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The Wave Equation The Guitar In the case of the guitar or the harpsi- chord, a string is pulled by a ﬁnger or hook to initial shape u (0 ,x ) = f ( x ) and then released with zero initial velocity [ g = 0 ]. Then the solution is u ( t,x ) = X n =1 a n sin ± nπx L ² cos ³ nπαt L ´ , where a n def = 2 L Z L 0 f ( x ) sin ± nπx L ² dx , n = 1 , 2 ...

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so that f ( x ) = X n =1 a n sin ± nπx L ² . Observe f ( x + at ) = X n =1 a n × ³ sin ± nπx L ² cos ´ nπαt L µ + sin ´ nπαt L µ cos ± nπx L ² f ( x - at ) = X n =1 a n × ³ sin ± nπx L ² cos ´ nπαt L µ - sin ´ nπαt L µ cos ± nπx L ² and u ( t,x ) = ˇ f ( x + αt ) + ˇ f ( x - αt ) 2 , 2
where ˇ f is the 2 L –periodic extension of the odd extension of f to [ - L,L ] . 3

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## This note was uploaded on 09/07/2010 for the course PHY 303L taught by Professor Turner during the Spring '08 term at University of Texas at Austin.

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