Topic List

Types of Waves

Longitudinal Waves

Transverse Waves

Surface Waves

Frequency

Wavelength

Period

Amplitude

Wave speed

Transmission of Waves

Reflection

Refraction

Superposition Principle

Interference

Diffraction

Standing Waves & Resonance

Waves & InformationMarvin the Martian would like to send a message from Mars to Earth. There are two ways of sending a message. He could enclose the message in a rocket and physically send it to Earth. Or, he could send some type of signal, maybe in the form of radio waves.

Information can be sent via matter or waves. If sent via waves, nothing material is actually transmitted from sender to receiver. If you talk to a friend, be it in person or on the phone, you are transmitting information via waves. Nothing is physically transported from you to your friend. This would not be the case, however, if you sent him a letter.

Waves & Energy Suppose Charlie Brown wants to wake up Snoopy. Some energy is required to rouse Snoopy from his slumber. Like information, energy can also be transmitted via physical objects or waves. Charlie Brown can transmit energy from himself to Snoopy via Woodstock: Woodstock flies over; his kinetic energy is physically transported in the form a little, yellow bird. Alternatively, Charlie Brown could send a pulse down a rope that’s attached to Snoopy’s dog house. The rope itself is not transported, but the pulse and its energy are!

pulse

Types of WavesA mechanical wave is just a disturbance that propagate through a medium. The medium could be air, water, a spring, the Earth, or even people. A medium is any material through which a wave travels. Mechanical wave examples: sound; water waves; a pulse traveling on a spring; earthquakes; a “people wave” in a football stadium.

An electromagnetic wave is simply light of a visible or invisible wavelength. Oscillating intertwined electric and magnetic fields comprise light. Light can travel without medium—super, duper fast.

A matter wave is a term used to describe particles like electrons that display wavelike properties. It is an important concept in quantum mechanics.

A gravity wave is a ripple in the “fabric of spacetime” itself. They are predicted by Einstein’s theroy of relativity, but they’re very difficult to detect.

Mechanical Waves: Three TypesMechanical waves require a physical medium. The particles in the

medium can move in two different ways: either perpendicual or parallel to direction of the wave itself.

In a longitudinal wave, the particles in the medium move parallel to the direction of the wave.

In a transverse wave, the particles in the medium move perpen-dicular to the direction of the wave.

A surface wave is often a combination of the two. Particles typically move in circular or elliptical paths at the surface of a medium.

Longitudinal Parallel

Transverse Perpendicular

Surface Combo

Longitudinal WavesA whole bunch of kids are waiting in line to get their picture taken with Godzilla. The bully in back pushes the kid in front of him, who bumps into the next kid, and so on down the line. A longitudinal pulse is sent through the line of kids. It’s longitudinal because as each kid gets bumped, he moves forwards, then backwards (red arrow), parallel to the direction of the pulse. The location of the pulse is the point where two kids are being compressed together. The next slide shows how the pulse progresses through the line.

pulse direction

Longitinal Waves (cont.)

Ouch!

Ouch!

Ouch!

C

C

CR

R

C = Compression (high kid density)

R = Rarefaction (low kid density)

The compression (the pulse) moves up the line, but each kid keeps his place in line.

I hope Godzilla eats

that bully!

Sound is a Longitudinal WaveAs sound travels through air, water, a solid, etc., the molecules of the medium move back and forth in the direction of the wave, just like the kids in the last example, except the molecules continually move back and forth for as long as the sound persists. If the bully kept shoving the kid in front of

him, a series of pulses would be generated. If he shoved with equal force each time and did this at a regular rate, we would call these pulses a wave. Similarly, when a speaker or a tuning fork vibrates, it repeatedly shoves the air in front of it, and a longitudinal wave propagates through to the air. The speaker shoves air molecules; the bully shoves people. In either case, the components of the medium must bump into their neighbors.

