Out at sea, the waves roll on.

Transcription

1 PAP Physics - Waves 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 o the ways that energy can be transerred rom one place to another. Out at sea, the waves roll on. Waves disturbances that travel through space transerring energy rom one place to another. Sound, light, and the ocean's sur are all examples o waves. There are two species o 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 ield (both o which you, lucky student, will get to study in the near uture). 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 itsel does not move. Waves carry energy should be obvious. Picture the waves on the ocean. Waves are generated ar out at sea mainly by the wind. The wave travels through the water or hundreds or even thousands o miles. Finally 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 o bump that travels through a medium. A good example o a traveling wave would be a pulse sent down a rope. Such a thing is shown in the drawing below.

2 Note that the rope itsel does not travel, just the pulse, which is the disturbance. You will have seen traveling waves in one o the class demonstrations on a long slinky spring. Another type o wave is the continuous wave. Sometimes these are called wave trains. A continuous wave requires a periodic source o energy and a medium or the wave to travel through. There are two types o 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 Transverse Wave Wave direction Disturbance direction Longitudinal Wave Wave direction Disturbance direction The Physics Kahuna will have shown you a lovely demonstration o these two types o waves. Here are some examples o these waves: Transverse waves water waves, waves on a string or spring, seismic waves, and electromagnetic waves. Longitudinal waves well, the best example would be sound waves, but you saw them on springs as well. crest A transverse wave is made up o a series o positive pulse and negative pulses. The positive pulses are called crests and negative pulses are called troughs. The height o the crest and the depth o the trough is o course the amplitude o the wave. trough

3 Longitudinal waves are a bit dierent. Basically the medium gets scrunched or pushed together and then pulled apart stretched out The areas o increased medium density are called compressions. The compression is surrounded on either side by an area where the medium is stretched out. These areas o low medium density are called rareactions. Rareaction Compression 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 or example. These waves can travel hundreds o miles across the sea. But only the wave travels. The disturbance in water waves is the changing height o the water the water goes up and it goes down. A duck sitting on the water would not be swept orward with the wave. Instead the duck would bob up as the crest o the wave passed beneath it and then the duck would go down as the trough went past. Water Wave Passing Under a Duck The requency o a traveling wave is simply the number o cycles divided by the time they occur within n Here is the requency, n is the number o cycles (and has no unit) and t t is the time. A speed boat zooms by you as you lie on your loating mattress. You ind yoursel 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 or them to go past. Six wave crests go by in ive seconds. So what is the requency? n 6.0 t 5.0 s 1.2 Hz Below is the plot o a transverse wave. The displacement is plotted on the y axis and distance is plotted on the x axis. The Y amplitude, A, is shown. This is the maximum displacement, just as it was or 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. A X

4 The wave is traveling at some velocity v. We know that velocity is given by this equation: x v t We also know that the wave travels a distance o in the period, T. We can plug these into the equation or velocity: x v so v t T T But we also know that the period is given by: T 1 so We can plug this in or T in the velocity equation we ve been working on: v T This gives us a very important equation: v A middle C note (notes are these musical requency kind o deals) has a requency o approximately 262 Hz. Its wavelength is 1.31 meters. Find the speed o sound. 1 v m 343 m s s A wave has a requency o 25.0 Hz. Find the (a) wavelength, (b) period, (c) amplitude, and (d) velocity o wave. A graph o this wave is shown below cm Y 12.0 cm X (a) Amplitude: We can read the amplitude directly rom the graph: 12.0 cm (b) Wavelength: This can be read directly rom the graph as well cm (c) Period: The period is the inverse o the requency, which we know. 1 1 T s (d) Velocity: We use the wave velocity equation s 1 v m 8.75 m s s

5 Here s another problem. The speed o light is 3.00 x 10 8 m/s. What is the wavelength or an FM radio signal broadcast at MHz? (Note, radio waves all travel at the speed o light.) v v 8m x x10 m 2.85 m s x10 s Wave Dynamics: When a wave is busy t raveling through a medium, it s a beautiul thing. But what happens when a wave travels rom one medium to another? What happens when two waves meet up? These are good questions and the answers are even better. Relection: When a wave traveling through a given medium encounters a new medium, two things happen: some o the energy the wave is carrying keeps going on into the new medium and some o the wave energy gets relected back rom whence it came. I the dierence in the wave velocity is large, then most o the wave will be relected. I the dierence in velocity is small, most o the wave will be transmitted into the new medium. The junction o the two mediums is called a boundary. I there is no relative motion between the two mediums, the requency will not change on relection. Also, and this is a key thing, the requency does not change when the wave travels rom one medium into another. It stays the same. This means that the wavelength does change. There are two types o relection. The type o relection depends on how the mediums at the boundary are allowed to move. The two types are: ixed end relection, and ree end relection. For ixed end relection think o the medium as being constrained in its motion. In the picture to the let you see a string that is securely ixed to the wall. The string (the old medium) is ree 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 beore. In ixed end relection, the wave that is relected back is out o phase by 180. In the drawing you see an erect pulse traveling down the string. When it is relected it ends up inverted. It will have the same speed going in as coming out. So in ixed end relection an erect pulse would be relected as an inverted pulse.

