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3_prelab_solutions - Mechanical Waves Prelab Figure 1 The...

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Mechanical Waves Prelab Figure 1: The Shive Wave Machine 1 Reflected Pulse A wave pulse with positive amplitude A travels from the origin in the + x direction and meets an unclamped end of the Shive Wave Machine, figure 1. Will there be a reflected wave pulse? Will the amplitude of the reflected pulse be positive or negative? There will be a reflected wave pulse. The end of the Shive Wave Machine is unclamped, which means that it is free to oscillate. When a wave pulse hits a free boundary, the amplitude of the reflected wave is upright with respect to the am- plitude of the incoming wave. Since the amplitude of the incoming wave pulse is positive, the amplitude of the reflected pulse will also be positive. If the end of the Shive Wave Machine is instead clamped down, then the boundary is fixed. In this case the amplitude of the reflected wave will be inverted with respect to the amplitude of the incoming wave. If the incoming wave has a positive amplitude, the outgoing wave will have a negative amplitude. 1
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2 Both Ends Fixed Using the formula for the wave function given in equation 10, show that the wave function for a machine which is clamped at both ends has nodes at both ends. The wave function for a machine which is clamped down at both ends is y m ( x, t ) = A m cos(2 πf m t ) sin mπx L The variable x describes the horizontal position along the standing wave. For example, x might describe exactly which rib of the Shive Wave Machine you are looking at. ( x = 0 would represent the first rib of the wave machine, and x = L would represent the last rib of the wave machine.) The variable t represents a moment in time. Finally, the wave function y ( x, t ) tells you the height y
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