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Chapter 12
LECTURE NOTES
A vibrating object can excite longitudinal waves in a surrounding medium, transmitting energy
by sound.
The velocity of sound in air is v = (331 + .6T) m/s where T is the air temperature in °C. Sound
travels more quickly at higher temperatures and in denser media; sound cannot travel in a
vacuum.
Humans can hear sound throughout the audible spectrum that extends in frequency from 20 Hz
to 20,000 Hz. Some animals emit and detect higher (ultrasonic) and lower (infrasonic)
frequencies.
The intensity of sound I in W/m2 is defined as power per unit area; I = P/A = P /4  r2 if sound
expands in spherical shells of radius r. The human ear is remarkably sensitive, able to detect tiny
amounts of energy. The ear (as other senses) responds in a roughly logarithmic fashion. We
define the intensity level we perceive as  = 10 log (I/I0) where I0 is the intensity of the faintest
sound we can hear. I0 = 1 x 10 -12 to W/m 2, the threshold of hearing.  is dimensionless, but we
label these numbers decibels (d  ) or tenths of a bel (named after Alexander Graham Bell,
inventor of the telephone).
The log in the formula above is log10. Intensity of 120 dB is at the threshold of pain. Prolonged
exposure to sound over 85dB may cause permanent hearing loss.
Example: The average choir student produces intensity I = 10-5 W/m2 at the podium. What is
the intensity level perceived by the choir director from the 32 students in his class?
 (32)(10 5 W / m 2 ) 
I
  10 log  10 log 
  85dB
12
2
I0
 10 W / m

Note: If half of the students stop singing the intensity level only drops 82dB. For every
doubling of intensity, intensity level increases 3dB.
Music
An octave is a doubling of frequencies. Various scales have been used throughout space and
time to produce musical sounds pleasing to the ear. The equally tempered chromatic scale
currently in use divides each octave into 12 tones each 12 2 times the frequency of the preceding
tone. The A in the middle of the piano is at frequency 440Hz.
We previously discussed strings. We found the frequency of the wave on the string to be
L
f
inversely proportional to L, the length of the string. We write 1  2 ; shorter strings produce
L 2 f1
higher frequencies.
Chapter 12
LECTURE NOTES
Note: The wave on the string produces the sound we hear – both must have the same frequency.
However, the velocity of sound in air will differ from the wave velocity on the string and the
wavelength of the sound in air will differ from the wavelength of the waves on the string.
Many musical instruments are based on resonance in pipes or tubes. If both ends of a tube are
open to the air, the tube is open. If one end of the tube is closed, the tube is closed. (Tubes with
both ends closed are not considered as they emit no sound.
Air can rush in and out of an open tube end; this is a displacement antinode (and a pressure
node).
In a closed tube, the fundamental tone is produced when a displacement node is at the open end
1
and a displacement antinode at the closed end. Thus the length of the tube L =  or
4
v
v
1  4 L and f1  
where v is the speed of sound in air. All overtones resonate with this
 4L
3
4
3v
pattern of node and antinode; L   2 or  2  L and f 2 
is the second harmonic. In
4
3
4L
nv
general, fn =
;n = 1, 3, 5, . . for closed tube.
4L
An open tube has a displacement antinode at each (open) tube end with at least 1 node between.

v
nv
The fundamental tone has L  or  1  2L and f 1 
. In general, fn =
; n = 1, 2, 3, . . for
2L
2L
2
open tube. Corrections to these formulas can be made to account for the tube diameter.
Beats
If two sound waves interfere, we may hear an average frequency getting louder, then softer. If
frequencies of 881Hz and 883Hz are heard together, we perceive a tone of 882Hz getting louder
and softer (beating) 2 times per second. The beat frequency = fhigher – flower. This can be used to
tune musical instruments. If the frequencies are too different, we do not hear beats; they blur
together.
Sound waves in general exhibit constructive and destructive interferences. Some concert hall
seats are ‘dead’. This is more evident when sounds of a single frequency interfere.
Doppler Effect
The frequency an observer hears depends on the frequency emitted by the source of sound and
the relative motion between source and observer. If an observer moves toward the source, they
encounter the sound waves more frequently than if they stood still; they hear a higher frequency.
Chapter 12
LECTURE NOTES
Let
Then f1 = f
v = velocity of sound
vs = velocity of source of sound
v0 = velocity of observer
f = Source frequency
f1 = frequency perceived by observer
v  v0
where the upper sign is used for approach and the lower sign for recession.
v  vs
Example: You drive at 30m/s toward an oboe at rest playing A(440Hz). What frequency do you
hear if the velocity of sound is 343 m/s?
m
m

 343  30.0 
s
s  = 478Hz
f1 = 440Hz 
m


343


s


This is above A# and is clearly heard to be different than A.
Shock Waves
If a source travels through a medium faster than the speed of sound in that medium, a shock
wave is produced. This is responsible for the sonic booms heard from supersonic aircraft.
The shock wave is a cone extending from the plane; the half angle  of the cone is given by
v
1
sin   sound 
.
v object Mach number
The Mach number is vobject/vsound; Mach 2 is twice the speed of sound.