Template:Separation of variables for a floating elastic plate
Separation of variables under the Plate
[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }[/math]
subject to the boundary conditions
[math]\displaystyle{ Z^{\prime}(-h) = 0 }[/math]
and
[math]\displaystyle{ \left(\beta \kappa^4 + 1 - \alpha\gamma\right)Z^{\prime}(0) = \alpha Z(0) }[/math]
(the first term comes from the beam eigenvalue problem, where [math]\displaystyle{ \partial_x^4 X_n = \kappa_n^4 X_n }[/math]). We then use the boundary condition at [math]\displaystyle{ z=-h \, }[/math] to write
[math]\displaystyle{ Z = \frac{\cos \kappa(z+h)}{\cos \kappa h} }[/math]
The boundary condition at the free surface ([math]\displaystyle{ z=0 }[/math]) is the Dispersion Relation for a Floating Elastic Plate
Solving for [math]\displaystyle{ \kappa }[/math] gives a pure imaginary root with positive imaginary part, two complex roots (two complex conjugate paired roots with positive imaginary part in most physical situations), an infinite number of positive real roots which approach [math]\displaystyle{ {n\pi}/{h} }[/math] as [math]\displaystyle{ n }[/math] approaches infinity, and also the negative of all these roots (Dispersion Relation for a Floating Elastic Plate) . We denote the two complex roots with positive imaginary part by [math]\displaystyle{ \kappa_{-2} \, }[/math] and [math]\displaystyle{ \kappa_{-1} \, }[/math], the purely imaginary root with positive imaginary part by [math]\displaystyle{ \kappa_{0} \, }[/math] and the real roots with positive imaginary part by [math]\displaystyle{ \kappa_{n} \, }[/math] for [math]\displaystyle{ n }[/math] a positive integer. The imaginary root with positive imaginary part corresponds to a reflected travelling mode propagating along the [math]\displaystyle{ x }[/math] axis. The complex roots with positive imaginary parts correspond to damped reflected travelling modes and the real roots correspond to reflected evanescent modes.
