Consider the problem of a string held at both ends and acted on by gravity. Take the coordinate x to be along the string with the ends of the string at x = 0 and x = L. Take t as time and the variable u(x, t) as the vertical position of the string. The equation governing the string motion is the wave equation
$$ \frac{∂^2u}{∂t^2} = \frac{c^2∂^2u}{∂x^2}+G $$
0 ≤ x ≤ L, t > 0
where G is a constant due to gravity and c is the constant wave speed. The ends of the string are held at different heights so that the boundary conditions are:
u(0,t)=0, t>0 u(L,t)=H, t>0
and the string is initially held completely flat and stationary:
u(x,0)=0, 0<x<L $$ \frac{∂u(x,0)}{∂t}0 , 0<x<L $$
(a) The steady-state solution u(x, t) = w(x) as t → ∞ is found by setting $ \frac{∂^2u}{∂t^2}= 0.$
Now consider u(x, t) = φ(x, t) + w(x). Obtain a homogenous wave-equation for φ with homogenous boundary conditions derived from the problem for u. Similarly, derive initial conditions for φ: one of these should be non-zero! Use separation of variables to find the solution for φ, and hence u.
for (a) I found $w(x) = \frac{Hx}{L}-G\frac{x(x-L)}{2c^2}$ but for (b) I'm not sure what it means for find a homogenous solution for φ. Does this mean I set u(x, t)=φ(x, t) , and w(x)=0 ? Not really sure what the question means.