Chapter 17
Waves (II)
Sound Waves
Key contents:
Sound Waves
Speed of Sound
Pressure Fluctuation in Sound Waves
Interference
Intensity and Sound Level
Sources of Musical Sound
Beats
Doppler Effect
Supersonic Speeds, Shock Waves
17.1 Sound Waves
Sound waves are longitudinal waves of density
and pressure fluctuations.
# In this chapter, we will focus on sound waves in the air.
# For human ears, audible in 20 Hz ~ 20000 Hz
17.2 Speed of Sound
 The wave speed on a string is
 The bulk modulus B is defined to be the elastic factor for a gas:
 With the density r being the inertia factor, the sound speed can be
most likely expressed as
17.2 Speed of Sound
A derivation:
We have:
Also,
And
Therefore,
But
Finally,
17.3 Pressure Fluctuation in Sound Waves
17.3 Pressure Fluctuation in Sound Waves
}
# Δs is the displacement
change at the two sides of
the element Δx, while Δp is
the pressure excess in Δx.
# Δp and s are always with a
90o phase difference.
¶Dp
¶2 s
¶2 s
= -r 2 = -B 2
¶x
¶t
¶x
# From
one may also see that
v=
B
r
,
Example, Pressure and Displacement Amplitudes
17.4 Interference
17.4 Interference
Phase difference f can be related to path length difference DL, by noting that a phase
difference of 2p rad corresponds to one wavelength.
Therefore,
Fully constructive interference occurs when f is zero, 2p, or any integer multiple
of 2p.
Fully destructive interference occurs when f is an odd multiple of p:
Example, Interference:
Example, Interference:
17.5 Intensity and Sound Level
The intensity I of a sound wave at a surface is the average rate
per unit area at which energy is transferred by the wave through
or onto the surface.
Therefore, I =P/A where P is the time rate of energy transfer (the
power) of the sound wave and A is the area of the surface
intercepting the sound.
The intensity I is related to the displacement amplitude sm of the
sound wave by
17.5 Intensity and Sound Level
Consider a thin slice of air of
thickness dx, area A, and mass dm,
oscillating back and forth as the
sound wave passes through it.
The kinetic energy dK of the slice
of air is
But,
Therefore,
And,
Then the average rate at which
kinetic energy is transported is
If the potential energy is carried along with
the wave at this same average rate, then the
wave intensity I, the average rate per unit
area at which energy of both kinds is
transmitted by the wave, is
17.5 Intensity and Sound Level
Another derivation:
the instantaneous power supplied to the wave element in the
following figure is
(p is the pressure excess)
s
s(x, t) = s0 sin(kx - w t)

1
2
Pavg = r Av(w s0 )
2
(recall
)
This should be compared with that on a string:
17.5 Intensity and Sound Level: Variation with Distance
17.5 Intensity and Sound Level: The Decibel Scale
Ex. For sound waves at 1 kHz in the air
Here dB is the abbreviation for
decibel, the unit of sound level.
(density is 1.29 kg/m3, sound speed 340 m/s)
I0 is a standard reference intensity
( 10-12 W/m2), chosen near the lower
limit of the human range of hearing.
@ 0 dB, s0 = 1.07 x 10-11 m
p0 = 2.96 x 10-5 Pa
(1 atm = 1.013 x 105 Pa)
For I =I0 , b =10 log 1 = 0,
(our standard reference level
corresponds to zero decibels).
@ 120 dB, s0 = 1.07 x 10-5 m
p0 = 29.6 Pa
# office conversation, 60 dB
heavy traffic (3m), 80 dB
loud rock music, 120 dB
Jet engine (20m), 130 dB
Example, Cylindrical Sound Wave:
Example, Decibel, Sound Level, Change in Intensity:
Many veteran rockers suffer from acute hearing
damage because of the high sound levels they
endured for years while playing music near
loudspeakers or listening to music on headphones.
Recently, many rockers, began wearing special
earplugs to protect their hearing during
performances. If an earplug decreases the sound
level of the sound waves by 20 dB, what is the ratio
of the final intensity If of the waves to their initial
intensity Ii?
17.6 Sources of Musical Sound
Musical sounds can be set up
by oscillating strings (guitar,
piano, violin), membranes
(kettledrum, snare drum), air
columns (flute, oboe, pipe
organ, and the digeridoo of
Fig.17-12), wooden blocks or
steel bars (marimba,
xylophone), and many other
oscillating bodies. Most
common instruments involve
more than a single oscillating
part.
17.6 Sources of Musical Sound
A. Pipe open at both ends
B. Pipe open at one end only
Example, Double Open and Single Open Pipes:
17.7 Beats
When two sound waves whose frequencies
are close, but not the same, are superimposed,
a striking variation in the intensity of the
resultant sound wave is heard. This is the beat
phenomenon. The wavering of intensity
occurs at a frequency which is the difference
between the two combining frequencies.
Example, Beat Frequencies:
17.8 Doppler Effect
When the motion of detector or
source is toward the other, the sign
on its speed must give an upward
shift in frequency. When the motion
of detector or source is away from
the other, the sign on its speed must
give a downward shift in frequency.
Here the emitted frequency is f, the
detected frequency is f’, v is the
speed of sound through the air, vD
is the detector’s speed relative to the
air, and vS is the source’s speed
relative to the air.
Example, Doppler Shift:
17.9 Supersonic Speeds, Shock Waves
Homework:
Problems 14, 22, 36, 49, 62