Transverse WavesAfter a great performance at a drum and bugle corps contest, the audience decides to start a wave in the stands. Each person rises and sits at just the right time so the effect is similar to the pulse in Charlie Brown’s rope. Like the Godzilla example, people make up the wave medium here. But this is a transverse wave because, as the wave moves across the stands, folks are moving up and down.

wave direction

Transverse Waves (cont.)In a transverse wave, molecules aren’t being compressed and spread out as they are in a longitudinal wave. The reason a transverse wave can propagate is because of the attraction between adjacent molecules. Imagine if each person in the stands on the last slide were connected to the person on his left and right with giant rubber bands. As soon the person on one end stood up, the band stretches. The tension in the band pulls his neighbor up, who, in turn, lifts the next guy.

The tension in the rubber bands is analogous to the forces connecting particles of the medium to their neighbors. The colored sections of rope tug on each other as the waves travels through them. If they didn’t, it would be as if the rope were cut, and no wave could travel through it.

Surface WavesBelow the surface fluids can typically only transmit longitudinal waves, since the attraction between neighboring molecules is not as strong as in a fluid. At the surface of a lake, water molecules (white dots) move in circular paths, which are partly longitudinal and partly transverse. The molecules are offset, though: when one is at the top of the circle, the one in front of it is near the top. As in any wave, the particles of the medium do not move along with the wave. The water molecules complete a circle each time a crest passes by. Animation

wave direction

Breaking WavesWaves break near the shore because the water becomes shallow. Close to the shore the ground beneath the water interferes with the circular motions of the water molecules as they participate in a passing wave. Sandbars further off shore can have the same effect, much to the delight of surfing enthusiasts like Bart.

Seismic WavesSeismic waves use Earth itself as their medium. Earthquakes produce them and so does a nation when it carries out an underground nuclear test. (Other countries can detect them.) Seismic waves can be longitudinal, transverse, or surface waves. P and S type waves are called body waves, since they are not confined to the surface. Rayleigh waves do most of the shaking during a quake.

Name Type Info

P Wave Longitudinal Also known as primary, compressional, or acoustic waves; fastest seismic wave

S wave Transverse Also known as secondary, or shear waves; do not travel through fluids;

Rayleigh Wave

Surface Rolls along surface like a water wave; large amplitude

Love Wave Surface Ground moves side to side as wave moves forward

“Mini Seismic” Waves

Though we might not refer to them as seismic, anything moving on the ground can transmit waves through the ground. If you stand near a moving locomotive or a heard of charging elephants, you would feel these vibrations. Even something as small as abeetle generates pulses when it moves. These pulses can be detected by a nocturnal sand scorpion. Sensors on its eight legs can detect both longitudinal and surface waves. The scorpion can determine the direction of the waves based on which legs feel the waves first. It can determine the distance of the prey based on the time delay between the fast moving longitudinal waves and the slower moving surface waves. The greater the time delay, the farther away the beetle. This is the same way seismologists determine the distance of a quake’s epicenter. Sand is not the best conductor of waves, so the scorpion will only be able to detect beetles within about a half meter.

Wave CharacteristicsAmplitude (A) – Maximum displacement of particle of the medium from its equilibrium point. The bigger the amplitude, the more energy the wave carries.

Wavelength () – Distance from crest (max positive displacement) to crest; same as distance from trough (max negative displacement) to trough.

Period (T) – Time it takes consecutive crests (or troughs) to pass a given point, i.e., the time required for one full cycle of the wave to pass by. Period is the reciprocal of frequency: T = 1 / f.

Frequency (f ) – The number of cycles passing by in a given time. The SI unit for frequency is the Hertz (Hz), which is one cycle per second.

Wave speed (v) – How fast the wave is moving (the disturbance itself, not how fast the individual particles are moving, which constantly varies). Speed depends on the medium. We’ll prove that v = f.

