Conceptually, a first order hyperbolic wave equation is the simplest wave propagation model. The characteristic solution  in the time--space domain for the homogeneous, constant velocity model is illustrated in Figure 2. The first order hyperbolic system, Equation 2, has an analytic solution, . The initial condition is and the slope of the characteristic solution is the media velocity c.
The sign of , of Equation 2 determines the propagation direction of the wave. If is positive, the wave propagates to the right. Bidirectional waves are a property of the second order hyperbolic wave equation.