Transcript Chapter 17

Chapter 17
Sound Waves
Introduction to Sound Waves
Waves can move through three-dimensional bulk media.
Sound waves are longitudinal waves.
They travel through any material medium.
 Commonly experienced as the mechanical waves traveling through air that
result in the human perception of hearing
 As the sound wave travels through the air, elements of air are disturbed from
their equilibrium positions.
 Accompanying these movements are changes in density and pressure of the
air.
The mathematical description of sinusoidal sound waves is very similar to
sinusoidal waves on a string.
Introduction
Categories of Sound Waves
The categories cover different frequency ranges.
Audible waves are within the sensitivity of the human ear.
Infrasonic waves have frequencies below the audible range.
Ultrasonic waves have frequencies above the audible range.
Introduction
Speed of Sound Waves
The diagram shows the motion of a
one-dimensional longitudinal sound
pulse moving through a long tube
containing a compressible gas.
The piston on the left end can be
quickly moved to the right to compress
the gas and create the pulse.
Before the piston is moved, the gas has
uniform density.
When the piston is suddenly moved to
the right, the gas just in front of it is
compressed.
 Darker region in b
 The pressure and density in this region are
higher than before the piston was pushed.
Section 17.1
Speed of Sound Waves, cont
When the piston comes to rest, the
compression region of the gas
continues to move.
 This corresponds to a longitudinal
pulse traveling through the tube
with speed v.
Section 17.1
Producing a Periodic Sound Wave
A one-dimensional periodic sound wave
can be produced by causing the piston
to move in simple harmonic motion.
The darker parts of the areas in the
figures represent areas where the gas
is compressed and the density and
pressure are above their equilibrium
values.
The compressed region is called a
compression.
Section 17.1
Producing a Periodic Sound Wave, cont.
When the piston is pulled back, the gas in front of it expands and the pressure
and density in this region ball below their equilibrium values.
The low-pressure regions are called rarefactions.
They also propagate along the tube, following the compressions.
Both regions move at the speed of sound in the medium.
The distance between two successive compressions (or rarefactions) is the
wavelength.
Section 17.1
Periodic Sound Waves, Displacement
As the regions travel through the tube, any small element of the medium moves
with simple harmonic motion parallel to the direction of the wave.
The harmonic position function is
s (x, t) = smax cos (kx – wt)
 smax is the maximum position of the element relative to equilibrium.
 This is also called the displacement amplitude of the wave.
 k is the wave number.
 ω is the angular frequency of the wave.
 Note the displacement of the element is along x, in the direction of the sound
wave.
Section 17.1
Periodic Sound Waves, Pressure
The variation in gas pressure, DP, is also periodic.
DP = DPmax sin (kx – wt)
 DPmax is the pressure amplitude.
 It is the maximum change in pressure from the equilibrium value.
 k is the wave number.
 w is the angular frequency.
The pressure can be related to the displacement.
 This relationship is given by DPmax = B smax k.
 B is the bulk modulus of the material.
Section 17.1
Periodic Sound Waves, final
A sound wave may be considered
either a displacement wave or a
pressure wave.
The pressure wave is 90o out of phase
with the displacement wave.
 The pressure is a maximum when
the displacement is zero, etc.
Section 17.1
Speed of Sound in a Gas
Consider an element of the gas
between the piston and the dashed line.
Initially, this element is in equilibrium
under the influence of forces of equal
magnitude.
 There is a force from the piston on
left.
 There is another force from the rest
of the gas.
 These forces have equal
magnitudes of PA.
 P is the pressure of the gas.
 A is the cross-sectional area of the tube.
Section 17.2
Speed of Sound in a Gas, cont.
After a time period, Δt, the piston has
moved to the right at a constant speed
vx.
The force has increased from PA to
(P+ΔP)A.
The gas to the right of the element is
undisturbed since the sound wave has
not reached it yet.
Section 17.2
Impulse and Momentum
The element of gas is modeled as a non-isolated system in terms of momentum.
The force from the piston has provided an impulse to the element, which
produces a change in momentum.
The impulse is provided by the constant force due to the increased pressure:
I  FDt   A DP Dt  ˆi
The change in pressure can be related to the volume change and the bulk
modulus:
v
DV
DP  B
B x
V
v
Therefore, the impulse is
v


