SAT II Physics (SN) Episode 2 Part 4 pps

The medium transmitting transverse waves oscillates in a direction perpendicular to the direction the wave is traveling.. By contrast, the medium transmitting longitudinal waves oscillat

There are two major kinds of waves: transverse waves and longitudinal waves The

medium transmitting transverse waves oscillates in a direction perpendicular to the direction the wave is traveling A good example is waves on water: the water oscillates up and down while transmitting a wave horizontally Other common examples include a wave on a string and electromagnetic waves By contrast, the medium transmitting longitudinal waves oscillates in a direction parallel to the direction the wave is traveling The most commonly discussed form of longitudinal waves is sound

Transverse Waves: Waves on a String

Imagine—or better yet, go grab some twine and set up—a length of string stretched between two posts so that it is taut Each point on the string is just like a mass on a spring: its equilibrium position lies on the straight line between the two posts, and if it is plucked away from its resting position, the string will exert a force to restore its

equilibrium position, causing periodic oscillations A string is more complicated than a simple mass on a spring, however, since the oscillation of each point influences nearby points along the string Plucking a string at one end causes periodic vibrations that eventually travel down the whole length of the string Now imagine detaching one end of the string from the pole and connecting it to a mass on a spring, which oscillates up and down, as in the figure below The oscillation at one end of the string creates waves that propagate, or travel, down the length of the string These are called, appropriately,

traveling waves Don’t let this name confuse you: the string itself only moves up and

down, returning to its starting point once per cycle The wave travels, but the medium—the string, in this case—only oscillates up and down

The speed of a wave depends on the medium through which it is traveling For a stretched string, the wave speed depends on the force of tension, , exerted by the pole on the string, and on the mass density of the string, :

The formula for the wave speed is:

EXAMPLE

A string is tied to a pole at one end and 100 g mass at the other, and wound over a pulley The string’s mass is 100 g, and it is 2.5 m long If the string is plucked, at what speed do the waves travel along the string? How could you make the waves travel faster? Assume the acceleration due to gravity is 10 m/s 2

Since the formula for the speed of a wave on a string is expressed in terms of the mass density of the string, we’ll need to calculate the mass density before we can calculate the wave speed

The tension in the string is the force of gravity pulling down on the weight,

The equation for calculating the speed of a wave on

a string is:

This equation suggests two ways to increase the speed of the waves: increase the tension

by hanging a heavier mass from the end of the string, or replace the string with one that is less dense

Longitudinal Waves: Sound

While waves on a string or in water are transverse, sound waves are longitudinal The

term longitudinal means that the medium transmitting the waves—air, in the case of

sound waves—oscillates back and forth, parallel to the direction in which the wave is moving This back-and-forth motion stands in contrast to the behavior of transverse waves, which oscillate up and down, perpendicular to the direction in which the wave is moving

Imagine a slinky If you hold one end of the slinky in each of your outstretched arms and then jerk one arm slightly toward the other, you will send a pulse across the slinky toward the other arm This pulse is transmitted by each coil of the slinky oscillating back and forth parallel to the direction of the pulse

When the string on a violin, the surface of a bell, or the paper cone in a stereo speaker oscillates rapidly, it creates pulses of high air pressure, or compressions, with low

pressure spaces in between, called rarefactions These compressions and rarefactions are the equivalent of crests and troughs in transverse waves: the distance between two compressions or two rarefactions is a wavelength

Pulses of high pressure propagate through the air much like the pulses of the slinky illustrated above, and when they reach our ears we perceive them as sound Air acts as the medium for sound waves, just as string is the medium for waves of displacement on a string The figure below is an approximation of sound waves in a flute—each dark area below indicates compression and represents something in the order of 1024 air molecules

Loudness, Frequency, Wavelength, and Wave Speed

Many of the concepts describing waves are related to more familiar terms describing

sound For example, the square of the amplitude of a sound wave is called its loudness,

or volume Loudness is usually measured in decibels The decibel is a peculiar unit

measured on a logarithmic scale You won’t need to know how to calculate decibels, but it may be useful to know what they are

The frequency of a sound wave is often called its pitch Humans can hear sounds with

frequencies as low as about 90 Hz and up to about 15,000 Hz, but many animals can hear

sounds with much higher frequencies The term wavelength remains the same for sound

waves Just as in a stretched string, sound waves in air travel at a certain speed This speed is around 343 m/s under normal circumstances, but it varies with the temperature and pressure of the air You don’t need to memorize this number: if a question involving the speed of sound comes up on the SAT II, that quantity will be given to you

