Exercise 5.10: String with friction

The effect of the friction on an element of string $(x,x+\Delta x)$ is to oppose its motion. Assume that the force of friction is proportional to the vertical velocity $\partial u/\partial t$ of the string element. This changes the wave equation to

$$\frac{\partial ^{2}u}{\partial t^{2}}+2\kappa \frac{\partial u}{\partial t}=% \frac{\partial ^{2}u}{\partial x^{2}},$$

where the constant $\kappa$ is proportional to the viscosity of the medium in which the string is vibrating, and is inversely proportional to the density of the string. Generalize the algorithm for the wave equation to include friction and observe the change in the wave behavior. Use the same parameters of the pervious example, with $\kappa =5$.