Transcript 16-7 Doppler Effect

Chapter 16
Sound
Copyright © 2009 Pearson Education, Inc.
Units of Chapter 16
Sections – 2, 4, 6, 7 only
• Mathematical Representation of Longitudinal
Waves
• Sources of Sound: Vibrating Strings
• Quality of Sound, and Noise; Superposition
• Interference of Sound Waves; Beats
• Doppler Effect
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16-2 Mathematical Representation of
Longitudinal Waves
Longitudinal waves are often
called pressure waves. The
displacement is 90° out of
phase with the pressure.
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16-2 Mathematical Representation of
Longitudinal Waves
By considering a small cylinder within the
fluid, we see that the change in pressure is
given by (B is the bulk modulus):
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16-2 Mathematical Representation of
Longitudinal Waves
If the displacement is sinusoidal, we have
where
and
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16-5 Quality of Sound, and Noise;
Superposition
So why does a trumpet
sound different from a
flute? The answer lies in
overtones—which ones are
present, and how strong
they are, makes a big
difference. The sound wave
is the superposition of the
fundamental and all the
harmonics.
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16-5 Quality of Sound, and Noise;
Superposition
This plot shows
frequency spectra for
a clarinet, a piano, and
a violin. The
differences in
overtone strength are
apparent.
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16-6 Interference of Sound Waves; Beats
Sound waves interfere in the
same way that other waves do
in space.
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16-6 Interference of Sound Waves; Beats
Example 16-12: Loudspeakers’
interference.
Two loudspeakers are 1.00 m apart. A
person stands 4.00 m from one speaker.
How far must this person be from the
second speaker to detect destructive
interference when the speakers emit an
1150-Hz sound? Assume the temperature
is 20°C.
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16-6 Interference of Sound Waves; Beats
Waves can also interfere in time, causing a
phenomenon called beats. Beats are the slow
“envelope” around two waves that are
relatively close in frequency.
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16-6 Interference of Sound Waves; Beats
If we consider two waves of the same
amplitude and phase, with different
frequencies, we can find the beat frequency
when we add them:
This represents a wave vibrating at the
average frequency, with an “envelope” at
the difference of the frequencies.
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16-6 Interference of Sound Waves; Beats
Example 16-13: Beats.
A tuning fork produces a steady 400-Hz
tone. When this tuning fork is struck and
held near a vibrating guitar string, twenty
beats are counted in five seconds. What are
the possible frequencies produced by the
guitar string?
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16-7 Doppler Effect
The Doppler effect occurs when a source of
sound is moving with respect to an observer.
A source moving toward an observer appears
to have a higher frequency and shorter
wavelength; a source moving away from an
observer appears to have a lower frequency
and longer wavelength.
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16-7 Doppler Effect
If we can figure
out what the
change in the
wavelength is, we
also know the
change in the
frequency.
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16-7 Doppler Effect
The change in the frequency is given by:
If the source is moving away from the
observer:
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16-7 Doppler Effect
If the observer is moving with respect to the
source, things are a bit different. The
wavelength remains the same, but the wave
speed is different for the observer.
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16-7 Doppler Effect
We find, for an observer moving toward a
stationary source:
And if the observer is moving away:
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16-7 Doppler Effect
Example 16-14: A moving siren.
The siren of a police car at rest emits at a
predominant frequency of 1600 Hz. What
frequency will you hear if you are at rest and
the police car moves at 25.0 m/s (a) toward
you, and (b) away from you?
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16-7 Doppler Effect
Example 16-15: Two
Doppler shifts.
A 5000-Hz sound wave is
emitted by a stationary
source. This sound wave
reflects from an object
moving toward the source.
What is the frequency of
the wave reflected by the
moving object as detected
by a detector at rest near
the source?
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16-7 Doppler Effect
All four equations for the Doppler effect
can be combined into one; you just have to
keep track of the signs!
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Summary of Chapter 16
• Sound is a longitudinal wave in a medium.
•The strings on stringed instruments produce a
fundamental tone whose wavelength is twice the
length of the string; there are also various
harmonics present.
• Sound waves exhibit interference; if two
sounds are at slightly different frequencies they
produce beats.
• The Doppler effect is the shift in frequency of a
sound due to motion of the source or the
observer.
