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Unformatted text preview: Math 124A November 16, 2010 Viktor Grigoryan 14 Waves on the finite interval In the last lecture we used the reflection method to solve the boundary value problem for the wave equation on the half-line. We would like to apply the same method to the boundary value problems on the finite interval, which correspond to the physically realistic case of a finite string. Consider the Dirichlet wave problem on the finite line ( v tt- c 2 v xx = 0 , < x < l, < t < , v ( x, 0) = ( x ) , v t ( x, 0) = ( x ) , x > , v (0 ,t ) = v ( l,t ) = 0 , t > . (1) The homogeneous Dirichlet conditions correspond to the vibrating string having fixed ends, as is the case for musical instruments. Using our intuition from the half-line problems, where the wave reflects from the fixed end, we can imagine that in the case of the finite interval the wave bounces back and forth infinitely many times between the endpoints. In spite of this, we can still use the reflection method to find the value of the solution to problem (1) at any point ( x,t ). Recall that the idea of the reflection method is to extend the initial data to the whole line in such a way, that the boundary conditions are automatically satisfied. For the homogeneous Dirichlet data the appropriate choice is the odd extension. In this case, we need to extend the initial data , , which are defined only on the interval 0 < x < l , in such a way that the resulting extensions are odd with respect to both x = 0, and x = l . That is, the extensions must satisfy f (- x ) =- f ( x ) and f ( l- x ) =- f ( l + x ) . (2) Notice that for such a function f (0) =- f (0) from the first condition, and f ( l ) =- f ( l ) from the second condition, hence, f (0) = f ( l ) = 0. Subsequently, the solution to the IVP with such data will be odd with respect to both x = 0 and x = l , and the boundary conditions will be automatically satisfied. Notice that the conditions (2) imply that functions that are odd with respect to both x = 0 and x = l satisfy f (2 l + x ) =- f (- x ) = f ( x ), which means that such functions must be 2 l- periodic. Using this we can define the extensions of the initial data , as ext ( x ) = ( ( x ) for 0 < x < l,- (- x ) for- l < x < , extended to be 2 l- periodic ,...
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