Superposition

Suppose that two experimenters, holding opposite ends of a stretched string, each shake their end of the string, sending wave crests toward each other What will happen in the middle of the string, where the two waves meet? Mathematically, you can calculate the displacement in the center by simply adding up the displacements from each of the two

waves This is called the principle of superposition: two or more waves in the same

place are superimposed upon one another, meaning that they are all added together Because of superposition, the two experimenters can each send traveling waves down the string, and each wave will arrive at the opposite end of the string undistorted by the

other The principle of superposition tells us that waves cannot affect one another: one wave cannot alter the direction, frequency, wavelength, or amplitude of another wave.

Destructive Interference

Suppose one of the experimenters yanks the string downward, while the other pulls up by exactly the same amount In this case, the total displacement when the pulses meet will

be zero: this is called destructive interference Don’t be fooled by the name, though:

neither wave is destroyed by this interference After they pass by one another, they will continue just as they did before they met

You may have noticed the phenomenon of interference when hearing two musical notes

of slightly different pitch played simultaneously You will hear a sort of “wa-wa-wa” sound, which results from repeated cycles of constructive interference, followed by

destructive interference between the two waves Each “wa” sound is called a beat, and

the number of beats per second is given by the difference in frequency between the two interfering sound waves:

Modern orchestras generally tune their instruments so that the note “A” sounds at 440 Hz If one violinist is slightly out of tune, so that his “A” sounds at 438 Hz, what will be the time between the beats perceived by someone sitting in the audience?

The frequency of the beats is given by the difference in frequency between the out-of-tune violinist and the rest of the orchestra: Thus, there will

be two beats per second, and the period for each beat will be T = 1/f = 0.5 s.

Standing Waves and Resonance

So far, our discussion has focused on traveling waves, where a wave travels a certain distance through its medium It’s also possible for a wave not to travel anywhere, but

simply to oscillate in place Such waves are called, appropriately, standing waves A

great deal of the vocabulary and mathematics we’ve used to discuss traveling waves applies equally to standing waves, but there are a few peculiarities of which you should be aware

Reflection

If a stretched string is tied to a pole at one end, waves traveling down the string will

reflect from the pole and travel back toward their source A reflected wave is the mirror

image of its original—a pulse in the upward direction will reflect back in the downward direction—and it will interfere with any waves it encounters on its way back to the source

In particular, if one end of a stretched string is forced to oscillate—by tying it to a mass on

a spring, for example—while the other end is tied to a pole, the waves traveling toward the pole will continuously interfere with their reflected copies If the length of the string is a multiple of one-half of the wavelength, , then the superposition of the two waves will result in a standing wave that appears to be still

The crests and troughs of a standing wave do not travel, or propagate, down the string

Instead, a standing wave has certain points, called nodes, that remain fixed at the

equilibrium position These are points where the original wave undergoes complete destructive interference with its reflection In between the nodes, the points that oscillate with the greatest amplitude—where the interference is completely constructive—are

called antinodes The distance between successive nodes or antinodes is one-half of the

wavelength,

Resonance and Harmonic Series

The strings on musical instruments vibrate as standing waves A string is tied down at both ends, so it can only support standing waves that have nodes at both ends, and thus can only vibrate at certain given frequencies The longest such wave, called the

fundamental, or resonance, has two nodes at the ends and one antinode at the center

Since the two nodes are separated by the length of the string, L, we see that the

fundamental wavelength is The string can also support standing waves with one, two, three, or any integral number of nodes in between the two ends This series of

standing waves is called the harmonic series for the string, and the wavelengths in the

series satisfy the equation , or:

In the figure above, the fundamental is at the bottom, the first member of the harmonic

series, with n = 1 Each successive member has one more node and a correspondingly

shorter wavelength

EXAMPLE

An empty bottle of height 0.2 m and a second empty bottle of height 0.4 m are placed next

to each other One person blows into the tall bottle and one blows into the shorter bottle What is the difference in the pitch of the two sounds? What could you do to make them sound at the same pitch?

