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Unformatted text preview: Math 124A November 09, 2010 Viktor Grigoryan 13 Waves on the half-line Similar to the last lecture on the heat equation on the half-line, we will use the reflection method to solve the boundary value problems associated with the wave equation on the half-line 0 < x < . Let us start with the Dirichlet boundary condition first, and consider the initial boundary value problem ( v tt- c 2 v xx = 0 , < x < , < t < , v ( x, 0) = ( x ) , v t ( x, 0) = ( x ) , x > , v (0 ,t ) = 0 , t > . (1) For the vibrating string, the boundary condition of (1) means that the end of the string at x = 0 is held fixed. We reduce the Dirichlet problem (1) to the whole line by the reflection method. The idea is again to extend the initial data, in this case , , to the whole line, so that the boundary condition is automatically satisfied for the solutions of the IVP on the whole line with the extended initial data. Since the boundary condition is in the Dirichlet form, one must take the odd extensions odd ( x ) = ( ( x ) for x > , for x = 0 ,- (- x ) for x < . odd ( x ) = ( ( x ) for x > , for x = 0 ,- (- x ) for x < . (2) Consider the IVP on the whole line with the extended initial data u tt- c 2 u xx = 0 ,- < x < , < t < , u ( x, 0) = odd ( x ) ,u t ( x, 0) = odd ( x ) . (3) Since the initial data of the above IVP are odd, we know from a homework problem that the solution of the IVP, u ( x,t ), will also be odd in the x variable, and hence u (0 ,t ) = 0 for all t > 0. Then defining the restriction of u ( x,t ) to the positive half-line x 0, v ( x,t ) = u ( x,t ) x , (4) we automatically have that v (0 ,t ) = u (0 ,t ) = 0. So the boundary condition of the Dirichlet problem (1) is satisfied for v . Obviously the initial conditions are satisfied as well, since the restrictions of odd ( x ) and odd ( x ) to the positive half-line are ( x ) and ( x ) respectively. Finally, v ( x,t ) solves the wave equation for x > 0, since u ( x,t ) satisfies the wave equation for all x R , and in particular for x > 0. Thus, v ( x,t ) defined by (4) is a solution of the Dirichlet problem (1). It is clear that the solution must be unique, since the odd extension of the solution will solve IVP (3), and therefore must be unique....
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