I   AB x Dt  ˆi
v


Section 17.2
Impulse and Momentum, cont.
The change in momentum of the element of gas of mass m is
Dp  mDv   pvv x ADt  ˆi
Setting the impulse side of the equation equal to the momentum side and
simplifying, the speed of sound in a gas becomes.
v
B
r
 The bulk modulus of the material is B.
 The density of the material is r
Section 17.2
Speed of Sound Waves, General
The speed of sound waves in a medium depends on the compressibility and the
density of the medium.
The compressibility can sometimes be expressed in terms of the elastic modulus
of the material.
The speed of all mechanical waves follows a general form:
v
elastic property
inertial property
For a solid rod, the speed of sound depends on Young’s modulus and the density
of the material.
Section 17.2
Speed of Sound in Air
The speed of sound also depends on the temperature of the medium.
 This is particularly important with gases.
For air, the relationship between the speed and temperature is
TC
273
 The 331 m/s is the speed at 0o C.
v  (331 m/s) 1 
 TC is the air temperature in Celsius.
Section 17.2
Relationship Between Pressure and Displacement
The pressure amplitude and the displacement amplitude are related by
ΔPmax = B smax k
The bulk modulus is generally not as readily available as the density of the gas.
By using the equation for the speed of sound, the relationship between the
pressure amplitude and the displacement amplitude for a sound wave can be
found.
ΔPmax = ρ v ω smax
Section 17.2
Speed of Sound in Gases, Example Values
Section 17.2
Energy of Periodic Sound Waves
Consider an element of air with mass
Dm and length Dx.
Model the element as a particle on
which the piston is doing work.
The piston transmits energy to the
element of air in the tube.
This energy is propagated away from
the piston by the sound wave.
Section 17.3
Power of a Periodic Sound Wave
The rate of energy transfer is the power of the wave.
Power  F v x
 Power avg 
1
r Avw 2smax 2
2
The average power is over one period of the oscillation.
Section 17.3
Intensity of a Periodic Sound Wave
The intensity, I, of a wave is defined as the power per unit area.
 This is the rate at which the energy being transported by the wave transfers
through a unit area, A, perpendicular to the direction of the wave.
I
Power avg
A
In the case of our example wave in air,
I = ½ rv(wsmax)2
Therefore, the intensity of a periodic sound wave is proportional to the
 Square of the displacement amplitude
 Square of the angular frequency
Section 17.3
Intensity, cont.
In terms of the pressure amplitude,
DPmax 

I
2
2 rv
Section 17.3
A Point Source
A point source will emit sound waves
equally in all directions.
 This can result in a spherical
wave.
This can be represented as a series of
circular arcs concentric with the source.
Each surface of constant phase is a
wave front.
The radial distance between adjacent
wave fronts that have the same phase
is the wavelength λ of the wave.
Radial lines pointing outward from the
source, representing the direction of
propagation, are called rays.
Section 17.3
Intensity of a Point Source
The power will be distributed equally through the area of the sphere.
The wave intensity at a distance r from the source is
I
Power avg Power avg
A

