One dimensitonal wave propogation described by the wave equation u(x, t) with utt(x, t) = c2 uxx(x, t) where c = 1 is drawn as t increases. The initial conditions are u(x, 0) = f(x) and ut(x, 0) = 0. Two cases, bounded and unbounded, are considered: (i) 0 ≦ x ≦ L < ∞ where the boundary conditions are u(0, t) = u(L, t) = 0 and (ii) -∞ < x < ∞ where no boundary conditions are imposed.
For the bounded case, f(x) is either Gaussian, of a box form or of a triangular form for a certain range of x. For the unbounded case, f(x) is either a cosine function (f(x) = 50 cos(2πx/L), a sine function (f(x) = 75sin(4πx/L + π/4)) or the average of them. D'Alembert solutions for both cases are illustrated. In particular, the former case turns out to be equivalent to the latter case where f(x) is replaced with F(x), which is an an odd and periodic extension of f(x) over an infinite horizon.