A plate is different from a membrane. Membrane (like a string) does not resist bending, and the only force taken into the account is the stretching force. The equation is \[u_{tt}=c^2\Delta u,\] or in one dimension (a string) \[u_{tt}=c^2u_{xx}.\] A plate (in one dimension this is called a rod) resists bending, it does not have to be stretched. The equation of a plate is of \(4\)-th order in the space variables: \[u_{tt}=k\Delta\Delta u,\] or in one dimension (the case of a rod): \[u_{tt}=ku_{xxxx}.\] You can separate the variables in the one dimensional equation (think what the appropriate boundary conditions are! There must be four of them in one dmensional case) and solve the corresponding eigenvalue-eigenfunction problem. Similarly, you can separate the variables for a rectangular or round plate. For the round plate the equation can be solved in Bessel functions.