2. (25 points) Consider a rectangular membrane with sides a and b and lying in the 2-y plane. The displacement of the membrane in the direction perpendicular to the x - y plane is given by us, y, t) where u obeys the wave equation D2 = 12 2 t2 v is a constant, and u vanishes on the boundaries at x = 0, x = a, y = 0, and y=b. (a) Use separation of variables to find solutions to the wave equation of the form u(x, y, t) = T(0)X(X)Y (y). Write out the general solution for u (b) Find the explicit solution given the initial conditions (11) (t, y, t=0) = 0 (10) for all x, y and u(x, y, t = 0) = A8 ( 2') 8 (y - y) where 0