Category:Shallow Depth - Revision history

Meylan at 00:12, 23 July 2009

2009-07-23T00:12:50Z

← Older revision		Revision as of 00:12, 23 July 2009
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	It removes the depth dependence in the equations of motion. The theory is sometimes referred to as ''Long Wave Theory''.		It removes the depth dependence in the equations of motion. The theory is sometimes referred to as ''Long Wave Theory''.

	== Derivation of the Equations ==		== Derivation of the Linear Equations ==
	The ~~theory~~ can be derived by a Taylor series expansion about the bottom surface. The equations for		The linear shallow water equations can be derived by a Taylor series expansion about the bottom surface. The equations for
	finite depth in the time domain are		finite depth in the time domain are
	<center><math>		<center><math>

Adi Kurniawan: /* Introduction */

2009-03-31T01:58:09Z

Introduction

← Older revision		Revision as of 01:58, 31 March 2009
Line 2:		Line 2:

	Shallow Depth occurs when the wavelength is much longer than the water depth (which is assumed constant).		Shallow Depth occurs when the wavelength is much longer than the water depth (which is assumed constant).
	It removes the depth dependence in the equations of motion. The theory is sometimes referred to as ''Long Wave Theory''		It removes the depth dependence in the equations of motion. The theory is sometimes referred to as ''Long Wave Theory''.

	== Derivation of the Equations ==		== Derivation of the Equations ==

Meylan at 05:58, 16 December 2008

2008-12-16T05:58:25Z

← Older revision		Revision as of 05:58, 16 December 2008
Line 50:		Line 50:
	For the case when the depth varies the equations become		For the case when the depth varies the equations become
	<center><math>		<center><math>
	\partial_t^2 \zeta - g \partial_x \left(h(x) \partial_x \zeta \right)		\partial_t^2 \zeta - g \partial_x \left(h(x) \partial_x \zeta \right).
	</math></center>		</math></center>
			It is standard to non-dimensionalise so that gravity <math>g=1</math>, and the equations
			then become
			{{shallow depth one dimension}}
			The equations in two dimensions are
			{{shallow depth two dimensions}}


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

Meylan: /* Equations in Depth and Surface Displacement only */

2008-12-16T05:48:38Z

Equations in Depth and Surface Displacement only

← Older revision		Revision as of 05:48, 16 December 2008
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	but we replace the second derivative by the Laplacian.		but we replace the second derivative by the Laplacian.

	== Equations in Depth ~~and Surface Displacement only~~ ==		== Equations for Variable Depth ==

	~~We can write~~ the ~~equations in terms of~~ the depth ~~and displacement only as~~		For the case when the depth varies the equations become
	<center><math>		<center><math>
	\~~frac{\partial~~^2\zeta}{\~~partial t^2} = -~~\zeta		\partial_t^2 \zeta - g \partial_x \left(h(x) \partial_x \zeta \right)
	</math></center>		</math></center>

Meylan: /* Derivation of the Equations */

2008-12-16T05:44:52Z

Derivation of the Equations

@@ Line 45: / Line 45: @@
 where the potential now depends only on <math>x</math>. In three-dimensions the formula is the same
 but we replace the second derivative by the Laplacian.
 [[Category:Linear Water-Wave Theory]]

Meylan at 05:25, 16 December 2008

2008-12-16T05:25:15Z

New page

== Introduction ==

Shallow Depth occurs when the wavelength is much longer than the water depth (which is assumed constant).
It removes the depth dependence in the equations of motion. The theory is sometimes referred to as ''Long Wave Theory''

== Derivation of the Equations ==
The theory can be derived by a Taylor series expansion about the bottom surface. The equations for
finite depth in the time domain are
<center><math>
\nabla^{2}\phi=0, \, -h<z<0,
</math></center>
<center><math>
\frac{\partial\phi}{\partial z}=0, \, z=-h,
</math></center>
<center><math>
\frac{\partial\phi}{\partial z} = \frac{\partial\zeta}{\partial t},\,z=0,
</math></center>
<center><math>
\frac{\partial\phi}{\partial t} = -\zeta,\,z=0,
</math></center>
where <math>\phi</math> is the velocity potential and <math>\zeta</math> is the surface displacement.

Since the depth is shallow we can perform a Taylor series expansion about the potential at the bottom
surface and obtain (retaining only three terms)
<center><math>
\phi(x,z) = \phi(z,-h) + (z+h)\phi_z(z,-h) + \frac{(z+h)^2}{2}\phi_{zz}(z,-h)
</math></center>
We now see that (under this approximation)
<center><math>
\left.\frac{\partial\phi}{\partial z}\right|_{z=0} = h\phi_{zz}(z,-h) =-h\partial _{x}^{2}\phi
</math></center>
Therefore, the change which occurs is that
<center><math>
\partial _{n}\phi =-h\partial _{x}^{2}\phi
</math></center>
This means that we are not required to solve Laplace's equation in the fluid, and leads to great simplifications.

The equations are therefore
<center><math>
-h\partial _{x}^{2}\phi = \frac{\partial\zeta}{\partial t}
</math></center>
<center><math>
\frac{\partial\phi}{\partial t} = -\zeta
</math></center>
where the potential now depends only on <math>x</math>. In three-dimensions the formula is the same
but we replace the second derivative by the Laplacian.
[[Category:Linear Water-Wave Theory]]