(adsbygoogle = window.adsbygoogle || []).push({}); 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) [tex]\int_0^\L [/tex](T[y_{x}(x,t)]^{2}+ [tex]\rho[/tex][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.

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[y_{t}(p,t)]^{2}do I get My_{t}(p,t)y_{tt}(p,t)?

Thanks :)

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# Homework Help: Wave equation and energy

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