7.10
3. State the beat frequency when the following pairs of frequencies
are heard together:
(a) 202 Hz, 200 Hz
(b) 341 Hz, 347 Hz
(c) 1003 Hz, 998 Hz
4. Use the principle of superposition to “add” these two sound
waves together on a piece of graph paper turned sideways:
Wave A: l = 4.0 cm; A = 1.0 cm; 5 wavelengths
Wave B: l = 5.0 cm; A = 1.0 cm; 4 wavelengths
Describe how the resulting pattern relates to the production of
beats.
5. A tuning fork with a frequency of 4.0 × 102 Hz is struck with a
second fork, and 20 beats are counted in 5.0 s. What are the possible frequencies of the second fork?
6. A third fork with a frequency of 410 Hz is struck with the second
fork in question 5, and 18 beats are counted in 3.0 s. What is the
frequency of the second fork?
7. A 440-Hz tuning fork is sounded together with a guitar string, and
a beat frequency of 3 Hz is heard. When an elastic band is
wrapped tightly around one prong of the tuning fork, a new beat
frequency of 2 Hz is heard. Determine the frequency of the guitar
string.
Making Connections
8. How would a piano tuner use a tuning fork or pitch pipe to tune
a piano by adjusting the tension of the strings?
7.10
The Doppler Effect and
Supersonic Travel
If you’ve ever been to an automobile race, you probably noticed that when a
racing car streaks past you, you can detect a change in frequency of the sound
from the car. As the car approaches, the sound becomes higher in frequency. At
the instant the car passes you, the frequency drops noticeably. The apparent
changing frequency of sound in relation to an object’s motion is called the
Doppler effect, named after Christian Doppler (1803–53), an Austrian physicist
and mathematician who first analyzed the phenomenon. You’ve probably heard
this effect from train whistles, car horns, or sirens on fire trucks, ambulances, or
police cruisers.
To understand why the Doppler effect occurs, look at Figure 1. Figure 1(a)
shows sound waves travelling outward from a stationary source. Figure 1(b)
shows the source of sound waves travelling to the left. As the waves approach
observer A, they are closer together than they would be if the source were not
moving. Thus, observer A hears a sound of higher frequency. Observer B, however, hears a sound of lower frequency because the source is travelling away, producing sounds of longer wavelength. A similar effect occurs when the source of
sound is stationary and the observer is moving toward or away from it.
Doppler effect: when a source of sound
approaches an observer, the observed
frequency increases; when the source
moves away from an observer, the observed
frequency decreases
Properties of Sound Waves 267
(a)
source of periodic waves
A
B A
Figure 1
The Doppler effect
(a) The source is stationary. Both observers A
and B hear the same frequency of sound.
(b) The source is moving to the left. Observer
A hears a higher frequency and Observer
B hears a lower frequency.
shorter wavelength
higher frequency
(b)
B
longer wavelength
lower frequency
It can be shown that the following relationship describes the effect on frequency. When a source either moves toward or away from a stationary observer,
v
f2 = f1 v ± vs
where v is the speed of sound in the medium and vs is the speed of the source
through the medium. The denominator v + vs is used if the source is moving away
from the observer, the denominator v – vs is used if the source is moving toward
the observer.
Sample Problem 1
A car travelling at 100.0 km/h sounds its horn as it approaches a hiker standing
on the highway. If the car’s horn has a frequency of 440 Hz and the temperature
of the air is 0°C, what is the frequency of the sound waves reaching the hiker
(a) as the car approaches?
(b) after it has passed the hiker?
Solution
t = 0°C
v = 332 m/s
vs = 100.0 km/h or 27.8 m/s
f2 = ?
(a)
v
f2 = f1 v ± vs
332 m/s
= 440 Hz 332 m/s – 27.8 m/s
f2 = 4.8 ×102 Hz
The frequency as the car approaches is 4.8 × 102 Hz.
268 Chapter 7
7.10
(b)
v
f2 = f1 v ± vs
332 m/s
= 440 Hz 332 m/s + 27.8 m/s
f2 = 4.0 ×
102
Hz
The frequency after the car passes the hiker is 4.0 × 102 Hz.
Practice
Understanding Concepts
1. You are standing at a railway crossing. A train approaching at
125 km/h sounds its whistle. If the frequency of the whistle is 442 Hz
and the air temperature is 20.0°C, what frequency do you hear when
the train approaches you? when the train has passed by you?
