1206 - Concepts in
Physics
Friday, November 20th 2009
Notes
•
Assignment #8 is posted on webpage, due
December 2nd - this is the last mandatory
assignment
•
There will be an Assignment #9, which I will supply
early next week. This will be due on Monday,
December 7th. It will be random topics from the
whole course, so a great way to prepare for the
final exam.
•
I strongly recommend, that everybody who missed
an assignment should do #9. Also, if you are
looking for some extra homework points, this is
your chance.
Waves
•
We discussed last time, that there are two types of waves: longitudinal and
transverse
•
These waves are called periodic waves because they consist of cycles or
patterns that are produced over and over again by the source.
•
•
Every segment of the slinky vibrates in simple harmonic motion
The wavelength λ is the horizontal length of one cycle of the wave.
T
time
one specific
position
A wave is a series of many cycles, the plot is like taking a picture of the whole structure at
one point in time. For one specific point “in the wave” watching it over time we obtain simple
harmonic motion - very similar picture. Note x axis labels!!!
The period T is commonly measured in seconds, which the frequency f = 1/T is
measured in cycles per second or Hertz (Hz). A simple relation exists between the
period, the wavelength, and the speed of a wave. Imaging waiting at a railroad crossing,
while a freight train moves by at a constant speed v. The train consists of a long line of
identical boxcars, each of which has a length λ an requires a time T to pass, so that the
speed is v = λ/T. This same equation applies for a wave, so we can write:
v = λ/T = fλ
This is true for both longitudinal and transverse waves.
Example: radio waves
AM And FM radio waves are transverse waves that consist of electric and magnetic
disturbances. These waves travel at a speed of 3.00 x 108 m/s (speed of light). A station
broadcasts an AM radio wave whose frequency is 1230 x 103 Hz and an FM radio wave
whose frequency is 91.9 MHz. Find the distance between adjacent crests in each wave
(the wave length).
The distance between adjacent crests is the wavelength λ. Since the speed of each
wave is v = 3.00 x 108 m/s and the frequencies are known, we can use v=fλ to
determine the wavelengths.
AM: λ = v/f = (3.00 x 108 m/s)/(1230 x 103 Hz) = 244 m
(Hz = 1/s)
FM: λ = v/f = (3.00 x 108 m/s)/(91.9 x 106 Hz) = 3.26 m
(Hz = 1/s)
Note! The AM wavelength is longer than 2.5 football fields. Also higher frequency
means shorter wavelength.
The speed of a wave on a string
The properties of the material or medium through which a wave travels determine the
speed of the wave. For example let’s look at a transverse wave on a string. As the wave
moves to the right, each particle is displaced, one after the other, from its undisturbed
position. Therefore the speed with which the wave moves to the right depends on how
quickly one particle of the string is accelerated upward in response to the net pulling
force exerted by its adjacent neighbors.
In accord with Newton’s second law, a stronger net force results in a greater
acceleration, and, thus a faster-moving wave. The ability of one particle to pull on its
neighbors depends on how tightly the string is stretched - that is, on the tension. The
greater the tension, the greater the pulling force the particles exert on each other, and
the faster the wave travels, other things being equal. Therefore, other things being equal,
a wave travels faster on a string whose particles have a small mass, or as it turns out, on
a string that has a small mass per unit length. The mass per unit length is called linear
density of the string. It is the mass m of the string divided by its length L, or m/L. The
effects of tension F and the mass per unit length can be pulled together in this
expression for the speed v of a small-amplitude wave on a string: v = sqrt(FL/m)
The motion of transverse waves along a string is important in the operation of musical
instruments, such as the guitar, the violin, the piano.
Example: guitar strings
Transverse waves travel on strings of an electric guitar after the string are plucked. The length
of each string between its two fixed ends is 0.628 m, and the mass is 0.208 g for the highest
pitched E string and 3.32 g for the lowest pitched E string. Each string is under a tension of
226 N. Find the speeds of the waves on the two strings.
The speed of a wave on a guitar string, as expressed in the previous slide, depends on the
tension F in the string and its linear density m/L. Since the tension is the same for both
strings, and smaller linear densities give rise to greater speeds, we expect the wave speed to
be greatest on the string with the smallest linear density.
The speeds of the waves are given:
High-pitched E: v = sqrt(FL/m) = sqrt{(226 N)(0.628 m)/(0.208 x 10-3 kg)} = 826 m/s
Low-pitched E: v = sqrt(FL/m) = sqrt{(226 N)(0.628 m)/(3.32 x 10-3 kg)} = 207 m/s
Note! These are quite high speeds.
Example: wave speed versus particle speed
Is the speed of a transverse wave on a string the same as the speed at which a particle on
the string moves?
The particle speed v(particle) specifies how fast the particle is moving as it oscillates up
and down, and it is different form the wave speed. If the source of the wave (for example
your hand) vibrates in simple harmonic motion, each string particle vibrates in a like
manner, with the same amplitude and frequency as the source. Moreover, the particle
speed, unlike the wav speed, is not constant. As for any object in simple harmonic motion,
the particle speed is greatest when the particle is passing through the undisturbed
position of the string an zero when the particle is at its maximum displacement.
