Template:Linear elastic plate on water time domain - Revision history

Meylan at 22:53, 12 November 2025

2025-11-12T22:53:51Z

← Older revision		Revision as of 22:53, 12 November 2025
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	\frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0; </math></center>		\frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0; </math></center>
	but the dynamic condition needs to be modified to include the effect of the the plate		but the dynamic condition needs to be modified to include the effect of the the plate
	<center><math> \rho g\zeta + \rho \frac{\partial\Phi}{\partial t}		<center><math> -\rho g\zeta + \rho \frac{\partial\Phi}{\partial t}
	= D \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}		= D \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
	, \ z=0; </math></center>		, \ z=0; </math></center>

Mike smith at 01:18, 3 July 2009

2009-07-03T01:18:14Z

← Older revision		Revision as of 01:18, 3 July 2009
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	but the dynamic condition needs to be modified to include the effect of the the plate		but the dynamic condition needs to be modified to include the effect of the the plate
	<center><math> \rho g\zeta + \rho \frac{\partial\Phi}{\partial t}		<center><math> \rho g\zeta + \rho \frac{\partial\Phi}{\partial t}
	= ~~\beta~~ \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}		= D \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
	, \ z=0; </math></center>		, \ z=0; </math></center>
	We also have [[Laplace's Equation\|Laplace's equation]]		We also have [[Laplace's Equation\|Laplace's equation]]

Mike smith at 01:17, 3 July 2009

2009-07-03T01:17:17Z

← Older revision		Revision as of 01:17, 3 July 2009
Line 5:		Line 5:
	but the dynamic condition needs to be modified to include the effect of the the plate		but the dynamic condition needs to be modified to include the effect of the the plate
	<center><math> \rho g\zeta + \rho \frac{\partial\Phi}{\partial t}		<center><math> \rho g\zeta + \rho \frac{\partial\Phi}{\partial t}
	= D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}		= \beta \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
	, \ z=0; </math></center>		, \ z=0; </math></center>
	We also have [[Laplace's Equation\|Laplace's equation]]		We also have [[Laplace's Equation\|Laplace's equation]]

Adi Kurniawan at 10:16, 7 April 2009

2009-04-07T10:16:44Z

← Older revision		Revision as of 10:16, 7 April 2009
Line 7:		Line 7:
	= D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}		= D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
	, \ z=0; </math></center>		, \ z=0; </math></center>
	We also have Laplace's equation		We also have [[Laplace's Equation\|Laplace's equation]]
	<center><math>		<center><math>
	\Delta \Phi = 0,\,\,-h<z<0		\Delta \Phi = 0,\,\,-h<z<0

Meylan: Template:Linear elastic thin plate on water moved to Template:Linear elastic plate on water time domain

2009-03-23T02:22:00Z

Template:Linear elastic thin plate on water moved to Template:Linear elastic plate on water time domain

← Older revision	Revision as of 02:22, 23 March 2009
(No difference)

Meylan at 02:20, 23 March 2009

2009-03-23T02:20:29Z

New page

We begin with the [[Linear and Second-Order Wave Theory| linear equations]] for a fluid.
The kinematic condition is the same
<center><math>
\frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0; </math></center>
but the dynamic condition needs to be modified to include the effect of the the plate
<center><math> \rho g\zeta + \rho \frac{\partial\Phi}{\partial t}
= D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
, \ z=0; </math></center>
We also have Laplace's equation
<center><math>
\Delta \Phi = 0,\,\,-h<z<0
</math></center>
and the usual non-flow condition at the bottom surface
<center><math>
\partial_z \Phi = 0,\,\,z=-h,
</math></center>
where
<math>\zeta</math> is the surface displacement, <math>\Phi</math> is the velocity potential,
and <math>\rho</math> is the fluid density.