Sound comes out of bottles when you blow on them because your breath creates a series

of standing waves inside the bottle The pitch of the sound is inversely proportional to the wavelength, according to the equation We know that the wavelength is directly proportional to the length of the standing wave: the longer the standing wave, the greater the wavelength and the lower the frequency The tall bottle is twice as long as the short bottle, so it vibrates at twice the wavelength and one-half the frequency of the shorter bottle To make both bottles sound at the same pitch, you would have to alter the

wavelength inside the bottles to produce the same frequency If the tall bottle were filled with water, the wavelength of the standing wave would decrease to the same as the small bottle, producing the same pitch

half-Pitch of Stringed Instruments

When violinists draw their bows across a string, they do not force the string to oscillate at any particular frequency, the way the mass on a spring does The friction between the bow and the string simply draws the string out of its equilibrium position, and this causes standing waves at all the different wavelengths in the harmonic series To determine what pitches a violin string of a given length can produce, we must find the frequencies

corresponding to these standing waves Recalling the two equations we know for the wave speed, and , we can solve for the frequency, , for any term, n, in the

harmonic series A higher frequency means a higher pitch

You won’t need to memorize this equation, but you should understand the gist of it This equation tells you that a higher frequency is produced by (1) a taut string, (2) a string with low mass density, and (3) a string with a short wavelength Anyone who plays a stringed instrument knows this instinctively If you tighten a string, the pitch goes up (1); the strings that play higher pitches are much thinner than the fat strings for low notes (2); and by placing your finger on a string somewhere along the neck of the instrument, you shorten the wavelength and raise the pitch (3)

The Doppler Effect

So far we have only discussed cases where the source of waves is at rest Often, waves are emitted by a source that moves with respect to the medium that carries the waves, like when a speeding cop car blares its siren to alert onlookers to stand aside The speed of the

waves, v, depends only on the properties of the medium, like air temperature in the case

of sound waves, and not on the motion of the source: the waves will travel at the speed of sound (343 m/s) no matter how fast the cop drives However, the frequency and

wavelength of the waves will depend on the motion of the wave’s source This change in

frequency is called a Doppler shift.Think of the cop car’s siren, traveling at speed ,

and emitting waves with frequency f and period T = 1/f The wave crests travel outward

from the car in perfect circles (spheres actually, but we’re only interested in the effects at

ground level) At time T after the first wave crest is emitted, the next one leaves the siren

By this time, the first crest has advanced one wavelength, , but the car has also traveled

a distance of As a result, the two wave crests are closer together than if the cop car had been stationary

The shorter wavelength is called the Doppler-shifted wavelength, given by the formula

The Doppler-shifted frequency is given by the formula:

Similarly, someone standing behind the speeding siren will hear a sound with a longer

You’ve probably noticed the Doppler effect with passing sirens It’s even noticeable with normal cars: the swish of a passing car goes from a higher hissing sound to a lower hissing sound as it speeds by The Doppler effect has also been put to valuable use in astronomy, measuring the speed with which different celestial objects are moving away from the Earth

EXAMPLE

A cop car drives at 30 m/s toward the scene of a crime, with its siren blaring at a frequency

of 2000 Hz At what frequency do people hear the siren as it approaches? At what frequency

do they hear it as it passes? The speed of sound in the air is 343 m/s.

As the car approaches, the sound waves will have shorter wavelengths and higher

frequencies, and as it goes by, the sound waves will have longer wavelengths and lower frequencies More precisely, the frequency as the cop car approaches is:

The frequency as the cop car drives by is:

Practice Questions

1 Which of the following exhibit simple harmonic motion?

I A pendulum

II A mass attached to a spring

III A ball bouncing up and down, in the absence of friction

m (C)

m (D)

m (E)

m

3 Two strings of equal length are stretched out with equal tension The second string is four

times as massive as the first string If a wave travels down the first string with velocity v,

how fast does a wave travel down the second string?

4 A piano tuner has a tuning fork that sounds with a frequency of 250 Hz The tuner strikes the fork and plays a key that sounds with a frequency of 200 Hz What is the frequency

of the beats that the piano tuner hears?

6 Two pulses travel along a string toward each other, as depicted above Which of the following diagrams represents the pulses on the string at a later time?

7 What should a piano tuner do to correct the sound of a string that is flat, that is, it plays

at a lower pitch than it should?

(A) Tighten the string to make the fundamental frequency higher

(B) Tighten the string to make the fundamental frequency lower

(C) Loosen the string to make the fundamental frequency higher

(D) Loosen the string to make the fundamental frequency lower

(E) Find a harmonic closer to the desired pitch

Questions 8 and 9 refer to a police car with its siren on, traveling at a velocity toward a person standing on a street corner As the car approaches, the person hears the sound at a frequency of Take the speed of sound to be v.