How to obtain w and Natural frequency as solution of unknown time function?

I have a problem with decoupling this PDE. The solutions of time unknown function are natural frequencies. Is it possible t obtain it. I believe that the calculating w is smaller problem then obtaining the solutions of time function. If somebody from the physics can help for decoupling PDE and then calculating code for values.

Everything known constants. Unknown x, t.

I1[x_] := I0/(1 + C0/E^((2*\[Alpha]*(x - xc))/h));

Conditions:

w[0, t] = 0;

w[L, t] = 0;

D[w[0, t], {x, 2}] = 0;

D[w[L, t], {x, 2}] = 0;

D[w[x, 0], {t, 2}] = 0;

D[w[x, 0], {t, 2}] = 0;

PDE

D[E0*(I0 - I1[x])*w[x, t], {x, 4}] + \[Rho]*A*D[w[x, t], {t, 2}] == 0

Re: How to obtain w and Natural frequency as solution of unknown time function?

I don't understand your notation. Is "D[f(x,t), {x,4}]" the **fourth** derivative of f with respect to x?

Re: How to obtain w and Natural frequency as solution of unknown time function?

Yes, this is code in Mathematica.