Phys 422 Spring 2023 Midterm Exam 4. Pulse evolution on string (30 pts) An elastic string of length L, tension T, and mass per unit length y is fixed at both ends. The string is distorted as shown below, where the height of the triangular pulse is H. Assuming small transverse motion of the string. At t = 0, the string is released from rest. L/4 x=0 x= L/2 x=L Figure 3: Accompanying Problem 4. (1) What are the normal modes and normal mode frequencies of the system? Show explicitly that they satisfy the wave equation and the boundary conditions for this system. (6 pts) (2) Find the Fourier series for the initial shape of the string at t = 0. (10 pts) Hint: the initial shape of the string can be expressed as: for