28 1. Simple Examples of Propagation In the limit as 0, one finds that both ∂tu and ∂xu are O(1/) while u = O(1) is negligibly small in comparison. Dropping this small term leads to the simplified equation for an approximation v , ∂tv + ∂xv = 0 , v t=0 = a(x) cos(x/) . The solution is v (t, x) = a(x t) cos ( (x t)/ ) , which misses the exponential decay. It is not a good approximation except for t 1. The two large terms compensate so that the small term is not negligible compared to their sum. 1.6. The law of reflection Consider the wave equation u = 0 in the half-space Rd := {x1 0}. At {x1 = 0} a boundary condition is required. The condition encodes the physics of the interaction with the boundary. Since the differential equation is of second order, one might guess that two boundary conditions are needed as for the Cauchy problem. An anal- ogy with the Dirichlet problem for the Laplace equation suggests that one condition is required. A more revealing analysis concerns the case of dimension d = 1. D’Alem- bert’s formula shows that at all points of space-time, the solution consists of the sum of two waves, one moving toward the boundary and the other toward the interior. The waves approaching the boundary will propagate to the edge of the domain. At the boundary one does not know what values to give to the waves which move into the domain. The boundary condition must give the value of the incoming wave in terms of the outgoing wave. That is one boundary condition. Factoring ∂2 t ∂2 x = (∂t ∂x)(∂t + ∂x) = (∂t + ∂x)(∂t ∂x) shows that (∂t ∂x)(ut + ux) = 0, so ut + ux is transported to the left. Similarly, ut ux moves to the right. Thus from the initial conditions, ut ux is determined everywhere in x 0, including the boundary x = 0. The boundary condition at {x = 0} must determine ut +ux. The conclusion is that half of the information needed to find all the first derivatives is already available and one needs only one boundary condition. Consider the Dirichlet condition, (1.6.1) u(t, x) x1=0 = 0 .
Previous Page Next Page