1.10 The speed of a wave

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1.10 The speed of a wave

• Assumed that a wave moves with a constant speed,
  v.
• How does this speed relate to the physical
  parameters of the medium in which the wave
  propagates?

• Let us consider a wave on a string. We will
  assume the following
• The string is under a constant tension F.
• The string has a mass per unit length µ.
• The string is uniform, perfectly elastic and
  flexible.
• Once the wave has passed the string returns to
  its original state.
• The amplitude of the wave is small.
• This means that the tension in the string does
  not change as wave moves along the string.
• We consider the string in the wave frame of
  reference.
• We sit on the wave and watch the string move.

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In the pulse frame of reference the string moves
from right to left with a speed v.
Take a small segment of the string AB which we
can treat as if it was an arc of circle of radius
R. The length of the segment is 2Rq and the mass,
MAB, will be
MAB 2Rqµ.
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What forces act on the string?
The tension in the string provides the
centripetal force Fc needed for circular motion.
In x direction the force is equal and
opposite. In y direction the force Fy, at each
end of segment is Fsinq. Total force in y
direction is 2Fsinq this drives the centripetal
force.
As amplitude of wave is small sinq q.
4
One end of a string of mass 2 g is attached to a
wall and the other end is allowed to hang over a
pulley. A mass of 3 Kg is attached to the free
end. A pulse is excited on the string. If the
sting is 1.5 m in length determine the speed of
the pulse.
The tension in the string is a result of the
weight hanging from the end of the string
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So F Mg
If the string has a mass ms and a length l then
the mass per unit length of the string is µ ms/l
Here M 3 Kg, ms 2 x 10-3 kg, l 1.5 m and g
9.81 ms-2 Hence v 150 ms-1