Difference between revisions of "Category:Shallow Depth"

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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
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  \partial_t^2 \zeta - g \partial_x \left(h(x) \partial_x \zeta \right)
 
</math></center>
 
</math></center>
  

Revision as of 05:48, 16 December 2008

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

[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]

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.

N

V

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