Wave propagation in a non-uniform string

In this article I present the behavior of the reflection and transmission coefficients of a pulse at a joint between two strings with mass densities mu(1) and mu(2). The joint is made of a string segment with mass density varying linearly from mu(1) to mu(2). It will be shown that the reflection of the pulse at the joint depends largely on the ratio between the pulse width and the length of the joint. Analogies with other physical systems such as antireflection coatings and tsunamis will be considered briefly.