Q

Amplitude & Wavelength

A

The red transverse wave has the same wavelength as the longitudinal wave in the spring. (P to Q is one full cycle.) Note that where the spring is most compressed, the red wave is at a crest, and where the spring is most stretched (rarified), the red wave is at a trough. The amplitude in the red wave is easy to see. In the longitudinal wave, the amplitude refers to how far a particle on the spring moves to the left or right of its equilibrium point. Often a graph like the red wave is used to represent a longitudinal wave. For sound, the y-axis might be pressure deviation from normal air pressure, and the x-axis might be time or position.

P

Frequency & PeriodAnswer: Riddle me this…

Why is the frequency of a wave the reciprocal of its

period?

Period = seconds per cycle.

Frequency = cycles per second.

They’re reciprocals no matter what unit we use for time. A sound wave that has a frequency of 1,000 Hz has a period of 1 / 1,000 of a second. This means that 1,000 high pressure fronts are moving through the air and hitting your eardrum each second.

Speed, Wavelength, & FrequencyBarney Rubble, a.k.a. “Barney the Wave Watcher,” is excited because

he just made a discovery: v = f. With some high tech, prehistoric equipment, Barney measures the wavelength of the incoming waves to be 18 ft. He counts 10 crests hitting the shoreline every minute. So,

18 ft

10 crests pass any given point in a time of one minute. But 10 crests corresponds to a distance of 180 ft, which means the wave is traveling at 180 ft / min. This result is the product of wavelength and frequency, yielding the result:

v = f

Harmonic Waves

m1

m2

m3

m4

m5

Imagine a whole bunch of equal masses hanging from identical springs. If the masses are set to bobbing at staggered time intervals, a snapshot of the masses forms a transverse wave. Each mass undergoes simple harmonic motion, and the period of each is the same. If the release of the masses is timed so that the masses form a sinusoid at each point in time, the wave is called harmonic. Right now, m4 is peaking. A little later m4 will be lower and m3 will be peaking. The masses (the particles of the medium) bob up and down but do not move horizontally, but the wave does move horizontally.

m6

m7

m8

m9

m10

wave direction

Making a Harmonic Wave

A generator attached to a rope moves up and down in simple harmonic motion. This generates a harmonic wave in the rope. Each little piece of rope moves vertically just like the masses on the last slide. Only the wave itself moves horizontally. The time it takes the wave to move from P to Q is the period of the wave, T. The distance from P to Q is the wavelength, . So, the wave speed is given by: v = / T = f (since frequency and period are reciprocals).

Since the generator moves vertically in SHM, the vertical position of the black doo-jobber is given by: y(t) = A cos t. The doo-jobber’s period is given by T = 2 / . This is also the period of the wave.

wave direction

P Q

Making a Non-harmonic Wave

wave direction

P Q

If the black doo-jobber does not move in SHM, the wave it generates will not be harmonic. As long as the generator has some sort of periodic motion, the wave generated will have a well defined period and wavelength. Here the generator pauses at the high and low points, causing the wave to flatten.

If the wave had moved at a constant speed and changed direction instantly, a saw-tooth wave would have been the result. Sound is not a transverse wave, but a graph of pressure vs. time as a sound waves pass by would look like a very few simple sinusoid in the case of a pure tone. It would be a very complicated wave if the sound is a musical instrument of someone’s voice.

Wave Speed on a Rope

m s

r

If a pulse is traveling along a rope to the right at a speed v, from its point of view it’s still and the rope is moving to the left at a speed v. As the red segment of rope of length s rounds the turn in the pulse, a centripetal force must act on it. The tension in the rope is F, and the downward components of the tension vectors add to make the centripetal force.

v (rope)F F

/ 2 / 2

FC = m v 2 / r 2 F sin ( / 2) = m v

2 / r 2 F ( / 2) = m v 2 / r

(since the sine of an angle the angle itself in radians)

F = m v

2 / r F r / m = v 2 F s / m = v

2

(since s = r )

(continued)

Wave Speed on a Rope (cont.)

m s

rv (rope)

F F / 2 / 2

If the rope is uniform density, then the mass per unit length is a constant. We’ll call this constant µ. Thus, µ = m / s. From the last slide we have:

v 2 = F s / m = F / µ

F µ

v =

This shows that waves travel faster in materials that are stiff (high tension) and light weight. Unit check: [N / (kg / m)]

½

= [N m / kg ] ½ = [(kg m / s

2) m / kg ] ½ = [m

2 / s

2] ½ = m / s.

