Physics II - Magnetism · Web viewPiano strings, the interior of a tuba, a flute, and violin strings all produce standing waves. Buildings being buffeted about by the wind also have

Physics Name____________________Gehman Date____________Per______

Waves & SoundClass Notes

After a fine romp through electricity and magnetism, we will now return to mechanics… specifically, let’s talk about energy. It’s time to build on the energy thing and keep the excitement moving along. This will surely happen with our next topic – WAVES. Waves turn out to be one of the ways that energy can be transferred from one place to another.

Waves disturbances that travel through space (or a medium) transferring energy from one place to another.

Sound, light, the ocean's surf, and earthquakes are all examples of waves.

A wave is a disturbance that travels through space, but what do that mean?

Think of how gossip travels through the country. This was an example that Einstein used to explain waves. A rumor starts in one of the math classes among the students – Mr. Gunderson says. “Hey, I’m thinking about getting a sports car.” One of the students hears the tail end of the statement and tells a friend out in the hall, “Mr. Gunderson just bought a sports car, I think he got a Ferrari.” The Physics Kahuna supposes that this is possible. Anyway, two hours later one of the students in the Kahuna Physics Institute tells the Physics Kahuna, “Hey, Mr. Gunderson just got a Ferrari.” This student was not in Mr. Gunderson’s class. So how did the gossip get from the math wing to the most southern part of the building? Very likely it happened through a series of conversations. Student A told student B who told student C and so on until the Physics Kahuna heard about it. No one student physically traveled the whole distance from one end of the school to the other with the information, but the scuttlebutt (the proper Navy term for gossip) did. This is how waves work. The medium would be the mass of students teeming in the halls of our beloved CCHS. The disturbances would be the little conversations between people.

Have you ever played the gossip game? You have a whole bunch of people and they are all placed in a line. The first person is given a bit of gossip and repeats it to the next person who repeats it to the next person and so on until it finally reaches the last person in the line. The object of the game is to see if there are any changes to the scuttlebutt as it travels down the line. But the Physics Kahuna’s object in bringing the thing up is to show you how it behaves like a wave – a bunch of disturbances that pass through a medium.

Well, back to the waves. Here is where we introduce the facts.

There are two species of waves, mechanical waves and electromagnetic waves. Mechanical waves require a medium that the wave will then travel through, or rather, the disturbance will travel through. Electromagnetic waves do not require a medium. The disturbance that travels is a changing magnetic and electric field. We’ll examine these waves later on in the course.

A key concept here is that the only things that moves is the disturbance. The medium itself does not move.

That waves carry energy should be obvious. Picture the waves on the ocean. Waves are generated far out at sea mainly by the wind. The wave travels through the water for hundreds or even thousands of miles. Finally

The notes presented in this packet are authored by Glenn Knapp, with editing from Mr. Gehman. 205

Physics Name____________________Gehman Date____________Per______it reaches the shore where the waves pound against the beach. They have enough energy to break down the coastline and erode away continents.

Traveling Waves: The traveling wave is a sort of bump that travels through a medium. A good example of a traveling wave would be a pulse sent down a rope. Such a thing is shown in the drawing below.

Note that the rope itself does not travel, just the pulse, which is the disturbance.You will have seen traveling waves in one of the class demonstrations on a long slinky spring.

Another type of wave is the continuous wave. Sometimes these are called wave trains. A continuous wave requires a periodic source of energy and a medium for the wave to travel through.

There are two types of mechanical waves, the transverse wave and the longitudinal wave.

Transverse Wave - The disturbance direction is perpendicular to the wave direction

Longitudinal Wave - The disturbance direction is parallel to the wave direction

The Physics Kahuna will have shown you a lovely demonstration of these two types of waves. Here are some examples of these waves:

Transverse waves – water waves, waves on a string or spring, “S” seismic waves, and electromagnetic waves.


D ist ur bance d ir ect ion

Wave directionTransverse Wave

L ongitudinal Wave

D ist ur bance d ir ect ion

Wave direction


Longitudinal waves –sound waves, “P” seismic waves; you saw them on springs as well.