So, the speed of a string particle is determined by the properties of the source creating
the wave and not by the properties of the string itself. In contrast, the speed of the wave
is determined by the properties of the string - that is the tension F and the mass per unit
length m/L.
Mathematical description of a wave
When a wave travels through a medium, it displaces the particles of the medium from their
undisturbed positions. Suppose a particle is locate at a distance x from a coordinate origin.
We would like to know the displacement y of this particle form its undisturbed position at
any time t as the wave passes. Fro periodic waves that result from simple harmonic motion
of the source, the expression for the displacement involves a sine or cosine, a fact that is
not surprising.
Wave motion toward +x: y = A sin (2πft - 2πx/λ)
Wave motion toward - x: y = A sin (2πft + 2πx/λ)
These equations apply to transverse or longitudinal waves and assume that y = 0 m for
x = 0 m and t = 0 s
Consider a transverse wave moving in the +x direction along a string. The term (2πf 2πx/λ) is called the phase angle of the wave. A string particle located at the origin ( x = 0
m) exhibits simple harmonic motion with a phase angle of 2πft, that is, its displacement as
a function of time is y = A sin (2πft). A particle located at a distance x also exhibits
simple harmonic motion, but its phase angle is
2πft - 2πx/λ = 2λf(t - x/(fλ)) = 2πf (t - x/v)
The quantity x/v is the time needed for the wave to travel the distance x. In other words,
the simple harmonic motion that occurs at x is delayed by the time interval x/v compared
to the motion at the origin.
Sound waves
Sound is a Pressure Wave
Sound is a mechanical wave which results from the back and forth
vibration of the particles of the medium through which the sound wave
is moving. If a sound wave is moving from left to right through air, then
particles of air will be displaced both rightward and leftward as the
energy of the sound wave passes through it. The motion of the particles
are parallel (and anti-parallel) to the direction of the energy transport.
This is what characterizes sound waves in air as longitudinal waves.
A vibrating tuning fork is capable of creating such a longitudinal wave.
As the tines of the fork vibrate back and forth, they push on
neighboring air particles. The forward motion of a tine pushes air
molecules horizontally to the right and the backward retraction of the
tine creates a low pressure area allowing the air particles to move back
to the left.
Because of the longitudinal motion of the air particles, there are regions
in the air where the air particles are compressed together and other
regions where the air particles are spread apart. These regions are known
as compressions and rarefactions respectively. The compressions are
regions of high air pressure while the rarefactions are regions of low air
pressure. The diagram below depicts a sound wave created by a tuning
fork and propagated through the air in an open tube. The compressions
and rarefactions are labeled.
The wavelength of a wave is merely the distance which a disturbance
travels along the medium in one complete wave cycle. Since a wave
repeats its pattern once every wave cycle, the wavelength is sometimes
referred to as the length of the repeating pattern - the length of one
complete wave. For a transverse wave, this length is commonly measured
from one wave crest to the next adjacent wave crest or from one wave
trough to the next adjacent wave trough. Since a longitudinal wave does
not contain crests and troughs, its wavelength must be measured
differently. A longitudinal wave consists of a repeating pattern of
compressions and rarefactions. Thus, the wavelength is commonly
measured as the distance from one compression to the next adjacent
compression or the distance from one rarefaction to the next adjacent
rarefaction.
Since a sound wave consists of a repeating pattern of high pressure and
low pressure regions moving through a medium, it is sometimes referred
to as a pressure wave. If a detector, whether it be the human ear or a
man-made instrument, is used to detect a sound wave, it would detect
fluctuations in pressure as the sound wave impinges upon the detecting
device. At one instant in time, the detector would detect a high pressure;
this would correspond to the arrival of a compression at the detector site.
At the next instant in time, the detector might detect normal pressure.
And then finally a low pressure would be detected, corresponding to the
arrival of a rarefaction at the detector site.
A sound wave is a pressure wave; regions of high
(compressions) and low pressure (rarefactions) are established
as the result of the vibrations of the sound source. These
compressions and rarefactions result because sound
a.) is more dense than air and thus has more inertia, causing
the bunching up of sound.
b.) waves have a speed which is dependent only upon the
properties of the medium.
c.) is like all waves; it is able to bend into the regions of space
behind obstacles.
d.) is able to reflect off fixed ends and interfere with incident
waves
e.) vibrates longitudinally; the longitudinal movement of air
produces pressure fluctuations.
A sound wave is a pressure wave; regions of high
(compressions) and low pressure (rarefactions) are established
as the result of the vibrations of the sound source. These
compressions and rarefactions result because sound
a.) is more dense than air and thus has more inertia, causing
the bunching up of sound.
b.) waves have a speed which is dependent only upon the
properties of the medium.
c.) is like all waves; it is able to bend into the regions of space
behind obstacles.
d.) is able to reflect off fixed ends and interfere with incident
waves
e.) vibrates longitudinally; the longitudinal movement of air
produces pressure fluctuations.
