Steady state solution for wave equation with gravity For the following wave equation and initial condition where G is a constant due to gravity, how would I go about finding the steady state solution:
$\frac{∂^2u}{∂t^2}= c \frac{∂^2u}{∂x^2}+ G, \quad 0\le x \le L,\: t\gt 0 $
$u(0, t) = 0 ,\quad t > 0,$ 
$u(L, t) = H ,\quad t > 0, $
$u(x, 0) = 0 ,\quad 0 < x < L$
$\frac{∂u}{∂t}(x, 0) = 0 ,\quad 0 < x < L$
I'm assuming that as t tends to infinity, $\frac{∂^2u}{∂t^2} = 0$ but I've never dealt with wave equations with gravity before so I'm not sure what to do with the G term. 

Steady-state generally means not depending on the time $t$, so you'd be looking at
the ODE $c \dfrac{d^2 u}{dx^2} + G = 0$ with boundary conditions $u(0)=0$, $u(L)=H$.  It's easy to solve...
However, you should be aware that in this case there is no dissipation, so 
the "transient" solutions don't die away as $t \to \infty$.