A transverse wave is made up of a series of positive pulse and negative pulses. The positive pulses are called crests and negative pulses are called troughs. The height of the crest and the depth of the trough is of course the amplitude of the wave.

Longitudinal waves are a bit different. Basically the medium gets scrunched or pushed together and then pulled apart – stretched out. The areas of increased medium density are called compressions. The compression is surrounded on either side by an area where the medium is stretched out. These areas of low medium density are called rarefactions.

A really important, like key-type concept is that it is only the disturbance that moves, not the medium. Take you your basic water waves, for example. These waves can travel hundreds of miles across the sea. But only the wave travels. The disturbance in water waves is the changing height of the water – the water goes up and it goes down. A duck sitting on the water would not be swept forward with the wave. Instead the duck would bob up as the crest of the wave passed beneath it and then the duck would go down as the trough went past. The Physics Kahuna has made a sequential type drawing of the very thing.

The frequency of a traveling wave is simply the number of cycles divided by the time they occur within (or

something that sounds more scientific than that, the old Physics Kahuna is stretching – at the limits of his writing skills).


Water Wave Passing U nder a Duck

crest

tr ough

R arefaction

C ompression


Here f is the frequency, n is the number of cycles (and has no unit) and t is the time.Example: A speed boat zooms by you as you lie on your floating mattress. You find yourself bobbing up and own on

the waves that the boat made. So, you decide to do a little physics experiment. You count the waves and time how long it takes for them to go past. Six wave crests go by in five seconds. So what is the frequency?

Below is the plot of a transverse wave. The displacement is plotted on the y axis and distance is plotted on the x axis. The amplitude, A, is shown. This is the maximum displacement, just as it was for periodic motion. The other thing that is shown on the graph is the wavelength, . The wavelength is the distance between two in phase points on the wave.

The wave is traveling at some velocity v. We know that velocity is given by this equation:

We also know that the wave travels a distance of in the period, T.

We can plug these into the equation for velocity:

But we also know that the period is given by:

so

We can plug this in for T in the velocity equation we’ve been working on:

This gives us a very important equation:

Examples: A middle C note (notes are these musical frequency kind of deals) has a frequency of approximately 262

Hz. Its wavelength is 1.31 meters. Find the speed of sound.


Y

X

A


A wave has a frequency of 25.0 Hz. Find the (a) wavelength, (b) period, (c) amplitude, and (d) velocity of wave. A graph of this wave is shown below.

(a) Amplitude: We can read the amplitude directly from the graph:

(b) Wavelength: This can be read directly from the graph as well.

(c) Period: The period is the inverse of the frequency, which we know.

(d) Velocity: We use the wave velocity equation.

Here’s another problem.

The speed of light is 3.00 x 108 m/s. What is the wavelength for an FM radio signal broadcast at 105.3 MHz? (Note, radio waves all travel at the speed of light.)

Wave Dynamics: When a wave is busy traveling through a medium, it’s a beautiful thing. But what happens when a wave travels from one medium to another? What happens when two waves meet up? These are good questions and the answers are even better.


YX

35.0 cm

12.0 cm

Physics Name____________________Gehman Date____________Per______Reflection: When a wave traveling through a given medium encounters a new medium, two things happen: some of the energy the wave is carrying keeps going on into the new medium and some of the wave energy gets reflected back from whence it came. If the difference in the wave velocity is large, then most of the wave will be reflected. If the difference in velocity is small, most of the wave will be transmitted into the new medium. The junction of the two mediums is called a boundary.

If there is no relative motion between the two mediums, the frequency will not change on reflection. Also, and this is a key thing, the frequency does not change when the wave travels from one medium into another. It stays the same. This means that the wavelength does change.

There are two types of reflection. The type of reflection depends on how the mediums at the boundary are allowed to move. The two types are: fixed end reflection, and free end reflection.

