Wave Equation

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Shashank Dwivedi

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A string of length L has its end at x=0 and x=L fixed. The mid point is taken to a small height h and released from rest at time t=0. Find the displacement of the string at a distance x from the end x=0 at time t.

I know that if u(x,t) is the displacement, then the boundary conditions are:
u(0,t)=0 and u(L,t)=0
But what will be the initial condition, u(x,0) here? I am provided with the value of the function at a point(i.e u(L/2,0)=h) in the domain, how am I supposed to derive the whole function from this hint?
 
"The mid point is taken to a small height h and released from rest at time t=0."

You are not just provided with a single point- grasping a string by a point and moving that point, the string makes two straight lines. One goes from one fixed point, (0,0), to the point at which the string is held, (L/2, h). The other goes from (L/2, h) to the other fixed point, (L, 0).

So, what is the equation of the line from (0,0) to (L/2, h)? What is the equation of the line from (L/2, h) to (L, 0)? Of course the other initial condition is "released from rest". That is, the derivative of u(x, 0), with respect to t, is 0: \(\displaystyle u_y(x, 0)= 0\).
 
Well I have tried. This is what I reached:

u(x,0)= \(\displaystyle \frac{hx}{a}\) if 0<= x <=a

\(\displaystyle \frac{h(2a-x)}{a} \) if a<= x <=2a

where 2a=L.

Am I correct?
 
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