Reflection of Waves

Whenever a wave encounters different medium, some of the wave may be reflected back, and some of the wave penetrate and be absorbed or transmitted through the new medium. Light waves reflects off of objects. If it didn’t, we would only be able to see objects that emitted their own light. We see the moon because it’s reflecting sunlight. Sound waves also reflect off of objects, creating echoes. Water waves, seismic waves, and waves traveling on a rope all can reflect.

Transmission & Reflection

Animation

Let’s look at 4 different scenarios of a waves traveling along a rope. The link below has an animation of each.

1. Hard boundary (fixed end): Reflected wave is inverted.

2. Soft boundary (free end): Reflected wave is upright.

3. Light rope to heavy rope: Reflected wave is faster and wider than transmitted wave. Transmitted wave is upright, but reflected wave is inverted (since to the thin rope, the thick rope is like a hard boundary).

4. Heavy rope to light rope: Transmitted wave is faster, wider, and has a greater amplitude than reflected wave. Both waves are upright. (The transmitted wave is upright this time since, to the thick rope, the thin rope is like a soft boundary).

Frequency of Transmitted Waves

The frequency of a transmitted wave is always unchanged. Say a wave with a frequency of 5 Hz is traveling along a rope that changes thickness at some point. Since 5 pulses hit this point every second, 5 pulses will be transmitted every second. Since the speed will vary depending on the thickness of the rope, the wavelength must vary too.

Here a wave travels from a thin rope to a thick one. Because µ is larger in the thick rope, the wave is slower there. This causes the waves to “bunch up,”which means a decrease in wavelength. (For clarity the reflected waves are not shown here.)

v v

Amplitude & EnergyThe energy carried by a wave is proportional to the square of its amplitude: E A

2

m1

m2

m3

m4

m5

Consider our masses on a string again. The amount of potential energy stored in a spring is given by: U = ½ k x

2, where k is the spring constant and x is the distance from equilibrium. For m1 or m4, U = ½ k A2. The other masses have kinetic energy but less potential. Since energy is conserved, the total energy any mass has

is ½ k A2. This shows that energy varies as the square of the amplitude. The constant of proportionality depends on the medium.

Amplitude of Reflected & Transmitted

Waves: Light to Heavy

Back to Animation

before

after

When a pulse on the light rope reaches the interface, the heavy rope offers a lot of resistance. The heavy rope is not affected much by the light rope, so the transmitted pulse has a smaller amplitude. The reflected pulse’s amplitude diminishes since some of the light rope’s energy it transmitted to the heavy rope.

incident pulse

inverted reflected pulse

transmitted pulse

Amplitude of Reflected & Transmitted Waves: Heavy to Light

When a pulse on the heavy rope reaches the interface, the light rope offers little resistance. The light rope is greatly affected by the heavy rope, so the transmitted pulse has a greater amplitude. The upright reflected pulse’s amplitude diminishes since some of the heavy rope’s energy it transmitted to the light rope.

before

after

Back to Animation

incident pulse

upright reflected pulse

transmitted pulse

RefractionWe’ve seen that when a wave reaches an interface (a change from one medium to another), part of the wave can be transmitted, and part can be reflected back. A rope is a 1-dimensional medium; in a 2-dimensional medium a transmitted wave can change direction. This is refraction—the bending of a wave as it passes from one medium to another. The most well know type of refraction is that of light bending as it passes from air to glass or water, which we’ll study in detail in a unit on light.