For fixed end reflection think of the medium as being constrained in its motion. In the picture to the left you see a string that is securely fixed to the wall. The string (the old medium) is free to move up and down, but at the boundary where it meets the new medium (the wall) it is constrained – the string can’t really move up and down like it could before. In fixed end reflection, the wave that is reflected back is out of phase by 180. In the drawing you see an erect pulse traveling down the string. When it is reflected it ends up inverted. It will have the same speed going in as coming out. So in fixed end reflection an erect pulse would be reflected as an inverted pulse.


Physics Name____________________Gehman Date____________Per______In free end reflection, the medium is free to move at the boundary. The reflected wave will be in phase. In the drawing on the right, you see an erect pulse traveling into the boundary being reflected with no phase change. The pulse went in erect and came out erect. Water waves reflecting off a solid wall are a good example of free end reflection.

Wave Speed: Wave speed is constant for any given medium—hence, the speed of sound in air, water, etc. If you change the medium (water to air, e.g.) or the characteristics of the medium (tension, temperature, etc.) you change the wave speed. For a wave on a string, the speed of the wave is directly proportional to the tension in the string. Increase the tension and the wave velocity will increase.

The speed of sound waves in air is directly proportional to air temperature and directly proportional to the air density. In other words, as the temperature of the air increases, the speed of sound increases. As the density of the air increases, the speed of sound also increases. For a given air temperature, the speed of sound would be less in Gillette than it is in Orlando because the air is less dense in Gillette than in Orlando due to the greater altitude of our fair city.

Principle of Superposition: What happens when two or more waves encounter each other as they travel through the same medium? The waves can travel right through each other. As they do this, they add up algebraically to form a resultant wave.

Toss a pebble in a pond and it makes a series of waves that spread out in expanding circles. You can see a drawing of the resultant wave pattern from such an event the waves travel outward in a series of expanding

wave fronts. In the drawing, each of the dark lines represents a wave crest. As the wave front expands, the energy of the wave gets spread out and the wave crest decreases in amplitude. Eventually the energy is so spread out and diluted that the wave will cease to exist. This decrease in amplitude from spreading is called dampening or attenuation. This is why you couldn’t hear your mom calling you home when you were a wee tyke several blocks away.



What happens when two pebbles hit the water? Both produce waves and where the waves meet they produce interference patterns.

Here is a drawing showing a set of interference patterns from the two pebbles in the water deal.

These interference patterns occur where the wave crests and troughs meet each other. The interactions behave according to the law of superposition

Law of superposition when 2 or more waves meet, the resulting displacement is the algebraic sum of the individual separate wave displacements.

These interference patterns will be of great importance later on when we study light.

Basically, the waves add up or cancel each other out. Waves can add up constructively - we get constructive interference, or they can add up destructively - destructive interference.

Where two crests meet, they add up to make an even larger crest. Where a crest and a trough meet, they add up destructively - subtract from each other. So if a wave meets another wave that has the same amplitude but is out of phase (crest to trough, so to speak) they will completely cancel each other out.


+ =

Const r uct ive int er f er ence

+ =

Dest r uct ive int er f er ence

cancellat ion


Standing waves: If you take a long, slinky spring and fix one end of it to a wall and then shake the free end you produce a pulse that travels down the spring. The pulse will be reflected when it reaches the end of the spring. This would be fixed end reflection so it would be out of phase. If you just wave the end of the spring up and down, you get a very confused, chaotic looking thing. But, if you wave the end of the spring at just the right frequency, you can produce a standing wave. You produce an incident wave that travels down the rope. If the frequency is the correct value, the incident wave and the reflected wave will alternately interfere with each other constructively and destructively. The effect is that parts of the spring will not move at all and other parts will undergo great motion. The two waves moving in opposite directions will form a standing wave. The law of superposition acts and we get constructive and destructive interference, which forms the standing wave.

You will have observed standing waves in class with some of the demonstrations that we did. The parts of the wave that don't seem to be doing much are called the nodes and the places where the wave is undergoing maximum movement are called antinodes. The end of a string with such a wave that is attached to the wall would have to be a node, would it not?


v v

v

v

M eetin g w a vesou t o f ph a se

M ee tin g w avesin o f p h ase

Physics Name____________________Gehman Date____________Per______You can produce a variety of standing waves by controlling the frequency of the wave. You will have seen a delightful demonstration of standing waves in action.

