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

timeone 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 wavesAM 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 stringThe 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 stringsTransverse 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 speedIs 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 WaveSound 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 rulerThis 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 ThumbThere 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.