A matrix DʹAlembert formula for coupled wave initial value problems

If C is an invertible matrix in , the coupled wave equation initial value problem utt = C2uxx, −∞ < x < ∞, t> 0, u(x, 0) = f(x), and ut(x, 0) = g(x) for −∞ < x < ∞ is studied. A matrix DʹAlembert formula for the closed form solution of the coupled wave equation is given. The approach is based on the Fourier transform, the holomorphic matrix functional calculus and some elements of complex variable functions. For the scalar case, the proposed DʹAlembert formula coincides with the classical one.