Musical instruments produce standing waves. Piano strings, the interior of a tuba, a flute, and violin strings all produce standing waves. Buildings being buffeted about by the wind also have standing waves. Both transverse waves and longitudinal waves can form standing waves.

The standing wave to the left represents one half of the wavelength

of the wave or ½ .

This would be a complete wave cycle or 2/2 or 1 .


1

2

3

4

5

6

7

8

Tim e L a pse View o f S tan d in g W ave


This would be 3/2 or 1 ½ wave.

The lowest frequency standing wave for a system is called the fundamental frequency or the first harmonic.

Fundamental Frequency Lowest frequency of vibration

Integer multiples of the fundamental frequency are called harmonics. The first harmonic is the fundamental frequency. The second harmonic is simply two times the fundamental frequency. The third harmonic is three times the fundamental harmonic. And so on.

AP Physics - Sound Sound is a longitudinal mechanical wave. For most of the time, what we will be talking about is a wave that travels through air. Sound can travel through other mediums - water, other liquids, solids, and gases. We can hear sounds that travel through other mediums than air – put your ear to the wall and hear the sounds on the other side. You hear sounds when you are under water - although not as well as in air. This is because our ears are set up for listening to sounds that move through the atmosphere.

The disturbance which travels through air is the compression of air molecules – they are squeezed together and pulled apart. Sound is a series of traveling high pressure and low pressure fronts.


fu n d a m e n ta l fre q u e n c y1 s t h a rm o n ic

2 n d h a rm o n ic

3 rd h a rm o n ic

Physics Name____________________Gehman Date____________Per______Sound waves are frequently graphed with pressure on the y axis and time on the x axis. This makes the wave look like a transverse wave - a sine wave shape on the graph. But in this depiction, changes in pressure are being plotted Vs time and it is not a depiction of the disturbance itself, which is longitudinal. So please, the Physics Kahuna begs of you, don't be confused about the thing.

Here we see a graph of pressure vs. time. The compressions are regions where the air pressure is greater than the ambient pressure of the air. The rarefactions are areas of lower pressure. These high and low pressure ridges travel outward in an expanding sphere from the sound source.The sound source is simply something that vibrates. It can be the clangor on an alarm clock, a window shade flapping in the wind, or your vocal cords vibrating because air is passing through them. The vibrating sound source collides with air molecules, causing them to scrunch together and pull apart. These scrunches travel through the air. But the air molecules do not physically travel across the room. They are excited by the sound source and gain kinetic energy. They move outward and have elastic collisions with other air molecules, which then gain energy, and so on.

The damping of a sound wave (decrease in amplitude) as it travels is called attenuation. Attenuation depends on the medium and the frequency of the sound. Low frequency sounds are attenuated less than high frequency sounds. Whales make very low frequency sounds (in the neighborhood of 1 - 10 Hz) which can travel hundreds of miles through the ocean. It has recently been found that elephants also employ similar low frequency sounds to communicate. In air, these sounds can travel many miles.



Audible Spectrum: The human ear is not the world's best sound receptor, although it does all right (if you take care of them). A typical person can hear sounds whose frequency ranges from 20 Hz to about 20 000 Hz. This is known as the audio spectrum. Sounds with a higher frequency are called ultrasonic sounds and sounds of a lower frequency are called infrasonic sounds.

Other animals have different hearing spectrums. Dogs can easily hear sounds up to 45 000 Hz. Whales and elephants hear very low frequency sounds (below10 Hz).

The Doppler Effect: Imagine a water bug floating motionless on the surface of a calm pond on a lovely summer day. The bug, bored out of its little bug brain, is tapping the water with a pair of its little segmented legs, making a series of waves that radiate outward on the surface - like the ripples on a pond (actually, they are ripples on a pond, ain’t they? Curious, what?). The bug is unwittingly producing a traveling wave. It would look like this:

The waves spread out in all directions. The distance between the wave crests is the wavelength, . This wavelength is the same in all directions. Now, imagine that the bug starts swimming in one direction, but it still makes its little periodic vibrations with its legs at a constant frequency. We would see a different wave pattern.

