Hyperbolic Waves

Abstract

When describing the motion of any matter, it is always required that the mass is conserved. This mass conservation law, whether conjugating with other conservation laws or not, puts a constraint on the material motion so that a material property of the matter, such as the density, moves along a real curve in the space (x, y, z, t). This curve is usually called the characteristic of the conservation law. The term “hyperbolic” is equivalent to the characteristic curves being real. In this chapter, we will describe some basic examples of the hyperbolic conservation laws and show how to use the characteristics method to solve initial value problems for hyperbolic conservation laws. Here, we emphasize finding solutions to certain simple problems and explaining the physical meaning of these solutions. Mathematical rigor and extensive studies may be found in more specialized books (such as that by (1983)) on hyperbolic conservation laws.