Chapter 12 Sound Waves
We study the properties and detection of a particular type of wave – sound waves.
A speaker generates sound. The density of the air changes as the wave propagates.
Notice that the displacement maxima and minima occur where the pressure variation is zero and
the pressure variation maxima and minima occur where the displacement is zero.
The range of frequencies that can be heard by humans is typically taken to be between 20 Hz and
20,000 Hz. Most people struggle to hear the highest frequencies and that ability lessens with
age.
Speed of Sound
Recall
v
Restoring Force
Inertia
In fluids
12-1
v
B

The speed of the wave in a fluid (especially air) depends on temperature. In solids
v
Y

Amplitude and Intensity of Sound Waves
A sound wave can be described either by talking about pressure or displacement. Since the
displacement creates the pressure change, there is a relationship between the amplitude of the
pressure p0 and the amplitude of the displacement s0. For a harmonic sound wave the relation is
p0  vs0
A larger amplitude wave appears louder, but the relation between amplitude and loudness is very
complicated. Loudness is subjective and depends on the response of the ear and the brain.
Usually the intensity and not the amplitude is used for loudness. Again for a harmonic wave
2
p
I 0
2 v
“The most important thing to remember is that intensity is proportional to the amplitude
squared, which is true for all waves, not just sound.” (p 425)
12-2
Decibel Scale
The perception of hearing is roughly proportional to the logarithm of the intensity. The lowest
intensity of sound that can be heard by most people is
I 0  1.0 1012 W/m 2
I0 is called the threshold of hearing. It is used as the reference level for measuring sound
intensity. The sound intensity level in decibels is defined as
  (10 dB) log10
I
I0
(Be sure to practice with the decibel scale. Logarithms can be tricky.) An intensity level of 0 dB
corresponds to the threshold of hearing.
For incoherent sound waves with intensities I1 and I2, the total intensity is
I  I1  I 2
If the sound waves are coherent, the waves can interfere and the intensity is between |I1 – I2| and
I1 + I2, depending on the phase relationship between the two waves.
Decibels can be used in a relative sense. The difference in two dB readings
12-3
I2
I
 (10 dB) log10 1
I0
I0
 2  1  (10 dB) log10
I2
I1
 (10 dB) log10
is related to the ratio of the intensities.
Standing Sound Waves
Recall that a standing wave is the superposition of two traveling waves. The wave reflects at the
boundary of the wave.
Pipe open at Both Ends
The boundary conditions are the same at both ends. Since the end is open to the atmosphere, the
pressure at the ends can not deviate much from atmospheric pressure. The ends are pressure
nodes. Pressure nodes are displacement antinodes.
From the diagram, the wavelengths satisfy
n 
2L
n
The frequencies
fn 
v
n
n
v
 nf1
2L
The index n is an integer and it can vary from 1, 2, etc.
12-4
Pipe Open at One End
The situation is different from the pipe opened at both ends. The closed end is a pressure
antinode. The air at the closed end is isolated from the atmosphere and the pressure can deviate
far from atmospheric. The air at the closed end is a displacement node since the rigid wall
prevents the air from moving.
From the diagram, the wavelengths satisfy
n 
4L
n
The frequencies
fn 
v
n
n
v
 nf1
4L
This time n has odd values only (1, 3, 5, etc.)
12-5
Problem 27 Two tuning forks A and B, excite the next-to-lowest resonant frequencies in two air
columns of the same length, but A’s column is closed at one end and B’s column is open at both
ends. What is the ratio of A’s frequency to B’s frequency.
Since A excites the pipe open at one end, only the odd harmonics are possible
fn 
v
n
n
v
 nf1
4L
Where n = 1, 3, 5, etc. Next to lowest resonant frequency refers to the second frequency. Here
that mean n = 3 and
fA  n
v
v
3
4L
4L 1
For B, all the harmonics are possible since it is exciting a pipe open at both ends.
fn 
v
n
n
v
 nf1
2L
n = 1, 2, 3, etc. Next to lowest in this sequence corresponds to n = 2,
fB  n
v
v
2
2L
2L
Forming a ratio
v
fA
3v 2 L 3
 4L 

v
fB 2
4 L 2v 4
2L
3
Timbre
In general, a musical instrument will produce sounds that are made of combinations of the
available frequencies. The lowest frequency in a complex sound wave is called the
fundamental and the other frequencies are called overtones. Since all the overtones are integral
multiples of the fundamental, they are also called harmonics.
The complex shape of the sound wave means that different instruments playing the same note
will have a different tone quality. You can recognize your favorite singer by the timbre of the
singer’s voice.
12-6
A complex periodic signal can be created by adding together a set of harmonic waves. The wave
having three frequencies 110, 165, and 220 Hz repeats at 55 Hz since each of these frequencies
are harmonics of 55 Hz.
In principle, any complex waveform can be decomposed into a series of harmonic waves. This is
called Fourier (or spectral) analysis. Limiting our study to harmonic waves actually includes all
waveforms.
12-7
Human Ear
The physiology of the ear is detailed in the text. Please read it.
A brief overview of the ear: http://www.youtube.com/watch?v=p3Oy4lodZU4
The perception of loudness depends on frequency.
Pitch is the perception of frequency. Higher pitch means higher frequency (and shorter
wavelengths).
Beats occurs when two sound waves are close in frequency. It is very useful for tuning
instruments. The beat frequency is the difference in the two frequencies
12-8
fbeat  f1  f 2
The Doppler Effect
Probably best explained in http://www.youtube.com/watch?v=Y5KaeCZ_AaY
Or not! Maybe this will work? http://www.youtube.com/watch?v=yWIMWqkcRDU
The Doppler effect is the change in observed frequency (pitch) resulting from the motion of the
sound source and/or sound observer. It can be used to measure the speed of a moving car (or
baseball).
In front of the moving source, the wave crests are closer together and the frequency is higher.
Behind the moving source, the wave crests are further apart and the frequency is lower.
 1 
 f S
f O  
1

v
/
v
S


Important: vS > 0 for a source moving in the direction of the wave.
12-9
An observer moving relative to a stationary source will experience a different frequency. An
observer moving towards the source will experience a higher frequency and an observer moving
away from the source will experience a lower frequency.
fO  (1  vO / v) f S
Important: vO > 0 for an observer moving in the direction of the wave.
If both the source and observer move
 1  vO / v 
 f S
f O  
 1  vS / v 
12-10
The sign convention is given for the individual cases. You must get the signs right to get the
correct answer!!
Shock Waves
When the source moves faster than the speed of sound, the wave crests pile on top of each other
and a large amplitude wave occurs. For airplanes, this is called a sonic boom.
Shock waves: http://www.youtube.com/watch?v=-d9A2oq1N38
Echolocation and Medical Imaging
Sound and echoes are used to locate objects. Here is an animation explaining sonar:
http://www.youtube.com/watch?v=w_q2dqUdi8U
Ultrasound can be used to image structures inside the body.
12-11
High frequency sound is used since higher frequencies will have shorter wavelengths. Short
wavelengths diffract less around small obstacles.
Time for the end of the semester music festival.
12-12