6 In ree end relection, the medium is ree to move at the boundary. The relected wave will be in phase. In the drawing on the right, you see an erect pulse traveling into the boundary being relected with no phase change. The pulse went in erect and came out erect. Water waves relecting o a solid wall are a good example o ree end relection. Wave Speed: For a wave on a string, the speed o the wave is directly proportional to the tension in the string. Increase the tension and the wave velocity will increase. The speed o sound waves in air is directly proportional to air temperature and directly proportional to the air density. In other words, as the temperature o the air increases, the speed o sound increases. As the density o the air increases, the speed o sound also increases. For a given air temperature, the speed o 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 o our air city. Principle o 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 orm a resultant wave. Toss a pebble in a pond and it makes a series o waves that spread out in expanding circles. You can see a drawing o the resultant wave pattern rom such an event the waves travel outward in a series o expanding wave ronts. In the drawing, each o the dark lines represents a wave crest. As the wave ront expands, the energy o 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 rom 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 intererence patterns. Here is a drawing showing a set o intererence patterns rom the two pebbles in the water deal.

7 These intererence patterns occur where the wave crests and troughs meet each other. The interactions behave according to the law o superposition Law o superposition when 2 or more waves meet, the resulting displacement is the algebraic sum o the individual separate wave displacements. These intererence patterns will be o 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 intererence, or they can add up destructively - destructive intererence. 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 rom each other. So i a wave meets another wave that has the same amplitude but is out o phase (crest to trough, so to speak) they will completely cancel each other out. + = Constructive intererence v + = cancellation Destructive intererence v v v Meeting waves out o phase Meeting waves in o phase

8 Standing waves: I you take a long, slinky spring and ix one end o it to a wall and then shake the ree end you produce a pulse that travels down the spring. The pulse will be relected when it reaches the end o the spring. This would be ixed end relection so it would be out o phase. I you just wave the end o the spring up and down, you get a very conused, chaotic looking thing. But, i you wave the end o the spring at just the right requency, you can produce a standing wave. You produce an incident wave that travels down the rope. I the requency is the correct value, the incident wave and the relected wave will alternately interere with each other constructively and destructively. The eect is that parts o the spring will not move at all and other parts will undergo great motion. The two waves moving in opposite directions will orm a standing wave. The law o superposition acts and we get constructive and destructive intererence, which orms the standing wave. You will have observed standing waves in class with some o the demonstrations that we did. The parts o 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 o a string with such a wave that is attached to the wall would have to be a node, would it not? You can produce a variety o standing waves by controlling the requency o the wave. You will have seen a delightul demonstration o standing waves in action Time Lapse View o Standing Wave

9 Musical instruments produce standing waves. Piano strings, the interior o a tuba, a lute, and violin strings all produce standing waves. Buildings being bueted about by the wind also have standing waves. Both transverse waves and longitudinal waves can orm standing waves. The standing wave to the let represents one hal o the wavelength o the wave or ½. This would be a complete wave cycle or 2/2 or 1. This would be 3/2 or 1 ½ wave. The lowest requency standing wave or a system is called the undamental requency or the irst harmonic. Fundamental Frequency Lowest requency o vibration Integer multiples o the undamental requency are called harmonics. The irst harmonic is the undamental requency. The second harmonic is simply two times the undamental requency. The third harmonic is three times the undamental harmonic. And so on. undamental requency 1st harmonic 2nd harmonic 3rd harmonic Sound Sound is a longitudinal mechanical wave. For most o 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 or listening to sounds that move through the atmosphere. Rareaction v Compression Wavelength

10 The disturbance which travels through air is the compression o air molecules they are squeezed together and pulled apart. Sound is a series o traveling high pressure and low pressure ronts. Sound waves are requently 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 o the disturbance itsel, which is longitudinal. Pressure Time Here we see a graph o pressure vs time. The compressions are regions where the air pressure is greater than the ambient pressure o the air. The rareactions are areas o lower pressure. These high and low pressure ridges travel outward in an expanding sphere rom the sound source. The sound source is simply something that vibrates. It can be the clangor on an alarm clock, a window shade lapping 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. Sound waves propagate as expanding sphere Sound source Air molecule gets excited, gains energy, collides with another molecule and transers energy to it and so on.

