4. Consider the wave equation for a vibrating rectangular membrane 0 < x <L,0< y < H, U₁ = c² (uxx + Uyy) subject to the initial condition u(x, y, 0) = f(x, y), and u(x, y, 0) = g(x, y). a) Solve the initial value problem if u(0, y, t) = 0, u(L, y, t) = 0, u(x, 0, t) = 0, u(x, H,t) = 0 Make sure to state clearly what the eigenvalues are for the two-dimensional eigenvalue problem. b) Draw the nodal lines for the first four eigenfunctions and explain what the different modes look like in general. c) Show that the sum fo the second and third eigenfunctions, (1, 2) and (2, 1), has a nodal line that goes from one corner to the opposite corner. [Hint: recall your double angle formulas]