Template:Equations for a beam

For a Bernoulli-Euler Beam on the surface of the water, the equation of motion is given by the following

[math]\displaystyle{ \partial_x^2\left(D\partial_x^2 \zeta\right) + \rho_i h \partial_t^2 \zeta = p }[/math]

where [math]\displaystyle{ D }[/math] is the flexural rigidity, [math]\displaystyle{ \rho_i }[/math] is the density of the plate, [math]\displaystyle{ h }[/math] is the thickness of the plate, [math]\displaystyle{ p }[/math] is the pressure and [math]\displaystyle{ \zeta }[/math] is the plate vertical displacement. Note that this equations simplifies if the plate has constant properties.

The edges of the plate can satisfy a range of boundary conditions. The natural boundary condition (i.e. free-edge boundary conditions).

[math]\displaystyle{ \partial_x^2 \zeta = 0, \,\,\partial_x^3 \zeta = 0 }[/math]

at the edges of the plate.

If we assume that the pressure is of the form [math]\displaystyle{ p(x,t) = e^{i\omega t} \bar{p}(x) }[/math] then it follows that [math]\displaystyle{ \zeta(x,t) = e^{i\omega t} \bar{\zeta}(x) }[/math] from linearity. In this case the equations reduce to

[math]\displaystyle{ \partial_x^2\left(D\partial_x^2 \bar{\zeta}\right) -\omega^2 \rho_i h \bar{\zeta} = \bar{p} }[/math]