lec16A - Math 124A November 23, 2009 Viktor Grigoryan 16...

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Unformatted text preview: Math 124A November 23, 2009 Viktor Grigoryan 16 Waves with a source, appendix In the lecture we used the method of characteristics to solve the initial value problem for the inhomo- geneous wave equation, u tt- c 2 u xx = f ( x,t ) ,- < x < ,t > , u ( x, 0) = ( x ) , u t ( x, 0) = ( x ) , (1) and obtained the formula u ( x,t ) = 1 2 [ ( x + ct ) + ( x- ct )] + 1 2 c x + ct x- ct ( y ) dy + 1 2 c t x + c ( t- s ) x- c ( t- s ) f ( y,s ) dy ds. (2) Another way to derive the above solution formula is to integrate both sides of the inhomogeneous wave equation over the triangle of dependence and use Greens theorem. H x 0, t L x- c t x + c t L L 1 L 2 D x t Figure 1: The triangle of dependence of the point ( x ,t ). Fix a point ( x ,t ), and integrate both sides of the equation in (1) over the triangle of dependence for this point. 4 ( u tt- c 2 u xx ) dxdt = 4 f ( x,t ) dxdt. (3) Recall that by Greens theorem D ( Q x- P t ) dxdt = D P dx + Qdt, where D is the boundary of the region D with counterclockwise orientation. We thus have 4 ( u tt- c 2 u xx ) dxdt = 4 (- c 2 u x ) x- (- u t ) t dxdt = 4- u t dx- c 2 u x dt. The boundary of the triangle of dependence consists of three sides, 4 = L + L 1 + L 2 , as can be seen in Figure 1, so 4 ( u tt- c 2 u xx ) dxdt = L + L 1 + L 2- u t dx- c 2 u x dt, and we have the following relations on each of the sides L : dt = 0 L 1 : dx =- cdt L 2 : dx = cdt 1 Using these, we get L- c 2 u x dt- u t dx =- x + ct x- ct u t ( x, 0) dx =- x + ct x- ct ( x ) dx, L 1- c 2 u x dt- u t dx = c L...
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lec16A - Math 124A November 23, 2009 Viktor Grigoryan 16...

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