Waves on a String Fixed at Both Ends

A string of mass density $\mu$ is stretched to a tension $T$ and fixed at $x = 0$ and $x = L$. The transverse string displacement is measured in the $y$ direction. All answers should be given in terms of these quantities or new quantities you define in terms of these quantities.

a) Write down the wave equation (the differential equation of motion) for waves on a string. You do not have to derive it.

b) Find $k_n, \omega_n, f_n, \lambda_n$ for the first four modes supported by the string. Sketch them in on the figure above, labelling nodes and antinodes.

c) Write down the equation for standing waves on this string, with mode index $n$, assuming that each mode is maximally displaced at $t = 0$.

d) Find the total energy in one of these modes, assuming that it has an amplitude $A$.