There are two families of characteristics: left-running x + at = const and right-running x − at = const.
It can be shown that the general solution of equation can be represented as
where F1 and F2 are functions determined by initial and boundary conditions. If we ignore the effect of boundaries, the solution for the
specified initial conditions u(x, 0) = f (x), ∂u(x, 0) = g(x) can be written explicitly in the d’Alembert form
The first part of the solution can be interpreted using the given below figure. The initial perturbation of u(x, 0) = f (x) around the point x0 (e.g., a localized deformation of a string) is split into halves, which propagate without changing their shape along the characteristics x + at = x0 and x − at = x0.
The second part of the solution (3.27) represents the response to the initial perturbation of ‘velocity’ ∂(t)u. If, for example, the initial velocity is a delta function g(x) = δ(x − x0), the solution evolves as given below figure. u(x, t ) is a constant equal to 1/2a within the cone between the left-running and right-running characteristics and zero outside this cone
An important feature of the hyperbolic systems is illustrated by below figure. The perturbations propagate in space with a finite speed. Let us consider the situation, when a source of perturbations suddenly appears at the time moment t(0) and space location x(0) (point P in Figure). An observer located at the distance L from the source will not notice the perturbations until the time t(0) + L/a. In general, the state of the solution at the point P only affects the solution within a cone between the left-running and right running characteristics intersecting at P. The cone is called the domain of influence. Similarly, the solution at P itself is affected only by the solution
within the domain of dependence
The behavior described by hyperbolic equations and, thus, determined by characteristics appears in many physical systems—for example, in supersonic flows, propagation of acoustic and electromagnetic waves, and flows in stratified systems such as waves on water surface of internal waves in the ocean. Navier Stokes contains this in form of some non linear terms.