Notice that the waves in front of the bug are pushed closer together. Behind the bug, the waves are stretched further apart. The waves in front have a shorter wavelength, the waves to the rear have a longer wavelength. Since the speed of the wave is a constant and equal to the wavelength multiplied by the frequency, this means that the frequency of the waves traveling in front of the bug is higher. The waves behind the bug are lower in frequency. We call this frequency change the Doppler shift or the Doppler effect.


Physics Name____________________Gehman Date____________Per______This happens because the bug makes a wave and then swims after it. So that, when he makes the next wave, it will start out closer to the first wave and so on. As the wave travels to the rear, it is already further away from the first wave, so the wavelength is longer and the frequency shorter. All the waves travel at the same speed so they can't make up the difference.

What happens if the bug swims at the same speed as the wave?

The bug is making a wave and then moving right along with it. So the bug is riding on top of the wave. Then the bug makes another wave that is on top of the first, and so on. The bug ends up riding an enormous wave because all the wave crests are in phase and add up. This would be tough swimming for the old water bug.

What happens if the bug swims faster than the waves?

The bug makes a wave and swims through it into clear water, then it makes the next wave, and so on. The bug is always in front of the waves in nice smooth water. The waves propagating behind the bug will have their crests in phase along a line to either side of the bug that trails back from the bug - they sort of overlap. They will constructively interfere with each other and form a V shaped bow wave. The bow wave will have a very large amplitude as it spreads out behind the bug. Boats and ships do this all the time. Many harbors have speed


Physics Name____________________Gehman Date____________Per______limits for ships because if the ship travels too fast it will generate a large bow wave that can damage property on either side of the vessel.

Doppler Shift and Sound: Sound, like all waves, undergoes this Doppler shift. A car moving towards you pushes its sound waves closer together in front so the sound you actually hear has a higher frequency. When it moves away from you its frequency is lower. You can hear this change when you are near traffic. You can listen to a car and tell from the change in pitch when it stops coming toward you and starts moving away. In order to experience the Doppler shift, there must be relative motion between the sound source and the listener.

If there was no motion between the sound source and the listener, then this equation would hold true:

But there is motion between them. What happens when the listener is moving towards the sound source?

The frequency heard must be:

The velocity is the sum of the velocity of the listener and the speed of sound.

We know that the wavelength is given by:

We can plug that into the equation we’ve derived for the new frequency:

So the new frequency heard by a moving listener closing on a stationary sound source is given by:

f’ is the new frequency heard, f is the frequency actually produced, v is the velocity of sound, and v0 is the velocity of the listener.

If the listener is moving away from the sound source. It is obvious that the new frequency is given by:



But what about if the sound source is moving and the listener is stationary?

If the sound source is moving towards the listener, the wavefronts that arrive at the listener are closer together because of the motion of the sound source. The wavelength measured by the listener is shorter than the wavelength that is actually produced by the sound source. Is this clear? Think about it and do not proceed until the previous statement is clear. The Physics Kahuna will patiently wait for you.

So ’ (wavelength collected by listener) is shorter than (wavelength that is produced). Each vibration or cycle takes a time that is the period of the wave, T. During this time T, the source moves a distance of;

Since the time is the period we get:

The period and frequency are related by:

We can put this together with the equation for distance and we get:

The distance is the change in wavelength, so we can write this as:

This is the change in wavelength that the listener observes.

The observed wavelength, the one the listener measures is the original wavelength minus the change in wavelength:

Because , we can write

We can solve for f’:

So, cleaning it up a bit, we get:

f’ is the frequency heard, f is the original frequency, v is the speed of sound, and vs is the speed of the sound source.

If the sound source is moving away from the stationary listener, the equation becomes:



When solving Doppler problems, we will assume that the speed of sound is 345 m/s.

