Hey guys, a wave question for you (Andy & Nick, it's Q2a from the problem sheet Waves 2).
We have a string of uniform linear density p which is stretched to a tension T and it's ends are fixed at x=0 and x=L. Using separation of variables, you end up with the general solution:
y(x,t) = (Acos.kx + Bsin.kx)(Ccos.kct + Dsin.kct)
But since y(0,t)=0 and y(L,t)=0, we get A=0 and kL=r.pi
thus y(x,t) = Bsin(rx.pi/L)[Ccos(rct.pi/L)+ Dsin(rct.pi/L)]
I know have to show that the following are both solutions, obeying the boundary conditions:
y(x,t) = Esin(rx.pi/L)sin(rct.pi/L)
y(x,t) = Fsin(rx.pi/L)cos(rct.pi/L)
I can get the second one by using the fact that it's initially at rest, so dy/dt=0, thus in my earlier solution, D=0 and F=BC. How do I obtain the first solution however? I clearly have to use different boundary conditions, but I can't see what they should be. Suggestions?