Since the particles of the medium vibrate in a longitudinal fashion, compressions and
rarefactions are created. (e) is correct ...
Air is a gas, and a very important property of any gas is the speed of sound through
the gas. Why are we interested in the speed of sound? The speed of "sound" is
actually the speed of transmission of a small disturbance through a medium. Sound
itself is a sensation created in the human brain in response to sensory inputs from the
inner ear.
Disturbances are transmitted through a gas as a result of collisions between the
randomly moving molecules in the gas. The conditions in the gas are the same before
and after the disturbance passes through. Because the speed of transmission
depends on molecular collisions, the speed of sound depends on the state of the gas.
The speed of sound is a constant within a given gas and the value of the constant
depends on the type of gas (air, pure oxygen, carbon dioxide, etc.) and the
temperature of the gas. An analysis based on conservation of mass and momentum
shows that the speed of sound a is equal to the square root of the ratio of specific
heats γ times the gas constant Rs times the temperature T.
a = sqrt [γ * R * T]
Notice that the temperature must be specified on an absolute scale (Kelvin). The
dependence on the type of gas is included in the gas constant Rs. which equals the
universal gas constant divided by the molecular weight of the gas, and the ratio of
specific heats.
The speed of sound in air depends on the type of gas and the temperature of the gas.
On Earth, the atmosphere is composed of mostly diatomic nitrogen and oxygen, and
the temperature depends on the altitude in a rather complex way. Scientists and
engineers have created a mathematical model of the atmosphere to help them account
for the changing effects of temperature with altitude. Mars also has an atmosphere
composed of mostly carbon dioxide. There is a similar mathematical model of the
Martian atmosphere.
The speed of sound in an ideal gas (in our previous notations) is:
v = sqrt(γkT/m) ,where k is the Boltzmann constant
and γ is cp/cV
Example: An ultrasonic ruler
This is the way bats avoid crashing into objects. They send out ultrasonic waves and
detect them when the get reflected back from objects. Ultrasonic rulers can be used to
measure the distance between itself and a target such as a wall. To initiate the
measurement, the ruler generates a pulse of ultrasonic sound that travels to the wall
and, like an echo, reflects form it. The reflected pulse returns to the ruler, which
measures the time it takes for the round-trip. Using a pre-set value for the speed of
sound, the unit determines the distance t the wall and displays it on a digital readout.
Suppose the round-trip travel time is 20.0 ms on a day when the air temperature is
23 °C. Assuming that air is an ideal gas for which γ = 1.40 and that the average
molecular mass of air is 28.9 u, find the distance x to the wall.
The distance between the rules and the wall is x = vt, where v is the speed of sound and t
is the time for the sound pulse to reach the wall. The time t is one-half the round-trip
time, so t = 10.0 ms. The speed of sound in air can be obtained directly provided the
temperature and mass are expressed in the SI units of Kelvins and kilograms respectively.
We need to convert the air temperature of 23 C to Kelvin and the mass of a molecule to
kg. So, T = 23 + 273.15 = 296 K. 1 u = 1.6605 x 10-27 kg
m = (28.9 u) (1.6605 x 10-27 kg/1u) = 4.80 x 10-26 kg
For the speed of sound: v = sqrt(γkT/m) = sqrt{(1.40)(1.38 x 10-23 J/K)(296K)/(4.80 x
10-26 kg)} = 345 m/s
The distance to the wall is: x = vt = (345 m/s)(10.0 x 10-3 s) = 3.45 m
Sonar (Sound Navigation Ranging) is a technique for determining water depth and
locating underwater objects, such as reefs, submarines, and schools of fish. The core of
a sonar unit consists of an ultrasonic transmitter and receiver mounted on the bottom
of a ship. The transmitter emits a short pulse of ultrasonic sound, and at a later time
the reflected pulse returns and is detected by the receiver. The water depth is
determined from the electronically measured round-trip time of the pulse and a
knowledge of the speed of sound in water; the depth registers automatically on an
appropriate meter.
Lightning, Thunder, and a Rule of Thumb
There is a rule of thumb for estimating how far away a thunderstorm is. After you see a
flash of lightning, count off the seconds until the thunder is heard. Divide the number of
seconds by five. The result gives the approximate distance (in miles) to the
thunderstorm. Why does this rule work?
When lightning occurs, light and sound (thunder) are produced very nearly at the same
instant. Light travels so rapidly -- v(light) = 3.0 x 108 m/s) that it reached the observer
almost instantaneously. It’s travel time (for 1 mile) is only (1.6 x 103 m)/(3.0 x 108 m/s)
= 5.3 x 10-6 s. In comparison, sound travels very slowly v(sound) = 343 m/s. The time
for the thunder to reach the person is (1.6 x 103 m)/(343 m/s) = 5 s. Thus, the time
interval between seeing the flash and hearing the thunder is about 5 seconds for every
mile of travel.