The Animated Telegraph Equation

This demonstration illustrates the behaviour of solutions of the telegraph equation

u_{tt} +(a+b)u_t+abu= 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=3. The initial speed is chosen to be g(x) = -cf'(x) - (a+b)f(x)/2, so that, when there is no dispersion the bump just translates with speed c and decays with rate (a+b)/2. The demonstration simultaneously plots, in gray, the solution to the wave equation (i.e. a=b=0) and, in black, the solution to the telegraph equation with the current values of a and b. The animation runs for a time interval of length 20. Once it stops you may change the current values of and b. To ensure that all modes remain underdamped, a and b are required to obey a>=0, b>=0 and a+b<=.6. You may also choose to "turn off the decay". This means plot exp(d*t)u(x,t), with d=(a+b)/2, rather than u(x,t).

The demonstration below is an applet. Google Chrome, Firefox, Safari and Microsoft Edge no longer execute applets natively because of security issues with NPAPI plugins. However applets can be played in the Google Chrome browser by using the CheerpJ Applet Runner extension. As well, the Pale Moon browser [ windows, linux, Mac ] still plays applets.