4 r 2
This is an inverse-square law.
 The intensity decreases in proportion to the square of the distance from the
source.
Section 17.3
Sound Level
The range of intensities detectible by the human ear is very large.
It is convenient to use a logarithmic scale to determine the intensity level, b
I 
b  10log  
 Io 
Section 17.3
Sound Level, cont
I0 is called the reference intensity.
 It is taken to be the threshold of hearing.
 I0 = 1.00 x 10-12 W/ m2
 I is the intensity of the sound whose level is to be determined.
b is in decibels (dB)
Threshold of pain: I = 1.00 W/m2; b = 120 dB
Threshold of hearing: I0 = 1.00 x 10-12 W/ m2 corresponds to b = 0 dB
Section 17.3
Sound Level, Example
What is the sound level that corresponds to an intensity of 2.0 x 10-7 W/m2 ?
b = 10 log (2.0 x 10-7 W/m2 / 1.0 x 10-12 W/m2)
= 10 log 2.0 x 105
= 53 dB
Rule of thumb: A doubling in the loudness is approximately equivalent to an
increase of 10 dB.
Section 17.3
Sound Levels
Section 17.3
Loudness and Frequency
Sound level in decibels relates to a physical measurement of the strength of a
sound.
We can also describe a psychological “measurement” of the strength of a sound.
Our bodies “calibrate” a sound by comparing it to a reference sound.
This would be the threshold of hearing.
Actually, the threshold of hearing is 10-12 W/m2 only for 1000 Hz.
There is a complex relationship between loudness and frequency.
Fig. 17.7 shows this relationship:
 The white area shows average human response to sound.
 The lower curve of the white area shows the threshold of hearing.
 The upper curve shows the threshold of pain.
Section 17.3
Loudness and Frequency, cont.
Section 17.3
The Doppler Effect
The Doppler effect is the apparent
change in frequency (or wavelength)
that occurs because of motion of the
source or observer of a wave.
 When the relative speed of the
source and observer is higher than
the speed of the wave, the
frequency appears to increase.
 When the relative speed of the
source and observer is lower than
the speed of the wave, the
frequency appears to decrease.
Section 17.4
Doppler Effect, Observer Moving
The observer moves with a speed of vo.
Assume a point source that remains
stationary relative to the air.
It is convenient to represent the waves
as wave fronts.
 These surfaces are called wave
fronts.
 The distance between adjacent
wave fronts is the wavelength.
Section 17.4
Doppler Effect, Observer Moving, cont
The speed of the sound is v, the frequency is ƒ, and the wavelength is l
When the observer moves toward the source, the speed of the waves relative to
the observer is v ’ = v + vo.
 The wavelength is unchanged.
The frequency heard by the observer, ƒ ’, appears higher when the observer
approaches the source.
 v  vo
ƒ'  
 v

ƒ

The frequency heard by the observer, ƒ ’, appears lower when the observer
moves away from the source.
 v  vo
ƒ'  
 v

ƒ

Section 17.4
Doppler Effect, Source Moving
Consider the source being in motion
while the observer is at rest.
As the source moves toward the
observer, the wavelength appears
shorter.
As the source moves away, the
wavelength appears longer.
Section 17.4
Doppler Effect, Source Moving, cont
When the source is moving toward the observer, the apparent frequency is
higher.
 v 
ƒ'  
ƒ
v

v
s 

When the source is moving away from the observer, the apparent frequency is
lower.
 v 
ƒ'  
ƒ
v

v
s 

Section 17.4
Doppler Effect, General
Combining the motions of the observer and the source
 v  vo 
ƒ'  
ƒ
v

v
s 

The signs depend on the direction of the velocity.
 A positive value is used for motion of the observer or the source toward the
other.
 A negative sign is used for motion of one away from the other.
Section 17.4
Doppler Effect, final
Convenient rule for signs.
 The word “toward” is associated with an increase in the observed frequency.
 The words “away from” are associated with a decrease in the observed
frequency.
The Doppler effect is common to all waves.
The Doppler effect does not depend on distance.
Section 17.4
Doppler Effect, Water Example
A point source is moving to the right .
The wave fronts are closer on the right.
The wave fronts are farther apart on the
left.
Section 17.4
Doppler Effect, Submarine Example
Sub A (source) travels at 8.00 m/s emitting at a frequency of 1400 Hz.
The speed of sound in the water is 1533 m/s.
Sub B (observer) travels at 9.00 m/s.
What is the apparent frequency heard by the observer as the subs approach
each other? Then as they recede from each other?
Section 17.4
Doppler Effect, Submarine Example cont.
Approaching each other:
 1533 m s   9.00 m s  
 v  vo 
ƒ'  
 (1400 Hz )
 ƒ  
v

v
1533
m
s


8.00
m
s


s 



 1416 Hz
Receding from each other:
 1533 m s   9.00 m s  
 v  vo 
ƒ'  
ƒ


 (1400 Hz )

 v  vs 
 1533 m s   8.00 m s  
 1385 Hz
Section 17.4
Shock Waves and Mach Number
The speed of the source can exceed
the speed of the wave.
The envelope of these wave fronts is a
cone whose apex half-angle is given by
sin q  v/vS.
 This is called the Mach angle.
The ratio vs / v is referred to as the
Mach number .
The relationship between the Mach
angle and the Mach number is
sinq 
vt
v

vst vs
Section 17.4
Shock Wave, final
The conical wave front produced when
vs > v is known as a shock wave.
 This is a supersonic speed.
The shock wave carries a great deal of
energy concentrated on the surface of
the cone.
There are correspondingly great
pressure variations.
Section 17.4