As ocean waves approach the shore at an angle, the part of the wave closer to shore begins to slow down because the water is shallower. This causes refraction, and the waves bend so that it the wave fronts (crests) come in nearly parallel with the shore. See pic on next slide. Even though the medium (water) doesn’t change, one of its properties does—the speed of the wave.

Refraction of Ocean WavesWave fronts are shown in white heading toward the beach. The water gets shallow at the bottom first, which causes the waves to slow down and bend, and the wavelength to decrease. By the time the waves reach shore, they’re nearly parallel to the shoreline. The effect can even be seen on islands, where winds nearly wrap around it and come toward the island from all sides.

SuperpositionCheck out this animation to see what happens when two pulses approach each other from opposite ends on a rope.

Superposition Animation

Note the following:

1. The waves pass through each other unaffected by their meeting.

2. As they’re passing through each other the waves combine to create a changing waveform.

3. The displacement of the rope at any point in this “combo wave” is the sum of the displacements of the displacements of the original waves. In other words, we add amplitudes. This is called superposition.

Constructive & Destructive Interference

Constructive Interference

Waves are “in phase.” By super-position, red + blue = green. If red and blue each have amplitude A, then green has amplitude 2A.

Destructive Interference

Waves are “out of phase.” By superposition, red and blue completely cancel each other out, if their amplitudes and frequencies are the same.

Interference

Interference Animation

Wave Interference

Like force vectors, waves can work together or opposition. Sometimes they can even do some of both at the same time. Superposition applies even when the waves are not identical.

Constructive interference occurs at a point when two waves have displacements in the same direction. The amplitude of the combo wave is larger either individual wave.

Destructive interference occurs at a point when two waves have displacements in opposite directions. The amplitude of the combo wave is smaller than that of the wave biggest wave.

Superposition can involve both constructive and destructive interference at the same time (but at different points in the medium).

DiffractionWhen waves bounce off a barrier, this is reflection. When waves bend due to a change in the medium, this is refraction. When waves change direction as they pass around a barrier or through a small opening, this is diffraction. Refraction involves a change in wave speed and wavelength; diffraction doesn’t.

Diffraction of water happens as waves bend around a boat in a harbor. This is different than the refraction of waves near shore because the depth of water does not decrease around the boat like it does near shore. Diffraction is most noticeable when the wavelength is large compared to the obstacle or opening. Thus, no noticeable diffraction may occur if the boat in the harbor is very big.

The sound waves from an owl’s hoot travel a greater distance in the forest than a song bird’s call, because a low pitch owl hoot has a longer wavelength than a high pitch songbird call, and the owl’s waves are able to diffract around trees. Pics on next slide

Diffraction PicsWhen waves pass a barrier they curve around it slightly. When they pass through a small opening, they spread out almost as if they had come from a point source. These effects happen for any type of wave: water; sound; light; seismic waves, etc.

Diffraction & BatsBats use ultrasonic sound waves (a frequency too high for humans to hear) to hunt moths. The reason they use ultrasound is because at lower frequencies much of the sound waves would have a wavelength close to the size of a moth, which means much of the sound would diffract around it.

Bats hunt by echolocation—bouncing sound waves off of prey and listening for the echoes, so they need to emit sound with a wavelength smaller that the typical moth, which means a high frequency is required. High frequency sound waves reflect off the moths rather than diffracting around them. If bats hunted bigger prey, we might have emitted sounds that we could hear.

We’ll learn more about diffraction when we study light.

Standing WavesWhen waves on a rope hits a fixed end, it reflects and is inverted. This reflected waves then combine with oncoming incident waves. At certain frequencies the resulting superposition yields a standing wave, in which some points on the rope called nodes never move at all, and other points called antinodes have an amplitude twice as big as the original wave.

A rope of given length can support standing waves of many different frequencies, called harmonics, which are named based on the number of antinodes.

1st Harmonic ( The Fundamental )

4th Harmonic

continued

2nd Harmonic

3rd Harmonic

Animations:

Standing Waves (cont.)