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Chapter 31
Maxwell’s Equations and
Electromagnetic Waves
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Units of Chapter 31
• Changing Electric Fields Produce Magnetic
Fields; Ampère’s Law and Displacement
Current
• Gauss’s Law for Magnetism
• Maxwell’s Equations
• Production of Electromagnetic Waves
• Electromagnetic Waves, and Their Speed,
Derived from Maxwell’s Equations
• Light as an Electromagnetic Wave and the
Electromagnetic Spectrum
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Units of Chapter 31
• Measuring the Speed of Light
• Energy in EM Waves; the Poynting Vector
• Radiation Pressure
• Radio and Television; Wireless
Communication
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31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
Ampère’s law
relates the
magnetic field
around a current
to the current
through a
surface.
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31-1 Changing Electric Fields Produce
Magnetic Fields; Ampère’s Law and
Displacement Current
In order for Ampère’s
law to hold, it can’t
matter which surface
we choose. But look
at a discharging
capacitor; there is a
current through
surface 1 but none
through surface 2:
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31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
Therefore, Ampère’s law is modified to include
the creation of a magnetic field by a changing
electric field – the field between the plates of the
capacitor in this example:
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31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
Example 31-1: Charging capacitor.
A 30-pF air-gap capacitor has circular plates of area
A = 100 cm2. It is charged by a 70-V battery through a
2.0-Ω resistor. At the instant the battery is connected,
the electric field between the plates is changing most
rapidly. At this instant, calculate (a) the current into
the plates, and (b) the rate of change of electric field
between the plates. (c) Determine the magnetic field
induced between the plates. Assume E is uniform
E
between
the plates at any instant and is zero at all
points beyond the edges of the plates.
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31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
The second term in Ampere’s law has the
dimensions of a current (after factoring out
the μ0), and is sometimes called the
displacement current:
where
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31-2 Gauss’s Law for Magnetism
Gauss’s law relates the electric field on a
closed surface to the net charge enclosed
by that surface. The analogous law for
magnetic fields is different, as there are no
single magnetic point charges
(monopoles):
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31-3 Maxwell’s Equations
We now have a complete set of equations
that describe electric and magnetic fields,
called Maxwell’s equations. In the absence of
dielectric or magnetic materials, they are:
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31-4 Production of Electromagnetic
Waves
Since a changing electric field produces
a magnetic field, and a changing
magnetic field produces an electric field,
once sinusoidal fields are created they
can propagate on their own.
These propagating fields are called
electromagnetic waves.
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31-4 Production of Electromagnetic
Waves
Oscillating charges
will produce
electromagnetic
waves:
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31-4 Production of Electromagnetic
Waves
Close to the antenna,
the fields are
complicated, and are
called the near field:
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31-4 Production of Electromagnetic
Waves
Far from the source, the waves
are plane waves:
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31-4 Production of Electromagnetic
Waves
The electric and magnetic waves are
perpendicular to each other, and to the
direction of propagation.
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
In the absence of currents and charges,
Maxwell’s equations become:
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
This figure shows an electromagnetic wave of
wavelength λ and frequency f. The electric and
magnetic fields are given by
.
where
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
Applying Faraday’s law to the rectangle of
height Δy and width dx in the previous figure
gives a relationship between E and B:
.
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
Similarly, we apply
Maxwell’s fourth
equation to the
rectangle of length Δz
and width dx, which
gives
.
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
Using these two equations and the
equations for B and E as a function of time
gives
.
Here, v is the velocity of the wave.
Substituting,
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
The magnitude of this speed is
3.0 x 108 m/s – precisely equal
to the measured speed of light.
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
Example 31-2: Determining E and B in EM
waves.
Assume a 60-Hz EM wave is a sinusoidal
wave propagating in the z direction with E
pointing in the x direction, and E0 = 2.0 V/m.
Write vector expressions for E and B as
functions of position and time.
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31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
The frequency of an electromagnetic wave
is related to its wavelength and to the
speed of light:
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31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
Electromagnetic waves can have any
wavelength; we have given different names to
different parts of the wavelength spectrum.
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31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
Example 31-3: Wavelengths of EM waves.
Calculate the wavelength
(a) of a 60-Hz EM wave,
(b) of a 93.3-MHz FM radio wave, and
(c) of a beam of visible red light from a
laser at frequency 4.74 x 1014 Hz.
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31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
Example 31-4: Cell phone antenna.
The antenna of a cell phone is often ¼
wavelength long. A particular cell phone has
an 8.5-cm-long straight rod for its antenna.
