Template:Separation of variables for a submerged dock

The depth above the plate is [math]\displaystyle{ d }[/math] and below the plate is [math]\displaystyle{ h-d }[/math]. We now introduce a new dispersion value [math]\displaystyle{ \mu_m }[/math] which is either [math]\displaystyle{ k_n^{d} }[/math] where [math]\displaystyle{ k_0^{d} }[/math] are the roots of the Dispersion Relation for a Free Surface with depth [math]\displaystyle{ d }[/math] for [math]\displaystyle{ 0 \leq n \leq N-M }[/math] or [math]\displaystyle{ n\pi/(h-d) }[/math] for [math]\displaystyle{ n }[/math] and integer. We also order the roots with the first the positive imaginary solution [math]\displaystyle{ k_0^{d} }[/math] and the second being zero, then order by increasing size. We then define a new function a new function

[math]\displaystyle{ \chi_n = \begin{cases} \psi_{n}(z),\,\,\,-h\lt z\lt -d \\ 0,\,\,\,-d\lt z\lt 0 \end{cases} }[/math]

or

[math]\displaystyle{ \chi_{n} = \begin{cases} 0,\,\,\,-h\lt z\lt -d \\ \phi_{n}^{d}(z),\,\,\,-d\lt z\lt 0 \end{cases} }[/math]

where

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

depending on whether the root [math]\displaystyle{ \mu_n }[/math] is above or below.

We define

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

where [math]\displaystyle{ B_{mn} }[/math] is either

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

or

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