Shock and traveling wave phenomena on an externally damped, non-linear string

We examine the propagation of shocks and traveling wave phenomena on a one-dimensional string that is executing finite-amplitude, transverse vibrations in a resisting medium. As part of our study, we develop an approach that allows us to describe, albeit approximately, the evolution and propagation of a shock front using analytical methods. In addition, exact traveling wave solutions, one of which involves the Lambert WW-function, of the string's equation of motion are determined and analyzed. Lastly, a possible new form of the solution to the linearized problem is presented and extensions and other applications of the present work are briefly discussed.