Abstract
This chapter discusses the second-order hyperbolic differential equations. The simplest nontrivial example of a hyperbolic equation is the wave equation. The chapter presents the solution of the Cauchy problem for the one-dimensional wave. The chapter also describes the physical interpretation of the solutions of the homogeneous wave equation in dimensions one, two, and three. Once the oncoming wave has reached a given point, it causes a residual disturbance at that point, even after the wave has passed beyond that point. In the case of an initial disturbance at a point at time t = t1, the energy dissipation of the wave at this point will continue for all t > t1. The domain bounded by the axes and the characteristic lines through the point (x, t) is called the domain of influence. The total disturbance at a given point is the sum of all the contributions because of the points of the wave front at t = 0. The chapter discusses the uniqueness of the solution of the wave equation in the region bounded by the characteristic cone and the plane t = 0.
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