It is important to understand that a standing is the result of the a wave interfering constructively and destructively with its reflection. Only certain wavelengths will interfere with themselves and produce a standing wave. The wavelengths that work depend on the length of the rope, and we’ll learn how to calculate them in the sound unit. (Standing waves are very important in music.)

Wavelengths that don’t work result in irregular patterns. A standing wave could be simulated with a series of masses on springs, as long as their amplitudes varied sinusoidally.

Standing Wave with Incident & Reflected Waves Shown Separately (scroll down)

Standing Wave with Superposition Shown(scroll down)

Animations:

ResonanceObjects that oscillate or vibrate tend to do so at a particular frequency called the natural frequency. For example, a pendulum will swing back and forth at a certain frequency that only depends on its length, and a mass on a spring will bob up in down at a frequency that depends on the mass and the spring constant. It is possible physically to grab hold of the pendulum or mass and force it to swing or bob at any frequency, but if no one forces them, each will swing of bob at its

m

M

own natural frequency. If left alone, friction will rob the masses of their energy, and their amplitudes will decay. If a periodic force, like an occasional push, matches the period of one of the masses, this is called resonance, and the mass’s amplitude will grow. (continued)

Resonance Animation

Resonance (cont.)Tarzan is swinging through the jungle, but he can’t quite make it across the river to the next tree. So, he asks Jane for a little help. She obliges by giving him a push every time he’s just about to swing away from her. In order to maximize his amplitude to get him across the river, her pushing frequency must match his natural frequency. This is resonance. When resonance occurs her applied force does the maximum amount of positive work. If she mis-times the push, she might do negative work, which would diminish his amplitude. The moral of the story is: Resonance involves timing and matching the natural frequency of an oscillator. When it happens, the oscillator’s amplitude increases.

F

x

Jane does positive work

F

x

Jane does negative work

Resonance QuestionExplain how you could get a 700 lb wrecking ball swing with a large amplitude only by pulling on it with a scrawny piece of dental floss. answer:

Wrecking Ball

Give the ball a little tug, as much as you can without breaking the floss. The ball with barely budge. Continue giving it tugs every time the ball is at its closest to you. If you match the natural frequency of the ball, its amplitude will slowly increase to the desired amount. In this way you are adding energy to the ball very slowly.

Tacoma Narrows BridgeEven bridges have resonant (natural) frequencies. The Tacoma Narrows bridge in Washington state collapsed due to the complicated effects of wind. One day in 1940 the wind blew at just the right speed. The wind was like Jane pushing Tarzan, and the bridge was like Tarzan. The bridge twisted and shook

violently for about an hour. Eventually, the vibrations caused the by wind grew in amplitude until the bridge was destroyed.

Click the pic to see the MPEG video clip.

CreditsThe following images were obtained for these websites:

Marvin the Martian http://store.yahoo.com/rnrdist/warnerbrothers.html

Charlie Brown & Snoopy http://www.snoopy.com/

Godzilla http://www.cinescape.com/godzilla/

Drum & Bugle Corps (Cavaliers of Rosemont, IL) http://www.cavaliers.org/

Sand Scorpion http://www.aps.org/meet/MAR00/baps/vpr/layy3-03-04.html

Beach pic http://www.ssdsupply.com/hawaii.htm Diffraction http://www.glenbrook.k12.il.us/gbssci/phys/Class/sound/u11l3d.html

http://hea-www.harvard.edu/ECT/the_book/Chap1/Chapter1.html

Credits

Wave movies: Dr. Ken Russel, Kettering University http://www.kettering.edu/~drussell/Demos.html

Standing wave animated gifs: Tom Henderson, Glenbrook South High School http://www.physicsclassroom.com/Class/waves/U10L4b.html

Tacoma Narrows Bridge: http://www.civeng.carleton.ca/Exhibits/Tacoma_Narrows/DSmith/fig06.gif