Category:Shallow Depth

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]