Sound – Beats
Superposition leads to beats – variation in loudness
f beat  f1  f 2
Resonance
Occurs in fixed media length L
Depends on boundary conditions
Occurs only for certain frequencies for which
an integral number of ½ or ¼ wavelengths
fit in medium length L
Affected by propagation speed in medium
Boundary Conditions
Ends of medium are either a node or antinode
Determined by the configuration of the system
Stringed instruments → both ends are nodes
Woodwinds, brass, organs → one end is always an
antinode, the other end can be either
Procedure
1.
2.
Identify physical situation (string or air column)
Identify boundary conditions (nodes or antinodes)
a.
b.
3.
4.
5.
6.
If the same use ½ wavelengths
If different use ¼ wavelengths
Sketch waveforms → determine how many ½ or ¼ λ fit
in length L
For each allowed n, solve for λn
Determine propagation speed v
Use v = f λ to find fn
Harmonic Series of Standing Waves on a
Vibrating String
2.
3.
4.
5.
6.
Both ends are nodes
(N) – use ½ λ s
Sketch
Solve for λn
If v not known use
Solve for fn
1 
L  n  n 
2 
2L
n 
n
v
T
 lin
v
nv
fn 

n 2 L
Harmonic Series of a Pipe Open at Both
Ends
2.
3.
4.
5.
6.
Both ends are antinodes
(A) – use ½ wavelengths
Sketch
Solve for λn
If v not known use
Solve for fn
1 
L  n  n 
2 
2L
n 
n
T
v  331
m/s
273K
v
nv
fn 

n 2 L
Harmonic Series of a Pipe Open at One End
2.
3.
4.
5.
6.
One end A, other end N
– use ¼ wavelengths
Sketch
Solve for λn
If v not known use
Solve for fn
1 
L  n  n  n odd
4 
4L
n 
n odd
n
T
v  331
m/s
273K
v
nv
fn 

n odd
n 4 L
Harmonics for Same Ends
Note that when ends are both nodes or both
antinodes, n = 1, 2, 3, 4, …
f2 = 2f1 → 2nd harmonic or 1st overtone
f3 = 3f1 → 3rd harmonic or 2nd overtone
f4 = 4f1 → 4th harmonic or 3rd overtone
f5 = 5f1 → 5th harmonic or 4th overtone
And so on …
fn = nf1
Harmonics for Different Ends
Note that when one end is N and the other is A, n =
1, 3, 5, 7, …
f3 = 3f1 → 3rd harmonic or 2nd overtone
f5 = 5f1 → 5th harmonic or 4th overtone
f7 = 7f1 → 7th harmonic or 6th overtone
f9 = 9f1 → 9th harmonic or 8th overtone
And so on … fn = nf1
→ only odd harmonics are allowed
Propagation Speed
In a string v depends on the
tension T and linear mass
density μlin
In an air column v depends
on temperature T
T (N)
v (m/s) 
lin (kg/m)
T (K)
v (m/s)  331
273K
Sound Levels
Decibel or dB scale
I 
sound level  10dB  log10  
 I0 
12
2
I 0  110 W/m
• I0 is the threshold of hearing
• Threshold of pain is 120 dB
The Doppler Effect




An observer of the wave crests behind or away from the
direction of motion would identify a lower frequency due
to a greater travel distance from the vibration source.
This apparent change in frequency is called the
DOPPLER EFFECT after Christian Doppler (1803 –
1853)
The greater the speed, the greater the Doppler effect
The police and weather reporters use the Doppler effect
of radar waves to measure the speeds of cars or water in
clouds.
The Doppler Effect
Vibrating in a stationary position
produces waves in concentric circles.
(because the wave speed is the same
in all directions)
Vibrating while moving at a speed less
than the wave speed produces
nonconcentric circles. The wave crests
in the direction of motion occur more
often (have a higher frequency) due to a
shorter travel distance.
Doppler Effect


If source (s) is moving, detected frequency (fd) changes
If detector (d) is moving, detected frequency changes
v ± vd
fd 
fs
v  vs
• Top sign when moving away from
• Bottom sign when moving towards
Astronomy and the Doppler Shift




Astronomers can measure the
Doppler Shift of a moving object.
The radio wave has a known
frequency for stationary objects.
If target is moving away, waves in
the signal will be stretched, and will
have a lower frequency.
The Doppler shift is called
“Red shift” if v > 0 (moving away)
since the wavelength is getting
longer and the frequency shorter
“Blue Shift” if v < 0 (moving closer)
since the wavelength is getting
shorter and the frequency higher
(recall that blue light is higher in
frequency than red)
TWO CLIPS ON THE DOPPLER
EFFECT
Bow Waves

When the vibration source is moving at the same speed
as the wave front, a bow wave is produced.

In aircraft this is called the “sound barrier”. It is not a
real barrier, but is where the wave crests build up on
each other at the speed of sound.

Speedboats and supersonic aircraft
travel in front of their bow waves.
Shock Waves

Shock waves produced by supersonic aircraft are the 3dimensional version of the V-shaped bow wave produced
by boats.

On the ground this is noted as a sonic boom.
BOOM !
SIMULATIONS FOR DOPPLER
EFFECT AND BOW WAVES
http://www.astro.ubc.ca/~scharein/a311
/Sim/doppler/Doppler.html