Transcript Chapter 17

Chapter 17
Sound Waves
17.1 Pressure variation in Sound Waves
17.2 speed of Sound Waves
17.3 Intensity of Periodic Sound Waves
17.4 The Doppler Effect
21 July 2015
W7
1
Objectives
1- Define the Speed of Sound Waves in different
media.
2- Study Periodic Sound Waves as a type of
longitudinal waves.
3- Derive the equation of Intensity of Periodic
Sound Waves
4- Study the phenomena of Doppler Effect
5- Apply the gained knowledge to understand the
idea of sound recording.
2
W7
21 July 2015
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.
20 Hz to 2o kHz
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
fall 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.
 Assignment
 Proof the equation
DP = DPmax sin (kx – wt)
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.
s (x, t) = smax cos (kx – wt)
DP = DPmax sin (kx – wt)
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.
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
 Assignment
 Proof the relation
v
B
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
v  (331 m/s) 1 
TC
273
 The 331 m/s is the speed at 0o C.
 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.
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.
DPmax 

I
2
2 rv
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
 In terms of the pressure amplitude,
Section 17.3
I
Power avg
A
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 
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 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
Assignment
Solve the following problems:
1, 3, 5, 7, 10, 11, 15, 17, 21, 24, 25, 28,
33, 35, 37, 39,