# The Animated Wave Equation

This demonstration illustrates the behaviour of solutions of the wave equation

u_{tt} = c^2 u_{xx}
u(0,t) = 0
u(l,t) = 0
u(x,0) = f(x)
u_t(x,0) = g(x)

In this example c=1, l=10, the intial amplitude consists of one bump centered on x=5 and the initial speed g(x) = -cf'(x). The demonstration plots the solution given by separation of variables that you have found in class. The demonstration animates the solution by successively plotting u(x,0), followed by u(x,dt), followed by u(x,2dt) and so on. Separation of variables expresses the solution as a sum

b_1(t) sin(pi x/l) + b_2(t) sin(2 pi x/l) + ...

of modes. The demonstration also plots the values of the first six coefficients b_k(t). What effect did my choice of g(x) = -cf'(x) have on the behaviour of the solution?

The demonstration below is an applet. Google Chrome, Firefox and Microsoft Edge no longer execute applets because of security issues with NPAPI plugins. Some browsers that still play applets are Internet Explorer, Safari and the Pale Moon browser [ windows, linux, Mac ].