A train is traveling at 125 km/h. It has a 550.0 Hz train whistle. What is frequency heard by a stationary listener in front of train?

First, convert the train’s speed to meters per second:

Then plug the data into the equation which you will have derived (as above). Make sure to use the proper sign. In this case the train is closing on the listener, so the negative sign is selected.

Supersonic travel: Supersonic motion means that the speed is greater than the speed of sound. (Figure that sound travels at 345 m/s.) In the past when one talked about supersonic motion, one was talking about flight, this is because airplanes were the main things that went faster than sound. (Bullets and projectiles also travel faster than sound.) That is no longer true as in the past few years goofy daredevils have managed to build cars that travel faster than sound. This was hard to do because the speed of sound is greater at the earth’s surface than it is at high altitudes.


Dubious Facts:

According to ancient Chinese astrologers, 70% of omens are bad. Airports that are at higher altitudes require a longer airstrip due to lower air density. Blue is the favorite color of 80% of Americans. California has issued 6 drivers licenses to people named Jesus Christ. Deaf people have safer driving records on average than hearing people in the U.S.A. Five percent of the people who use personal ads for dating are already married.


Supersonic motion is a lot like the deal with the bug swimming faster than the waves it makes. Supersonic airplanes fly faster than the speed of sound, so the sound the plane makes expands outward as do all sounds from a sound source, except that the sound source is always in front. The effect is to form a “sound wake” where the compressions of the sound are constructively reinforced. This creates a “cone” of sound energy that trails behind the aircraft. This cone packs a lot of sound energy, so when it goes by a listener, a really loud, intense sound is heard. This sound wake thing is called a shock wave or sonic boom. Sonic booms can break windows, scare babies and animals, and crack mirrors. For this reason, airplanes are not allowed to go faster than sound over populated areas.

Resonance: The word ‘resonance’ means “resound”. This is an important topic with physics – let’s develop the thing a bit.

One of the goofy demonstrations the Physics Kahuna performed involved a tiny little speaker. It was the cheapest and smallest speaker that he could find. When it was hooked up to a tape player, the sound quality was terrible. It sounded tinny and weak and really lousy -suitable only for rap music. But then the Physics Kahuna pulled out a big piece of cardboard that had a small hole in it. He held the speaker up to the hole and suddenly the sound was ever so rich and nice and much louder too! So how come that happened?

Natural Frequency: Every object has a natural frequency at which it will vibrate. How loud this sound is depends on the elasticity of the material, how long it can sustain a vibration, how well the whole object can vibrate, how big it is, etc. Some materials vibrate better than others. For example, a piece of metal, if excited (say you hit it with something), will vibrate. The vibrations will spread throughout the piece and the whole thing will vibrate. Think of a bell. On the other hand, a piece of Styrofoam, to look at the other extreme, is not nearly so good at vibrating. You can bang on it all day and get nothing better than the odd dull thud kind of sound. So bells are made of metal and not polystyrene foam. At any rate when you bang on an object, it will vibrate at its natural frequency. This principle is used in many musical instruments – xylophones immediately come to mind.

Forced Vibrations: The tinny little speaker did produce sound – bad as it was. Speakers have a small coil that is set to vibrating by the electrical output of an amplifier. Attached to the coil is a paper cone. The vibrating coil then forces the paper cone to vibrate. Air molecules in contact with the speaker cone are then set to vibrating which creates sound waves in the air. These waves travel through the air to your ears. The cone in


Physics Name____________________Gehman Date____________Per______this cheap speaker has a very small area and does not do a very good job of transferring the sound energy into the air. So it sounds lousy and weak.

When the big piece of cardboard was brought out and the speaker was pressed onto it, the speaker forced the cardboard to vibrate. Since it was mechanically connected to the cardboard, the vibrations were easily transferred. The cardboard had a very large area in contact with the air, so it was more efficient at sending the sound into the air. This is why the sound suddenly sounded so much better when the cardboard made its appearance.We call this phenomenon forced vibration.

Forced vibration The vibration of an object that is made to vibrate by another vibrating object in contact.

