A note on the propagation of the bulk of a disturbance for a hyperbolic equation
Our arguments can be extended to other linear hyperbolic equations with nonconstant coefficients but, for the sake of brevity, we consider here only the damped wave equation (I)If the initial data correspond to a disturbance that is confined, in the sense that / and g have compact support, then we can ask at what rate the disturbance propagates. The conventional answer is that the disturbance propagates at unit speed, i.e., in such a way that the distance travelled equals the time elapsed. The ground for this assertion is, of course, the entirely correct one that if the supports of / and g are contained within the compact interval a < x < (3 then, for each t > 0, the support of u{x, t) is contained within the interval a — t < x < (3 +1. Our purpose is to point out that the conventional answer nonetheless fails to tell us how the bulk of the disturbance propagates and that, in fact, the bulk propagates in such a way that the distance travelled is proportional to the square root of the time elapsed, as is the case for the parabolic heat equation. (See, for example, Fichera [1], Day [2], [3]. See also the discussion of what happens for the Fokker-Planck equation with periodic coefficients in Day and Saccomandi [4].)