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# L25x - 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74...

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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 cos nπαt L , where a n def = 2 L Z L 0 f ( x ) sin nπx L dx , n = 1 , 2 . . .

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so that f ( x ) = X n =1 a n sin nπx L . Observe f ( x + at ) = X n =1 a n × sin nπx L cos nπαt L + sin nπαt L cos nπx L f ( x - at ) = X n =1 a n × sin nπx L cos nπαt L - sin nπαt L cos nπx L 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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