A tuning fork makes a very weak sound - you can barely hear the thing. Place a vibrating tuning fork against a window or desktop, however, and the sound will become much louder.

Another example of this that you experienced was the coat hanger on a string deal. When the clothes hanger was hung from a piece of string and struck with something, you couldn’t really hear anything (unless you placed your ear very close to the hanger). But if you pressed the string into your ear, you heard a deep resonant gong/bell type sound when the hanger was tapped. The string was forced to vibrate and conducted the sound to your ears.

People do not recognize their own voice. Have you ever heard a tape recording of your voice? Did it sound like you? Probably not. This is because the sounds that you make travel to your ears via your skull and not through the air. The vibrations that reach your ear through your bones and tissue sound slightly different than the vibrations that travel through the air.

Forced vibration is very important in music. Many instruments have sounding boards which are forced to vibrate to make the instrument sound louder - pianos are a good example of this. Other instruments have bodies that act as sounding boards. Guitars, violins, banjos, mandolins, and ukuleles fit into this category. The vibrating strings of these instruments produce a very pitiful weak sound, but place the same string on your average Martin guitar, and you get a whole different deal. Much of what makes up the quality of sound produced by an instrument depends on how well it can transfer sound energy into the air. We've all heard of these fabulous old violins from Italy - the best known are the Stradivarius violins - which produce really exquisite sound. Modern violinists claim that no modern violin can even come close to the quality of sound that these violins produce. So a Stradivarius violin can sell (if anyone is willing to sell theirs) for millions of dollars. How these violins were made - the secret that gives them their rich sound is not known. All sorts of people are desperately trying to duplicate the feat, but so far, no one has succeeded. At least according to the music experts. The Physics Kahuna himself cannot tell the difference.

Resonance Discussed: Resonance is sometimes called sympathetic vibration. It means to "re sound" or "sound again". If two objects which have the same natural frequency are placed near each other, and one is set to vibrate, the other one will begin to vibrate as well.

What happens is that the first instrument forces the air to vibrate at its natural frequency. These sound waves travel to the other object and causes it to vibrate at that very frequency. But this is also its natural frequency. So the waves induce a vibration. Each compression arrives in phase with the vibrations of the object and adds


Physics Name____________________Gehman Date____________Per______to its energy, and causes it to build up. So the second object will begin to vibrate and then vibrate stronger and stronger.

A common demonstration of resonance can be done with a book. The book is hung from a bar. A person applies a puff of air to the book, causing it to swing a little. When it comes all the way back from the swing, another puff is applied. It swings just a bit farther out. Apply yet another puff, it swings more. Eventually you get the book to really swing. What you are doing is applying energy at the resonant frequency of the system. So the motion builds up and becomes greater and greater. For this to happen, however, the energy must be fed in at the resonant frequency of the object. We can associate these resonant waves with standing waves in the object. If you blow randomly, this will not work.

The other interesting thing is that you can do it at a harmonic frequency. Blow every other swing or every third swing. Do you see how this would work?

Resonant Air Columns: Have you ever blown into a pop bottle and gotten the thing to make a nice, deep, melodious sound? Bottles can do this because they will resonate. When you blow across the top of the bottle, you create turbulence – burbles of air – which occur at a broad band of frequencies. This is called the edge effect. One of those frequencies is the bottle’s resonant frequency. A standing wave forms in the bottle’s interior. As energy is fed in from the blowing thing, the standing wave gains energy until it is loud enough to hear.

Close Ended Pipes: The reason that the bottle resonates is that a standing wave forms in it. The wavelength of the standing wave has to "fit the bottle", so only the one frequency (or its harmonics) will resonate and be heard. The other frequencies aren't loud enough to be audible. The closed end of the pipe is a displacement node because the wall does not allow for the longitudinal displacement of the air molecules. As a result, the reflected sound pulse from the closed end is 180 out of phase with the incident wave. The closed end corresponds to a pressure antinode.

The open end of the pipe is, for all practical purposes, a displacement antinode and a pressure node. The reflected wave pulse from an open end of the pipe is reflected in phase. The open end of a pipe is essentially the atmosphere, so no pressure variations take place. The reflection actually takes place a slight distance outside the pipe, but we will ignore that.

