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).