Here we consider a uniform string of length 1, 0 < x < 1, with constant tension and density (c = 1). Initial shape is f(x) = x(1 - x), and the initial velocity is g(x) = (x - 1/2)^2.  Below we animate the solution to the wave equation (no external forces). Only the first 20 terms in the infinite series solution are used.

Now we change the initial shape to f(x) = sin(4 Pi x)/50 and keep everything else the same.