Lec7 - x = 0 Therefore the slope at x = 0 is sin(kx = 0...

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x = 0. Therefore the slope at x = 0 is sin( kx ) = 0 without restriction. At x = L , the slope is sin( kL ) = 0, so that kL = f = n 2 L · v as in the two Fxed endpoints case. These considerations are important, say, for wind instruments, such as organ pipes, ±utes, etc. Question : What are the boundary conditions for wind instruments for closed or open pipes? Periodic boundary conditions : We now want y ( x = 0) = y ( x = L ). We don’t want the spatial function to be sin( kx ), because it would always vanish. Rather, at x = 0 we should have a general value for the wave function. We therefore take y ( x, t )= A cos[ k ( x - x 0 )] sin( ωt ), where x 0 is the location of the Frst anti-node. (Of course, we could use sine function, and then x 0 would measure the distance of x = 0 from a node.) Therefore, y (0) = A cos( kx 0 ) sin( ωt ), and y ( L )= A cos[ k ( L - x 0 )] sin( ωt ). The periodicity condition that yields cos( kx 0 ) = cos[ k ( L - x 0 )], whose solution is k ( L - x 0 ) = 2 π n ± kx 0 . We choose the negative solution, and obtain kL =2 πn , or L =2 πn/ (2 π/λ )= = nv/f . Therefore, f = n v L , and the fundamental frequency is f = v L . We will do something very similar when we discuss the Bohr model in atomic physics. Beats
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Lec7 - x = 0 Therefore the slope at x = 0 is sin(kx = 0...

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