The Method of Characteristics: A Study of Waves

The method of characteristics is a powerful and intuitive tool for obtaining solutions to the PDEs that model various waves. This method takes advantage of the underlying geometry of the physical phenomena to find a family of curves which can be used to construct the final solution surface. During the course of this paper, this method will be showcased using the advection equation in its various forms: homogeneous, nonhomogeneous, and generalized linear. In each case, new properties of the method of characteristics will be explained, culminating in the application of the theory to the nonlinear advection equation to describe shockwaves and their properties.