Elastic Plate on Shallow Water

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We derive the equations for an elastic plate on shallow water. These equations can be found in Stoker 1957. The plate is infinite in the [math]\displaystyle{ y }[/math] direction, so that only the [math]\displaystyle{ x }[/math] and [math]\displaystyle{ z }[/math] directions are considered. The [math]\displaystyle{ x }[/math] direction is horizontal, the positive [math]\displaystyle{ z }[/math] axis points vertically up, and the plate covers the region [math]\displaystyle{ -b\leq x\leq b. }[/math] The water is of uniform depth [math]\displaystyle{ h }[/math] which is small enough that the water may be approximated as shallow. The amplitudes are assumed small enough that the linear theory is appropriate, and the plate is sufficiently thin that the Shallow Draft approximation may be made

The kinematic condition is

[math]\displaystyle{ \partial _{t}\zeta =-h\partial _{x}^{2}\phi , \,\,\,(1) }[/math]

where [math]\displaystyle{ \phi }[/math] is the velocity potential of the water (averaged over the depth) and [math]\displaystyle{ \zeta }[/math] is the displacement of the water surface or the plate (from the shallow draft approximation). The equation derived by equating the pressure at the free surface is

[math]\displaystyle{ -\rho g\zeta -\rho \partial _{t}\phi =\left\{ \begin{matrix} 0,\;\;x\notin (-b,b), \\ D\partial _{x}^{4}\zeta +\rho ^{\prime }d\partial _{t}^{2}\zeta ,\;\;x\in (-b,b), \end{matrix} \right. \,\,\, (2) }[/math]

where [math]\displaystyle{ D }[/math] is the bending rigidity of the plate per unit length, [math]\displaystyle{ \rho }[/math] is the density of water, [math]\displaystyle{ \rho ^{\prime } }[/math] is the average density of the plate, [math]\displaystyle{ g }[/math] is the acceleration due to gravity, and [math]\displaystyle{ d }[/math] is the thickness of the plate [math]\displaystyle{ . }[/math] At the ends of the plate the free edge boundary conditions

[math]\displaystyle{ \lim_{x\downarrow -b}\partial _{x}^{2}\zeta =\lim_{x\uparrow b}\partial _{x}^{2}\zeta =\lim_{x\downarrow -b}\partial _{x}^{3}\zeta =\lim_{x\uparrow b}\partial _{x}^{3}\zeta =0 \,\,\,(3) }[/math]

are applied, however the theory which will be developed applies equally to any of the energy-conserving edge conditions such as clamped or pinned and there is no need for the boundary conditions to be symmetric. Equation (3) gives the following implied boundary conditions for [math]\displaystyle{ \phi }[/math]

[math]\displaystyle{ \lim_{x\downarrow -b}\partial _{x}^{4}\phi =\lim_{x\uparrow b}\partial _{x}^{4}\phi =\lim_{x\downarrow -b}\partial _{x}^{5}\phi =\lim_{x\uparrow b}\partial _{x}^{5}\phi =0 \,\,\,(4) }[/math]