Problem. Consider a violin string of length L = 20 cm with fixed ends at x = 0 andx = L.(a) Find the particular solution u(x, t) given that at t = 0 the string was ‘plucked’in the middle and released, tha t is:u(x, 0) =Dx for 0 ≤ x < L/2D(L − x) for L/2 ≤ x < L, (6)with D = 0.5 cm.1 (b) Find the frequency of the ‘fundamental no te’, ν1= ω1/(2π) assuming a velocityc = 500 m/s.(c) The ‘power’ of the sound at a particular radian frequency ωnis given by thesquared magnitude of the amplitude of the wave-component bnat that frequency, that is P (ωn) = |bn|2. Find the ratio between the power of the first harmonic(that is, at radian frequency ω2) and of the fundamental at f requency ω1. Thatis, find | b2/b1|2. This will show how the power of the higher harmonics decreasesas the frequency increases.(d) Does the ratio |b2/b1|2depend on the amplitude D of the initial displacement(that is, on how hard we pluck the string)?