Wave Equation and Energy Conservation for a Stretched String

[Kate2010]

Homework Statement

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[y_x(x,t)]² + \rho[y_t(x,t)]²) dx + (1/2)M[y_t(p,t)]²
3. Deduce, using the wave equation and the boundary conditions, that dE/dt = 0 so the energy is constant.

Homework Equations

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[y_t(p,t)]² do I get My_t(p,t)y_tt(p,t)?

Thanks :)

 

[vela]

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.

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.

 

[vela]

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

Yup!

 

[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.