Answers
1. 492 Hz; 401 Hz
2. 13 m/s
2. A car sounds its horn (502 MHz) as it approaches a pedestrian by the
side of the road. The pedestrian has perfect pitch and determines that
the sound from the horn has a frequency of 520 Hz. If the speed of
sound that day was 340 m/s, how fast was the car travelling?
Doppler Shift
Although the Doppler effect was first explained in relation to sound waves, it
may be observed in any moving object that emits waves. The change in frequency
and resulting change in wavelength is called the Doppler shift. Astronomers use
the Doppler effect of light waves to estimate the speed of distant stars and
galaxies relative to that of our solar system.
Short-range radar devices, such as those used by the police, work on the
Doppler shift principle to determine the speed of a car (Figure 2). Radar waves
from a transmitter in the police car are reflected by an approaching car and arrive
back at a radar receiver in the police car with a slightly higher frequency. The
original waves and the reflected waves are very close together in frequency, and
beats are produced when the two are combined. The number of beats per second
is directly related to the speed of the approaching car. This beat frequency is electronically translated into kilometres per hour and displayed on a meter or paper
chart in the police car. Radar devices automatically correct for the movement of
the police car. Similar techniques are used to track the path of satellites circling
the Earth, to track weather systems, in ultrasonic and infrared detectors used in
home security systems, and to measure the speed of a pitched baseball.
(a)
police cruiser
(b)
police cruiser
Figure 2
The Doppler effect is applied in the use of
radar to determine the speed of vehicles on a
highway. The radar system in a moving police
cruiser can determine the speed of a car
ahead or behind, travelling in (a) the same
direction or (b) in opposite directions.
Properties of Sound Waves 269
Supersonic Travel
subsonic speed: speed less than the
speed of sound in air
Mach number: the ratio of the speed of
an object to the speed of sound in air;
speed of object
Mach number = speed of sound
supersonic speed: speed greater than
the speed of sound in air
Objects travelling at speeds less than the speed of sound in air have subsonic
speeds. When the speed of an object equals the speed of sound in air at that location, the speed is called Mach 1. The Mach number of a source of sound is the
ratio of the speed of the source to the speed of sound in air at that location. Thus,
at 0°C near the surface of Earth, Mach 2 is 2 × 332 m/s = 664 m/s. Speeds greater
than Mach 1 are supersonic. Speeds for supersonic aircraft, such as the Concorde
(Figure 3) and fighter aircraft, are given in terms of Mach number rather than
kilometres per hour.
Sample Problem 2
The speed of sound at sea level and 0°C is 332 m/s, or approximately 1200 km/h.
At an altitude of 10 km, it is approximately 1060 km/h. What is the Mach number
of an aircraft flying at an altitude of 10 km with a speed of 1800 km/h?
Solution
Figure 3
A Concorde passenger airplane
speed of object
Mach number = speed of sound
1800 km/h
= 1060 km/h
Mach number = 1.7
The Mach number of an aircraft flying at 1060 km/h is 1.7.
Practice
Understanding Concepts
Answers
3. (a) 1.2
(b) 0.77
4. 2.4 × 103 km/h
5. (a) 3.0
(b) 0.50
(c) 1.5
DID YOU KNOW ?
Mach Number
This ratio is named after Ernst Mach
(1838–1916), an Austrian physicist and
philosopher.
sound barrier: a high-pressure region
produced as an airplane approaches a speed
of Mach 1
270 Chapter 7
3. What is the Mach number of an aircraft travelling at sea level at 0°C
with a speed of
(a) 1440 km/h?
(b) 920 km/h?
4. A military interceptor airplane can fly at Mach 2.0. What is its speed
in kilometres per hour at sea level and at 0°C?
5. What is the Mach number of a plane travelling at each of the
following speeds at sea level in air with a temperature of 12°C?
(a) 1020 m/s
(b) 170 m/s
(c) 1836 km/h
Breaking the Sound Barrier
A static, or stationary, source radiates sound waves in concentric spheres (Figure
4(a)). An airplane radiates spheres of sound waves from successive positions.
Sphere 1 in Figure 4(b) was produced by a subsonic airplane at position 1, sphere 2
at position 2, and so on. Note that because the aircraft was moving, the wavefronts were farther apart behind it than they were in front of it.
