Consider vibrating strings of uniform density ρ0 and tension T0. *(a) What are the...

Consider vibrating strings of uniform density ρ₀ and tension T₀.

*(a) What are the natural frequencies of a vibrating string of length L fixed at both ends?

*(b) What are the natural frequencies of a vibrating string of length H, which is fixed at x = 0 and “free” at the other end [i.e., ∂u/∂x(H, t) = 0]? Sketch a few modes of vibration as in Fig. 4.4.1.

(c) Show that the modes of vibration for the odd harmonics (i.e., n = 1, 3, 5, . . .) of part (a) are identical to modes of part (b) if H = L/2. Verify that their natural frequencies are the same. Briefly

explain using symmetry arguments.