PDE's Solving the wave equation with gravity

Hiya,

I'm trying to solve this wave equation:

d^2u/dt^2 = c^2*(d^2u/dx^2) + G 0<= x <= L

With boundary condition:

u(0, t) = 0 ; t > 0; (2)

u(L, t) = H ; t > 0

And initial condition:

u(x, 0) = 0 ; 0 < x < L

du/dt (x, 0) = 0 ; 0 < x < L

So I have worked out the steady state of a solution w(x) with the information [ d^2u/dt^2 = 0 ] and now I need to consider u(x,t) = Y(x,t) + w(x) and find a homogeneous equation for Y which can be derived from u and solve it using separation of variable. I can do this but the gravity constant means that when I try and separate out the variable I can't because I have both variable on both sides.

I'm sorry if this doesn't make a lot of sense, but if there is anyone here who could help me it would be really useful!

Thank you!!