Stationary Waves

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Stationary Waves
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• When two continuous similar waves are travelling
  in opposite directions, they can superpose to
  form a stationary wave. A stationary wave is a
  fixed pattern of vibration. Unlike a progressive
  wave no energy is transferred along the wave.
• To superimpose the two waves must
• Have equal frequency
• Nearly the same amplitude
• Travel at the same speed
• Travel in opposite directions

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Unlike progressive waves there are some points on
a standing wave which do not vibrate at all.
These are called nodes. At a node the waves add
together to give zero displacement. The distance
between nodes is always half a wavelength. At
places between the nodes there are points where
the displacement is a maximum. These points are
called antinodes. Fig 31 page 26
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• Musical instruments depend on standing waves
• In a string, for example guitar, pianoforte,
  violoncello.
• In a column of air, e.g. clarinet, tuba, organ.

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Waves on Strings

• Stationary waves on the strings of an instrument
  are the source of vibration for musical notes.
  When the string is made to vibrate, reflections
  from either end superimpose to cause the
  stationary wave.
• Fig 32 p 26
• Since the string is fixed at either end, there is
  a node at each end. The simplest way it can
  vibrate is with one antinode in between. This
  wave pattern is known as the fundamental.

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The frequency of the fundamental mode, f0, is
given by c/?. If the string has a length, l, the
wavelength is 2l, since there is half a
wavelength on the string. This gives the
fundamental frequency as f0 c/2l Where c is
the speed of the wave along the string. The
string can also have an oscillation which has a
node in the centre of the string. This is called
the first overtone or the second harmonic. There
is now a whole wave on the string so ? l. The
frequency of the first overtone is f1 c/l
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The first overtone (2nd harmonic) -Fig 33 p27
The second overtone (3rd harmonic)
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In general the expression for the frequency fn of
the nth mode (or the nth harmonic, or (n-1)th
overtone) is fn nc / 2l
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Stationary Waves in Pipes

• Stationary waves are also formed by vibrating
  columns of air in pipes.
• There is always a node at a closed end of a pipe,
  since there can be little vibration there. At the
  open end of the pipe there is always an antinode.

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Pipes closed at one end

• The fundamental mode of vibration for a pipe that
  is closed at one end has one quarter of a
  wavelength along its length. Therefore, the
  wavelength is ? 4l, where l is the length of
  the pipe. The fundamental frequency is therefore
  f0 c/4l.
• Fig34 p28

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The first overtone occurs when there is ¾ of a
wave in the pipe. This gives a wavelength of ?
4l/3, and the frequency is given by 3c/4l. Fig
34 p28
The general expression for the frequency fn (the
nth harmonic or the (n-1)th overtone) is fn
(2n-1)c / 4l
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Pipes with two open ends

• At each end of an open pipe there is an antinode.
  This makes it similar to the vibrations of a
  string that is fixed at both ends. The
  fundamental frequency is f0c/2l, the frequency
  of the first overtone is f c/l.
• Fig 35-p29

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Example

• Find the frequency of the first overtone in an
  organ pipe, closed at one end, that has a length
  of 2.00m. (Take the speed of sound to be 340ms-1)
• Answer
• The first overtone will have a frequency of f
  3c/4l
• f 3 x 340 / 4 x 2
• f 127.5Hz

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More Work!

• From AS/A2 Textbook p188 copy 2nd example
• From p188 Answer the Now Its Your Turn
• From P.I.P

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Work for Friday

• In pairs produce either a powerpoint presentation
  or poster on one of the following
• Meldes apparatus
• Kunsts tube