Template:Frequency domain equations for a floating plate

If we make the assumption of Frequency Domain Problem that everything is proportional to [math]\displaystyle{ \exp (-\mathrm{i}\omega t)\, }[/math] the equations become

[math]\displaystyle{ \begin{align} -\mathrm{i}\omega\zeta &= \partial_z\phi , &z=0 \\ -\rho g\zeta - \mathrm{i}\omega\rho \phi &= D \partial_x^4 \zeta -\omega^2 \rho_i h \zeta, &z=0 \\ \Delta \phi &= 0, &-h\lt z\lt 0 \\ \partial_z \phi &= 0, &z=-h, \end{align} }[/math]

where [math]\displaystyle{ \zeta }[/math] is the surface displacement and [math]\displaystyle{ \phi }[/math] is the velocity potential in the frequency domain.

These equations can be simplified by defining [math]\displaystyle{ \alpha = \omega^2/g }[/math], [math]\displaystyle{ \beta = D/\rho g }[/math] and [math]\displaystyle{ \gamma = \rho_i h/\rho }[/math] to obtain

[math]\displaystyle{ \begin{align} \Delta \phi &= 0, &-h \lt z \leq 0 \\ \partial_z \phi &= 0, &z = - h \\ \beta \partial_x^4 \zeta + \left( 1 - \gamma\alpha \right) \zeta &= -\mathrm{i} \sqrt{\alpha}\phi, &z = 0 \\ -\mathrm{i}\omega\zeta &= \partial_z\phi , &z=0 . \end{align} }[/math]