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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 updown 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.
 Fall '07
 MichaelFowler
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