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# prexam5a - 1 The following expression is proposed as a...

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1. The following expression is proposed as a solution of the wave equation: y ( x, t ) = A sin( kx + φ ) cos( ωt ) , where A , k , ω , and φ are constants. a) Show that it is a solution of the wave equation, or more precisely, find the condition (involving some of the four constants) that en- sures it is a solution. b) Suppose that this expression represents the motion of a string that is tied down at two ends (a standing wave). One end is at x = 0 and the coordinate y is zero at that point for all times. What does this condition determine about any of the four constants? c) The other end of the string at x = L is also tied down so that y is zero for all times. What does this condition determine about any of the four constants? Solution: a) The wave equations is on the front cover: 2 y ∂x 2 = 1 v 2 2 y ∂t 2 ∂y ∂x = kA cos( kx + φ ) cos( ωt ); 2 y ∂x 2 = - k 2 A sin( kx + φ ) cos( ωt ) = - k 2 y ( x, t ) ∂y ∂t = - ωA sin( kx + φ ) sin( ωt ); 2 y
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