-ships-and-offshore-energy-systems-13-022-spring-2002/lecture-notes/lecture4.pdf here]

Adi Kurniawan: /* Energy in Linear Plane Progressive Regular Waves */

2009-10-19T14:41:15Z

Energy in Linear Plane Progressive Regular Waves

← Older revision		Revision as of 14:41, 19 October 2009
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	</math></center>		</math></center>

	<center><math> = \frac{1}{2} \rho \int_{-\infty}^0 \left( ~~\overline{~~(\partial_x\Phi)^2 + (\partial_z\Phi)^2 } \right) \mathrm{d}z + O (A^3) </math></center>		<center><math> = \frac{1}{2} \rho \int_{-\infty}^0 \overline{ \left( (\partial_x\Phi)^2 + (\partial_z\Phi)^2 \right) } \mathrm{d}z + O (A^3) </math></center>

	<center><math> = \rho \frac{\omega^2 A^2}{4k} = \frac{1}{4} \rho g A^2 , \qquad \mbox{for} \ k=\omega^2/g </math></center>		<center><math> = \rho \frac{\omega^2 A^2}{4k} = \frac{1}{4} \rho g A^2 , \qquad \mbox{for} \ k=\omega^2/g </math></center>

Adi Kurniawan: /* Energy in Linear Plane Progressive Regular Waves */

2009-10-19T14:40:49Z

Energy in Linear Plane Progressive Regular Waves

← Older revision		Revision as of 14:40, 19 October 2009
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	</math></center>		</math></center>

	<center><math> = \frac{1}{2} \rho \int_{-\infty}^0 \left( \overline{(\partial_x\Phi)^2 + (\partial_z\Phi)^2 \right)} \mathrm{d}z + O (A^3) </math></center>		<center><math> = \frac{1}{2} \rho \int_{-\infty}^0 \left( \overline{(\partial_x\Phi)^2 + (\partial_z\Phi)^2 } \right) \mathrm{d}z + O (A^3) </math></center>

	<center><math> = \rho \frac{\omega^2 A^2}{4k} = \frac{1}{4} \rho g A^2 , \qquad \mbox{for} \ k=\omega^2/g </math></center>		<center><math> = \rho \frac{\omega^2 A^2}{4k} = \frac{1}{4} \rho g A^2 , \qquad \mbox{for} \ k=\omega^2/g </math></center>