Hello everybody, the problem reads like this:

A piano wire of mass M and length L is made vibrating and it is subject to a tension F.
The wire at rest lies on the x axis, and its displacement on the y axis is set as y(0)=y(L).

Its energy is expressed by the following integral:

E= 1/2 Integral_from_0_to_L_of [ dx ((M/L) (dy/dt)^2 + F(dy/dx)^2) ]

where the displacement is written in terms of Fourier series as:

y(x,t)= Sum_over_n_of (A_n (t) sin(x n pi/L) )

Show that the energy can be written as:

E= Sum_over_n_of (M/4 (dA_n/dt)^2 + (npi)^2 * F/4L * (A_n)^2)

(I have shown this simply by substituting the y on the energy expression and performed the integral)

The other part of the question is:

a) What is the internal energy of the wire and hence its heat capacity?

Then Calculate the average values of A_n, (A_n^)2 and A_n A_m with n different than m.

c) Finally, calculate the average value of y^2, from this determine the mean square displacement of the wire at x=L/2 (You can use without proof that
Sum_over_n_of n^-2= pi^2/p .)

Thank you very much for your help.

I think you mean π²/6.

As for the internal energy, this system does or is the subject of no work, so you can set dE/dt=0 and extract the frequency of each A_n which has the form A_n = B_n cos ω_n t + C_n sin ω_n t where B_n and C_n are simple numbers. Which means E ∝ ∑ [(B_n)^2 + (C_n)^2] n^2 and so having reduced the description of the state to numbers, I think you could start with that to do a cannonical ensemble calculation to get the rest.