Boundary Conditions

Boundary Conditions - previous index next Boundary Conditions at the End of the String Michael Fowler Adding Opposite Pulses Our first move in

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previous index next Boundary Conditions: at the End of the String Michael Fowler 3/12/07 Adding Opposite Pulses Our first move in working with waves was to jiggle the end of a string (or spring) and generate a pulse that we saw traveled along with no perceptible change in shape. We showed that our observation could be expressed mathematically: taking the string initially at rest along the x -axis, its displacement y at point x at time t was evidently described by a function of the form . This function keeps its shape, but as t progresses it moves to the right with speed v . ( yfxv t =− ) We next analyzed the dynamics of the vibrating string by applying Newton’s Laws of Motion to a little bit of string. This reveals an equation, the wave equation , that any vibration of the string must obey. Reassuringly, our observed form for the moving pulse, ( ) t , does in fact satisfy the wave equation. The wave equation has one very important property: if you add two solutions to the wave equation, the sum is another solution to the wave equation . This means that if you and a friend send pulses down a rope from the opposite end, the pulses will go right through each other, and when they’re on top of each other, the total displacement of the rope will be just the sum of the displacements corresponding to the individual pulses. We shall see that this gives an important clue for understanding what happens when a pulse reaches the end of the string. Pulse Reflection What happens when the pulse gets to the end of the string depends on the end of the string: there are two possibilities: (a) the end of the string is fixed, (b) the end of the string is free to move up and down (the pulse corresponds to the string moving in an up-down way).
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This note was uploaded on 12/07/2011 for the course PHYSICS 152 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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Boundary Conditions - previous index next Boundary Conditions at the End of the String Michael Fowler Adding Opposite Pulses Our first move in

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