1. Wave equation. Consider the wave equation on the finite interval (0, L): Utt - c²uxx = 0, ur(0,t) = u(L,t) = 0, PDE BC (1) (2) where Neumann boundary conditions are specified. Physically, with Neumann boundary conditions, u(x, t) could represent the height of a fluid that sloshes between two walls. (a) Find the general Fourier series solution by repeating the derivation from class, now considering Neumann instead of Dirichlet boundary conditions. Your final solution should be 1 1 nπct u(x,t) = 40+ Bot+A cos nлct ппх + Bn sin COS L (3) L L n=1 (b) Consider the following general initial conditions: u(x, 0) = g(x), u₁(x, 0) = h(x). IC IC (4) (5) Derive formulas that relate the Fourier coefficients An, Bn, ŷn, ĥn- (c) Consider the following specific initial conditions: u(x, 0) = 1 -1000 u₁(x, 0) = 0. 2πα IC (6) L IC (7) Find the solution u(x, t).