The diagrams below show beats that occur when two different pairs of waves are added. For which of the two is the difference in frequency of the original waves greater?

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The diagrams below show beats that occur when two
different pairs of waves are added. For which of
the two is the difference in frequency of the
original waves greater?
A. Pair 1 B. Pair 2 C. The frequency
differences are the same. D. You need more
information to answer this question.
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Same frequency different A
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• Einstein and Infield, in their book The Evolution
  of Physics, describe a wave in this way
• "The wind passing over a field of grain, sets up
  a wave which spreads out across the whole field.
  Here again we must distinguish between the motion
  of the wave and the motion of separate plants,
  which undergo only small oscillations....The
  essentially new thing here is that for the first
  time we consider the motion of something which is
  not matter, but energy propagating through
  matter.
• A mechanical wave is a disturbance moving through
  a medium.

SB 16
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WAVE EQUATION (SB 16)

• The wave equation is a linear second-order
  partial differential equation. For a disturbance
  ?, which might represent the pressure in a sound
  wave, or the electric field in a radio wave, the
  amount of disturbance varies from place to place
  and from time to time. That is, ? is a function
  of a spatial variable, x, and time, t. The wave
  equation is

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linear second order partial differential equation

• linear because ? appears to the first power
  (linearly) throughout the equation
• second order because the highest (indeed, the
  only) derivatives which appear are ?2
• partial because ? is a function of two variables,
  x and t, and therefore we have to use partial
  rather than total derivatives.
• The constant v is the speed of the wave.

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• displacement of string sideways
• vertical movement of a small volume of water
• change in pressure for sound waves in a gas
• electric/magnetic field for light or radio waves

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Most general form of solution
Where h and g are any continuous,
twice-differentiable functions.
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Single-frequency waves
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PHASE VELOCITY
In a sinusoidal wave, this quantity gives the
speed at which peaks and troughs (points of
constant phase) move through the medium it is
called the phase velocity.
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LINEARITY/SUPERPOSITION

• As the wave equation is linear, we may superpose
  solutions and still get a solution which is a
  solution of the wave equation.