Difference between revisions of "Category:Shallow Depth"

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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).
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It removes the depth dependence in the equations of motion. The theory is sometimes referred to as ''Long Wave Theory''
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It removes the depth dependence in the equations of motion. The theory is sometimes referred to as ''Long Wave Theory''.
  
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== Derivation of the Equations ==
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== Derivation of the Linear Equations ==
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The theory can be derived by a Taylor series expansion about the bottom surface. The equations for
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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>
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For the case when the depth varies the equations become
 
For the case when the depth varies the equations become
 
<center><math>
 
<center><math>
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  \partial_t^2 \zeta - g \partial_x \left(h(x) \partial_x \zeta \right)
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  \partial_t^2 \zeta - g \partial_x \left(h(x) \partial_x \zeta \right).
 
</math></center>
 
</math></center>
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It is standard to non-dimensionalise so that gravity <math>g=1</math>, and the equations
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then become
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{{shallow depth one dimension}}
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The equations in two dimensions are
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{{shallow depth two dimensions}}
  
  
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]

Latest revision as of 00:12, 23 July 2009

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 Linear Equations

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

[math]\displaystyle{ \nabla^{2}\phi=0, \, -h\lt z\lt 0, }[/math]
[math]\displaystyle{ \frac{\partial\phi}{\partial z}=0, \, z=-h, }[/math]
[math]\displaystyle{ \frac{\partial\phi}{\partial z} = \frac{\partial\zeta}{\partial t},\,z=0, }[/math]
[math]\displaystyle{ \frac{\partial\phi}{\partial t} = -\zeta,\,z=0, }[/math]

where [math]\displaystyle{ \phi }[/math] is the velocity potential and [math]\displaystyle{ \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)

[math]\displaystyle{ \phi(x,z) = \phi(z,-h) + (z+h)\phi_z(z,-h) + \frac{(z+h)^2}{2}\phi_{zz}(z,-h) }[/math]

We now see that (under this approximation)

[math]\displaystyle{ \left.\frac{\partial\phi}{\partial z}\right|_{z=0} = h\phi_{zz}(z,-h) =-h\partial _{x}^{2}\phi }[/math]

Therefore, the change which occurs is that

[math]\displaystyle{ \partial _{n}\phi =-h\partial _{x}^{2}\phi }[/math]

This means that we are not required to solve Laplace's equation in the fluid, and leads to great simplifications.

The equations are therefore

[math]\displaystyle{ -h\partial _{x}^{2}\phi = \frac{\partial\zeta}{\partial t} }[/math]
[math]\displaystyle{ \frac{\partial\phi}{\partial t} = -\zeta }[/math]

where the potential now depends only on [math]\displaystyle{ x }[/math]. In three-dimensions the formula is the same but we replace the second derivative by the Laplacian.

Equations for Variable Depth

For the case when the depth varies the equations become

[math]\displaystyle{ \partial_t^2 \zeta - g \partial_x \left(h(x) \partial_x \zeta \right). }[/math]

It is standard to non-dimensionalise so that gravity [math]\displaystyle{ g=1 }[/math], and the equations then become

[math]\displaystyle{ \rho(x)\partial_t^2 \zeta = \partial_x \left(h(x) \partial_x \zeta \right). }[/math]

subject to the initial conditions

[math]\displaystyle{ \zeta_{t=0} = \zeta_0(x)\,\,\,{\rm and}\,\,\, \partial_t\zeta_{t=0} = \partial_t\zeta_0(x) }[/math]

where [math]\displaystyle{ \zeta }[/math] is the displacement, [math]\displaystyle{ \rho }[/math] is the string density and [math]\displaystyle{ h(x) }[/math] is the variable depth (note that we are unifying the variable density string and the wave equation in variable depth because the mathematical treatment is identical). The equations in two dimensions are

[math]\displaystyle{ \partial_t^2 \zeta = \nabla \left(h(x,y) \nabla \zeta \right). }[/math]

Subcategories

This category has only the following subcategory.

S

Pages in category "Shallow Depth"

The following 2 pages are in this category, out of 2 total.

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V

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