First-Order Wave Equation

Before we proceed further, let us look at the system of first-order equations which are frequently encountered in a class of fluid flow problems. Consider the second-order wave equation

(8.1)

Here c - wave speed and
       u - wave amplitude.

This can be written as a system of two first-order equations. If  and then we may write and

Rather, the system of equations may be written as

which is a first-order equation.

It is implicit that and The eigenvalues of the matrix are found by

det    or

Roots of the characteristic equation are and representing two traveling waves with speeds given by

and

The system of equations in this example is hyperbolic and it has also been seen that the eigenvalues of the A matrix represent the characteristics differential representation of the wave equations. Euler's equation may be treated as a system of first-order wave equations. For Euler's equations, in two dimensions, we can write a system of first order as

(8.2)

where

and S