# Wave equation - stretched string with mass at each end

1. Feb 4, 2010

### Kate2010

1. The problem statement, all variables and given/known data

A string is stretched to a tension T, its ends x=0 and x=L are attached to rings with mass M which are able to slide on parallel smooth wires perpendicular to the string. Show:
1) The transverse displacement must satisfy Mytt = Tyx at x=0, Mytt = -Tyx at x=0
2) The normal frequencies are
3) What are the normal frequencies in their limiting cases as M tends to 0 and infinity?

2. Relevant equations

y(x,t) = $$\sum$$ sin(n$$\pi$$x/l) (ancos(n$$\pi$$ct/l) + bnsin(n$$\pi$$ct/l)
But I think this is only for fixed ends which mine isn't?

3. The attempt at a solution

1. I need to use Newtons 2nd Law I think.
At x=0 Mytt (i+j) = T(i+yxj), so Mytt = Tyx
At x=L Mytt (i+j) = -T(i+yxj), so Mytt = -Tyx
This doesn't feel right.

2. I think I need more conditions to be able to solve this. The ends aren't fixed so I can't say y(0,t)=y(l,t)= 0. But can I say anything about the velocity?