4) (Strings) Consider the wave equation მე2 მ2 (-)-0. 12) 4 = 0, (4) for the string with x [0, a] and Dirichlet boundary conditions (t, 0) = (t, a) = 0. (a) Write down the most general solution for this Dirichlet string using the sine Fourier series. (b) Find the solution with initial condition (0,2) = sin (*) and (0,2) = 0, where o is a constant. (c) Find the solution with initial condition (0, x) = 0 and (0, x) = 0 sin (2x). (d) Find the solution with initial condition (0,2) = 0 and hr (0,x): { for 0≤x≤b h(a-z) a-b for b<<a (5) where b and h are constants. Plot the amplitude bk/h of the kth eigenmode as a function of b/a [0.1] for k 1,2,3,4 and discuss.