Standing Waves in Musical Pipes

1
Standing Waves in Musical Pipes

• Just like on a guitar, standing
  waves can be produced in a
  pipe (or air column) like an
  organ pipe, flute, or
  saxophone.
• Recall that changing the effective length of a
  guitar string changes the _________ (_____) of
  the wave. In the same way, changing the ________
  of a pipe changes the frequency of the sound wave
  that will resonate in the pipe.
• On a guitar, the two ends are always fixed, but
  in pipes, the ends can be either open or closed.

frequency
pitch
length
2
Closed Pipes

• Closed pipes are actually closed at one end and
  open at the other.
• At the closed end, there can be NO _______ of
  the air (the air molecules cant move through the
  closed end of the pipe!) At that point, there is
  a Displacement _________.
• At the open end, there is maximum ___________
  of the air, so at that point, there is a
  Displacement _____________.

motion
node
motion
antinode
3
A closer look
Imagine a vibrating tuning fork creating a sound
wave
(Animation)
Now imagine the tuning fork producing a vibration
that enters a closed pipe
4
We can represent this situation graphically
antinode
This standing wave will only be created if the
wavelength of the sound wave is suitably matched
to the length of pipe. In this situation, the
reflected sound waves traveling to the right in
the pipe interfere with the sound waves traveling
to the left, producing a standing wave pattern.
Recall that a standing wave ___________ the
amplitude of a wave, which in this case amplifies
the emitted sound wave (louder).
increases
5
Harmonics in Closed Pipes

• This standing wave pattern is called the
  ____________ frequency, or ____ harmonic.
• It represents ____ of a wavelength.

fundamental
1st
¼th
Recall that the higher harmonics are always
__________ of the 1st harmonic. So the 2nd
harmonic is _________ the frequency of the 1st,
but ______ the wavelength. So, what would these
harmonics look like?...
multiples
double
half
6

• There is something wrong with two of these
  harmonics! Can you see what is wrong?
• At the open end, there must be a displacement
  __________, which means the ___ and the ___
  harmonics __________________ Only ______
  harmonics will resonate in a closed pipe!
• This is the same result as the possible standing
  waves on a spring that is fixed at one end and
  free at the other Demo with Coil Spring!

antinode
2nd
4th
cannot exist!!
odd
7
Harmonics vs. Overtones
The term overtone is used to describe the
particular harmonics (resonant frequencies) that
will produce standing waves above or _____ the
____________________________. For closed pipes,
then
fundamental frequency (f1)
over
2nd overtone
1ST overtone
8
Example Closed Pipe Calculation

• Calculate the fundamental frequency and the
  first two overtones that would resonate in an
  organ pipe of length 66.4 cm.

Fundamental frequency
L66.4 cm
Before we can find f, we need to know the speed,
v
the speed of sound 340 m/s
a low C note
9
1st Overtone
L66.4 cm
3rd harmonic!
2nd Overtone
L66.4 cm
5th harmonic!
odd
Again, closed pipes only resonate ________
harmonics!
10
Harmonics in Open Pipes

• This standing wave pattern is called the
  ____________ frequency, or ____ harmonic.
• It represents ____ of a wavelength.

Displacement Antinode
fundamental
1st
½
Displacement Antinode
An Animation
Again, recall that the higher harmonics are
always __________ of the 1st harmonic. So, what
would these harmonics look like?...
multiples
11
Fundamental Frequency or 1st
Harmonic
2nd Harmonic
3rd Harmonic
Is there anything wrong with these harmonics?
____! So, unlike a closed pipe, an open pipe will
resonate _______________. The 1st overtone is
then the ____ harmonic. The 2nd overtone is the
____ harmonic, etc.
NO
ALL harmonics
2nd
3rd
12
Open Pipes vs. Strings
The standing waves on a string and in an open
pipe are very much related they produce the
same __________. To help us see why, recall that
one wavelength looks like
harmonics
½l
½l
l
l
13
Demos
1. PVC pipe Musical Instrument- Open pipe
fundamental frequency is _________ the frequency
of the Closed pipe fundamental
frequency
( _______________). Hold hand over bottom of pipe
to make closed pipe. 2. Small model of Organ
Pipe The pipe length can be changed
(similar to moving a trombone slide
out), which then changes
the fundamental
frequency (pitch)
double
1 octave higher
Also, play open vs. closed
14
3. Singing Rods When you hold a thin metal rod
in the middle (forcing a node there), and your
other hand slides across the rod, the rod begins
to sing loudly! If the hand is sticky, it
works better (more friction). Rosin can be used
to increase the stickiness.
Whats going on?
The vibrations created by the hand slipping over
the metal rod produce longitudinal vibrations in
the rod. When the vibration reaches the end of
the rod, some of the energy is reflected back,
and a standing wave results. Holding your hand
in the middle forces a node there and the 2 ends
would be antinodes. Holding 1/4th of the way in
forces a node there, resulting in a frequency
(pitch) that is __________________!
twice as high
15
In Summary
Harmonics for a fixed string
When a string is plucked, only standing waves
corresponding to resonant frequencies persist for
long.
16
Summary Contd
For an open pipe
For a closed pipe
Remember that although the amplitudes of the
graphs are shown vertically (as transverse
waves), the actual motion of the air molecules is
horizontal along the tube.