Template:Separation of variables for a dock

Separation of variables

We now separate variables and write the potential as

[math]\displaystyle{ \phi(x,z)=\zeta(z)\rho(x)\, }[/math]

Applying Laplace's equation we obtain

[math]\displaystyle{ \zeta_{zz}+\mu^{2}\zeta=0.\, }[/math]

We then use the boundary condition at [math]\displaystyle{ z=-h }[/math], which is the same for all [math]\displaystyle{ x }[/math] to write

[math]\displaystyle{ \zeta=\cos\mu(z+h)\, }[/math]

where the separation constant [math]\displaystyle{ \mu^{2} }[/math] must satisfy different equations depending on whether we are under the free-surface or the dock covered region. For the free surface

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

which is the Dispersion Relation for a Free Surface and for the dock covered region

[math]\displaystyle{ \kappa\tan(\kappa h)=0, }[/math]

Note that we have set [math]\displaystyle{ \mu=k }[/math] under the free surface and [math]\displaystyle{ \mu=\kappa }[/math] under the plate. We denote the positive imaginary solution of free-surface equation by [math]\displaystyle{ k_{0} }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} }[/math], [math]\displaystyle{ m\geq1 }[/math]. The solutions of dock equation are [math]\displaystyle{ \kappa_{m}=m\pi/h }[/math], [math]\displaystyle{ m\geq 0 }[/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 and

[math]\displaystyle{ \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad m\geq 0 }[/math]

as the vertical eigenfunction of the potential in the dock covered region. For later reference, we note that:

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

where

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

and

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

where

[math]\displaystyle{ B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin \kappa_{m}h}{\left( \cos k_{n}h\right) \left( k_{n} ^{2}-\kappa_{m}^{2}\right) } }[/math]

and

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

where

[math]\displaystyle{ C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h }[/math]