When an airplane is flying at the speed of sound, the wavefronts in front of
the airplane pile up, producing an area of very dense air, or intense compression,
called the sound barrier. To exceed the speed of sound, extra thrust is needed
until the aircraft “breaks through” the sound barrier. Unless the aircraft has been
designed to cut through this giant compression, it will be buffeted disastrously.
Only specially constructed aircraft can withstand the vibrations caused in
7.10
(a)
(b)
(c)
4
4
4
3
3
2
1
4
4
1
2
2
1
2
3
3
2
3
1
1
Figure 4
(a) Stationary source
(b) Subsonic source
(c) Supersonic source
breaking through the sound barrier to reach supersonic speeds. In present-day
supersonic aircraft, such as the Concorde, only slight vibration is noticed when
the sound barrier is crossed.
At supersonic speeds, the spheres of sound waves are left behind the aircraft
(Figure 4(c)). These sound waves interfere with one another constructively, producing large compressions and rarefactions along the sides of an invisible double
cone extending behind the airplane, from the front, and from the rear. This
intense acoustic pressure wave sweeps along the ground (Figure 5) in a swath
having a width of approximately five times the altitude of the aircraft. This is
DID YOU KNOW ?
Record-Breaking Speed
The record high speed of a fixed-wing aircraft
is approximately Mach 25. It is held by the
space shuttle Columbia and was achieved
where the speed of sound was about
300 m/s.
noise cone
zone where noise is heard
Figure 5
The acoustic pressure waves trail behind a
supersonic aircraft.
Properties of Sound Waves 271
sonic boom: an explosive sound that
radiates from an aircraft travelling at
supersonic speeds
DID YOU KNOW ?
Cracking the Whip
When a circus or rodeo performer snaps a
whip quickly enough, a loud cracking sound is
produced. This sound is actually a small sonic
boom, created when the tip of the whip
moves through the air faster than the speed
of sound.
usually referred to as a sonic boom. The sonic boom is heard as two sharp cracks,
like thunder or a muffled explosion.
For an airplane flying faster than the speed of sound at a height of 12 km,
the sonic boom is produced for 30 km on either side of the flight path. Unless it
comes from a supersonic aircraft at a low altitude, the sonic pressure wave is not
strong enough to cause any damage on the ground, although the sudden noise
may startle or frighten human beings and animals.
It is believed that most ecosystems can tolerate random sonic booms.
Recurring booms over a long period, on the other hand, might upset them.
Supersonic commercial aircraft, as a result, are restricted by many countries to
subsonic speeds except over water. For example, the Concorde flying from
London to New York does not go faster than the speed of sound until it is well
over the ocean, and it reduces speed below Mach 1 when it approaches land.
Most of the training of fighter pilots occurs in areas where there are few inhabitants or over water. Aside from the annoyance and distraction factor, the
breaking of the sound barrier can also break windows!
SUMMARY
The Doppler Effect and Supersonic Travel
• Doppler shift is the perceived frequency shift when a source of sound
moves relative to an observer.
• Doppler shift is used to measure the speed of moving objects.
• The Mach number is used to measure supersonic speeds.
• Objects exceeding the speed of sound break the sound barrier, creating
sonic booms.
• Sonic booms create disturbances for animals and humans.
Section 7.10 Questions
Understanding Concepts
1. State what happens to the apparent frequency of a sound source
in each of the following situations:
(a) The listener is stationary and the source is approaching.
(b) The listener is stationary and the source is receding.
(c) The source is stationary and the listener is approaching.
(d) The source is stationary and the listener is receding.
2. A car’s horn is pitched at 520 Hz. If the car travels by someone at
26 m/s, at what frequency will the person hear it as the car
moves away (speed of sound is 344 m/s)?
3. Assuming that the speed of sound at a certain altitude is 330 m/s,
calculate the speed of an airplane that is travelling at
(a) Mach 0.70
(b) Mach 4.2
4. A plane travelling at 260 m/s has a speed of Mach 0.80. What is
the speed of sound in the air?
5. Why is it difficult for an aircraft to break through the sound barrier?
Applying Inquiry Skills
6. You are travelling in a car near a speeding train. The train whistle
blows, but you fail to hear the Doppler effect. What conditions
might prevent you from hearing it?
272 Chapter 7