Template:Separation of variables for a free surface

Separation of variables for a free surface

The equation

[math]\displaystyle{ - \frac{1}{Z(z)} \frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} = \eta^2. }[/math]

subject to the boundary conditions

[math]\displaystyle{ \frac{dZ}{dz}(-h) = 0 }[/math]

and

[math]\displaystyle{ \frac{dZ}{dz}(0) = \alpha Z(0) }[/math]

We then use the boundary condition at [math]\displaystyle{ z=-h }[/math] to write

[math]\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }[/math]

The boundary condition at the free surface ([math]\displaystyle{ z=0 }[/math]) is

[math]\displaystyle{ k\tan\left( kh\right) =-\alpha, }[/math]

which is the Dispersion Relation for a Free Surface We denote the positive imaginary solution of this equation by [math]\displaystyle{ k_{0} }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} }[/math], [math]\displaystyle{ m\geq1 }[/math]. We define

[math]\displaystyle{ \phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }[/math]

as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that

[math]\displaystyle{ \int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{n}\delta_{mn} }[/math]

where

[math]\displaystyle{ A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{m}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right) }[/math]

.

The equation

[math]\displaystyle{ - \frac{1}{Z(z)} \frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} = \eta^2. }[/math]

is the equation for separation of variables for a free surface. In the setting of water of finite depth, the general solution [math]\displaystyle{ Z(z) }[/math] can be written as

[math]\displaystyle{ Z(z) = F \cos \big( \eta (z+d) \big) + G \sin \big( \eta (z+d) \big), \quad \eta \in \mathbb{C} \backslash \{ 0 \}, }[/math]

since [math]\displaystyle{ \eta = 0 }[/math] is not an eigenvalue. To satisfy the bed condition, [math]\displaystyle{ G }[/math] must be [math]\displaystyle{ 0 }[/math]. [math]\displaystyle{ Z(z) }[/math] satisfies the free surface condition, provided the separation constants [math]\displaystyle{ \eta }[/math] are roots of the equation

[math]\displaystyle{ - F \eta \sin \big( \eta (z+d) \big) - \alpha F \cos \big( \eta (z+d) \big) = 0, \quad z=0, }[/math]

or, equivalently, if they satisfy the Dispersion Relation for a Free Surface

[math]\displaystyle{ \alpha + \eta \tan \eta d = 0\,. }[/math]

This equation has an infinite number of real roots, denoted by [math]\displaystyle{ k_m }[/math] and [math]\displaystyle{ -k_m }[/math] ([math]\displaystyle{ m \geq 1 }[/math]), but the negative roots produce the same eigenfunctions as the positive ones and will therefore not be considered. It also has a pair of purely imaginary roots which will be denoted by [math]\displaystyle{ k_0 }[/math]. Writing [math]\displaystyle{ k_0 = - \mathrm{i} k }[/math], [math]\displaystyle{ k }[/math] is the (positive) root of the Dispersion Relation for a Free Surface

[math]\displaystyle{ \alpha = k \tanh k d,\, }[/math]

again it suffices to consider only the positive root of this equation. The solutions can therefore be written as

[math]\displaystyle{ Z_m(z) = F_m \cos \big( k_m (z+d) \big), \quad m \geq 0. }[/math]

It follows that [math]\displaystyle{ k }[/math] is the previously introduced wavenumber and the Dispersion Relation for a Free Surface gives the required relation between the radian frequency and the wavenumber.