# Wave equation and energy

1. Feb 17, 2010

### Kate2010

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

The ends (x=0,x=L) of a stretched string are fixed, the string is loaded by a particle with mas M at the point p (0<p<L).

1. What are the conditions that the transverse displacement y must satisfy at x=0, x=p and x=L?
2. Show that the energy of the system is E(t) = (1/2) $$\int_0^\L$$(T[yx(x,t)]2 + $$\rho$$[yt(x,t)]2) dx + (1/2)M[yt(p,t)]2
3. Deduce, using the wave equation and the boundary conditions, that dE/dt = 0 so the energy is constant.

2. Relevant equations

3. The attempt at a solution

1. I think y(0,t) = y(L,t) = 0 and y(p,t) = f(t) but I'm not too sure.

2. I think I have done this by considering the energy of the string and that of the mass separately.

3. This is where I'm really struggling. If I have the correct boundary conditions for x=0 and x=L we have worked through an example in lectures where the integral comes out to be 0 using Leibniz. However, a hint to answering this question is to break the integral into two, integrating between 0 and p, then p and L, so I think it can't be 0 as we must need a term to cancel out with the final term of the energy when differentiated. So, I think my boundary conditions may be incorrect.

Also, I think I have got a bit confused about partial differentiation. When I differentiate (1/2)M[yt(p,t)]2 do I get Myt(p,t)ytt(p,t)?

Thanks :)

2. Feb 17, 2010

### vela

Staff Emeritus
What's f(t)?

I think when you hang a mass on the string at x=p, you get a kink in the string or something like that. Try figuring out some condition describing a discontinuity in $\partial y/\partial x$ at x=p.

3. Feb 17, 2010

### vela

Staff Emeritus
Yup!

4. Feb 17, 2010

### Kate2010

I just meant to mean f(t) to be some function that depended only on time as I couldn't think of anything more specific than that. However, I will have another think tomorrow about how I could use the boundary condition to describe a discontinuity in the string.