1- Consider waves propagating in a vibrating quarter-circular membrane: a²u(r,e,t) at² = c²²u, r<a, 0 < 0 < x/2 0<t The displacement u(r, 0, t) is zero on the entire boundary at all times. a) Write down explicitly the three boundary conditions expressed above. b) Starting by the method of separation of variables, find the solution and show that it is given by un (r,e,t) =ΣΣn (√nct) sin(10) [Ancos(√√^nct)+Bin sin(√√^nct)], where I is a positive even integer, and n is a positive integer. c) Find an expression for the allowed frequencies of vibration. [Hint: The frequencies are found from the time-dependent part]. d) For the initial conditions IC1: u(r, 0, 0) = a(r, 0) and IC2: ou(r, 0, 0)/ot 0 determine the coefficients AI, and Bin in your solution [Hint: apply IC2 first, then IC1].