Estimate the operating frequency of this
phone.
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31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
Example 31-5: Phone call time lag.
You make a telephone call from New York
to a friend in London. Estimate how long it
will take the electrical signal generated by
your voice to reach London, assuming the
signal is (a) carried on a telephone cable
under the Atlantic Ocean, and (b) sent via
satellite 36,000 km above the ocean.
Would this cause a noticeable delay in
either case?
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31-7 Measuring the Speed of Light
The speed of light
was known to be
very large,
although careful
studies of the
orbits of Jupiter’s
moons showed
that it is finite.
One important
measurement, by
Michelson, used a
rotating mirror:
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31-7 Measuring the Speed of Light
Over the years, measurements have become
more and more precise; now the speed of light
is defined to be
c = 2.99792458 × 108 m/s.
This is then used to define the meter.
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31-8 Energy in EM Waves; the
Poynting Vector
Energy is stored in both electric and magnetic
fields, giving the total energy density of an
electromagnetic wave:
Each field contributes half the total energy
density:
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31-8 Energy in EM Waves; the
Poynting Vector
This energy is
transported by
the wave.
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31-8 Energy in EM Waves; the
Poynting Vector
The energy transported through a unit area
per unit time is called the intensity:
Its vector form is the Poynting vector:
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31-8 Energy in EM Waves; the
Poynting Vector
Typically we are interested in the average
S
value of S:
.
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31-8 Energy in EM Waves; the
Poynting Vector
Example 31-6: E and B from the Sun.
Radiation from the Sun reaches the Earth
(above the atmosphere) at a rate of about
1350 J/s·m2 (= 1350 W/m2). Assume that this
is a single EM wave, and calculate the
maximum values of E and B.
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31-9 Radiation Pressure
In addition to carrying energy, electromagnetic
waves also carry momentum. This means that a
force will be exerted by the wave.
The radiation pressure is related to the average
intensity. It is a minimum if the wave is fully
absorbed:
and a maximum if it is fully reflected:
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31-9 Radiation Pressure
Example 31-7: Solar pressure.
Radiation from the Sun that reaches
the Earth’s surface (after passing
through the atmosphere) transports
energy at a rate of about 1000 W/m2.
Estimate the pressure and force
exerted by the Sun on your
outstretched hand.
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31-9 Radiation Pressure
Example 31-8: A solar sail.
Proposals have been made to use the
radiation pressure from the Sun to help
propel spacecraft around the solar
system. (a) About how much force
would be applied on a 1 km x 1 km
highly reflective sail, and (b) by how
much would this increase the speed of
a 5000-kg spacecraft in one year? (c) If
the spacecraft started from rest, about
how far would it travel in a year?
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31-10 Radio and Television; Wireless
Communication
This figure illustrates the process by which a
radio station transmits information. The audio
signal is combined with a carrier wave.
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31-10 Radio and Television; Wireless
Communication
The mixing of signal and carrier can be done
two ways. First, by using the signal to modify
the amplitude of the carrier (AM):
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31-10 Radio and Television; Wireless
Communication
Second, by using the signal to modify the
frequency of the carrier (FM):
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31-10 Radio and Television; Wireless
Communication
At the receiving end, the wave is received,
demodulated, amplified, and sent to a
loudspeaker.
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31-10 Radio and Television; Wireless
Communication
The receiving
antenna is
bathed in
waves of many
frequencies; a
tuner is used to
select the
desired one.
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31-10 Radio and Television; Wireless
Communication
A straight antenna will have a current induced
in it by the varying electric fields of a radio
wave; a circular antenna will have a current
induced by the changing magnetic flux.
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31-10 Radio and Television; Wireless
Communication
Example 31-9: Tuning a station.
Calculate the transmitting wavelength
of an FM radio station that transmits
at 100 MHz.
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Summary of Chapter 31
• Maxwell’s equations are the basic equations
of electromagnetism:
Copyright © 2009 Pearson Education, Inc.
Summary of Chapter 31
• Electromagnetic waves are produced by
accelerating charges; the propagation speed
is given by
• The fields are perpendicular to each other
and to the direction of propagation.
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Summary of Chapter 31
• The wavelength and frequency of EM waves
are related:
• The electromagnetic spectrum includes
all wavelengths, from radio waves through
visible light to gamma rays.
• The Poynting vector describes the
energy carried by EM waves:
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