Let's look at a simple pipe that has a standing wave within it. There has to be a displacement node at the closed end and a displacement antinode at the open end. With this in mind, we can draw in the various standing waves that can form within the pipe. The first one is a quarter of a wave. This is the lowest resonant


F irst harmonic

Third harmonic

F ifth harmonic

14

34

54

Physics Name____________________Gehman Date____________Per______frequency that can form a standing wave in the tube. Note that the closed end reflects the sound wave out of phase - like a fix-ended wave is reflected.

Anyway, the pipe length turns out to be about ¼ of the wavelength. The lowest frequency is called the fundamental frequency. Its wavelength is essentially ¼ of the length of the pipe. The next possible frequency will have a wavelength that is ¾ of the pipe's length, then 5/4 of the length, and so on. You can see that only odd harmonics are resonant in the close-ended pipe.

Only the odd harmonics are present in a resonating close-ended pipe.

The equation that relates wavelength, frequency and wave speed is:For the fundamental frequency (the first harmonic), the wavelength is:

The frequency in the system must be:

If we want the frequency of the third or fifth or whatever harmonic, we would get:

Here fn is the harmonic frequency that resonates in the pipe, v is the speed of sound, L is the length of the pipe, and n is an integer for the harmonic that you want.

The wavelength for any harmonic would be:

Open Ended Pipes: Open-ended pipes can also resonate. At both ends of the pipe, the wave is reflected in phase. The fundamental wave and associated harmonics would look like this:

The wavelength is approximately twice the length of the tube. Note also that the open ended pipe has all harmonics present.

Using the same method of derivation as we did with the close-ended pipe, we can develop an equation for the wavelength for the fundamental frequency.

Here is the equation. See if you can derive it yourself.

A critical difference between the open and close-ended pipes is that the open-ended pipe can have all harmonics present. The close-ended pipe is limited to the odd harmonics.

All harmonics can be present in a resonant open-ended pipe.The notes presented in this packet are authored by Glenn Knapp, with editing from Mr. Gehman. 225

F irst harmonic

S econd harmonic

Third harmonic

12

22

32


A pipe is closed at one end and is 1.50 m in length. If the sound speed is 345 m/s, what are the frequencies of the first three harmonics that would be produced?

Use the close ended pipe formula to find the first harmonic (the fundamental frequency):

Recall that close ended pipes only have the odd harmonics, so the next two would be the third and fifth harmonics:

Musical instruments play things we call notes. A note is a specific frequency from a thing called a scale, which is a collection of eight notes called an octave. The notes are: A, B, C, D, E, F, and G. A C, for example, on some scales is 262 Hz. The interesting thing is that if you multiple the frequency of a note, you get the same note, but it is a higher harmonic. When you play a 262 Hz tone and a 524 Hz tone at the same time, they would give you the same musical sense, but would sound richer and fuller. The reason that different musical instruments sound different – think piano and mandolin, even when they are playing the same note, is that each instrument has its own set of harmonics. Some instruments only produce a fundamental frequency – flutes often do this, while other instruments produce a bunch of harmonics.

In the examples below, you see a pressure Vs. time graph for different things and instruments. A tuning fork, the first example, produces a single frequency, so its graph resembles a sine wave.

The flute and clarinet do not appear to be sine waves. This is because of the presence of harmonics and the law of superposition.


Tuning Fork

F lute

C lar inet


The last graph (above) shows the intensity of the different harmonics for the same instruments. The tuning fork only has the first harmonic. The flute has a strong 2nd and 4th harmonic. These are stronger than the fundamental frequency. The clarinet has a strong 5th and 1st harmonic. This is why they each sound different to our ears.


1 2 3 4 5 6Har monics

Intensity



7 7 8 9

Tuning For k F lute C lar inet

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Physics II - Magnetism · Web viewPiano strings, the interior of a tuba, a flute, and violin strings all produce standing waves. Buildings being buffeted about by the wind also have