11 The damping o a sound wave (decrease in amplitude) as it travels is called attenuation. Attenuation depends on the medium and the requency o the sound. Low requency sounds are attenuated less than high requency sounds. Whales make very low requency sounds (in the neighborhood o 1-10 Hz) which can travel hundreds o miles through the ocean. It has recently been ound that elephants also employ similar low requency 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 (i you take care o them). A typical person can hear sounds whose requency ranges rom 20 Hz to about Hz. This is known as the audio spectrum. Sounds with a higher requency are called ultrasonic sounds and sounds o a lower requency are called inrasonic sounds. Other animals have dierent hearing spectrums. Dogs can easily hear sounds up to Hz. Whales and elephants hear very low requency sounds (below10 Hz). The Doppler Eect: Imagine a water bug loating motionless on the surace o a calm pond on a lovely summer day. The bug, bored out o its little bug brain, is tapping the water with a pair o its little segmented legs, making a series o waves that radiate outward on the surace - 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: Wave Pattern Created By A Stationary Bug wiggling its Legs in the Water

12 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 requency. We would see a dierent wave pattern. Notice that the waves in ront o the bug are pushed closer together. Behind the bug, the waves are stretched urther apart. The waves in ront have a shorter wavelength, the waves to the rear have a longer wavelength. Since the speed o the wave is a constant and equal to the wavelength multiplied by the requency, this means that the requency o the waves traveling in ront o the bug is higher. The waves behind the bug are lower in requency. We call this requency change the Doppler shit or the Doppler eect. This happens because the bug makes a wave and then swims ater it. So that, when he makes the next wave, it will start out closer to the irst wave and so on. As the wave travels to the rear, it is already urther away rom the irst wave, so the wavelength is longer and the requency shorter. All the waves travel at the same speed so they can't make up the dierence. What happens i 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 o the wave. Then the bug makes another wave that is on top o the irst, 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 or the old water bug. What happens i the bug swims aster 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 ront o the waves in nice smooth water. The waves propagating behind the bug will have their crests in phase along a line to either side o the bug that trails back rom the bug - they sort o overlap. They will constructively interere with each other and orm 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 limits or ships because i the ship travels too ast it will generate a large bow wave that can damage property on either side o the vessel.

13 Doppler Shit and Sound: Sound, like all waves, undergoes this Doppler shit. A car moving towards you pushes its sound waves closer together in ront so the sound you actually hear has a higher requency. When it moves away rom you its requency is lower. You can hear this change when you are near traic. You can listen to a car and tell rom the change in pitch when it stops coming toward you and starts moving away. In order to experience the Doppler shit, there must be relative motion between the sound source and the listener. I there was no motion between the sound source and the listener, then this equation would hold true: v v v so and But there is motion between them. What happens when the listener is moving towards the sound source? The requency heard must be: v' v v0 ' The velocity is the sum o the velocity o the listener and the speed o sound. v We know that the wavelength is given by: We can plug that into the equation we ve derived or the new requency: v v0 v v0 v v0 ' v v So the new requency heard by a moving listener closing on a stationary sound source is given by: v v0 ' v is the new requency heard, is the requency actually produced, v is the velocity o sound, and v 0 is the velocity o the listener. I the listener is moving away rom the sound source. It is obvious that the new requency is given by: v v0 ' v But what about i the sound source is moving and the listener is stationary? I the sound source is moving towards the listener, the waveronts that arrive at the listener are closer together because o the motion o 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 or you.

14 So (wavelength collected by listener) is shorter than (wavelength that is produced). Each vibration or cycle takes a time that is the period o the wave, T. During this time T, the source moves a distance o; x v t x vst Since the time is the period we get: x vst vst The period and requency are related by: 1 T We can put this together with the equation or distance and we get: x vst 1 vs vs x 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 vs ' ' v We can solve or : vs v v v ' ' v vs ' s ' v v v v v v v ' s So, cleaning it up a bit, we get: v ' v vs s is the requency heard, is the original requency, v is the speed o sound, and v s is the speed o the sound source. I the sound source is moving away rom the stationary listener, the equation becomes: v s v ' v vs

15 When solving Doppler problems, we will assume that the speed o sound is 345 m/s. A train is traveling at 125 km/h. It has a Hz train whistle. What is requency heard by a stationary listener in ront o train? First, convert the train s speed to meters per second: km 1 h 1000 m m h s 1km s 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. m 345 v ' Hz s 612 Hz v v m m s s s Supersonic travel: Supersonic motion means that the speed is greater than the speed o sound. (Figure that sound travels at 345 m/s.) In the past when one talked about supersonic motion, one was talking about light, this is because airplanes were the main things that went aster than sound. (Bullets and projectiles also travel aster than sound.) That is no longer true as in the past ew years gooy daredevils have managed to build cars that travel aster than sound. This was hard to do because the speed o sound is greater at the earth s surace than it is at high altitudes. Supersonic motion is a lot like the deal with the bug swimming aster than the waves it makes. Supersonic airplanes ly aster than the speed o sound, so the sound the plane makes expands outward as do all sounds rom a sound source, except that the sound source is always in ront. The eect is to orm a sound wake where the compressions o the sound are constructively reinorced. This creates a cone o sound energy that trails behind the aircrat. This cone packs a lot o 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 aster than sound over populated areas.

16 Resonance: The word resonance means resound. This is an important topic with physics let s develop the thing a bit. Natural Frequency: Every object has a natural requency at which it will vibrate. How loud this sound is depends on the elasticity o 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 o metal, i excited (say you hit it with something), will vibrate. The vibrations will spread throughout the piece and the whole thing will vibrate. Think o a bell. On the other hand, a piece o Styrooam, 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 o sound. So bells are made o metal and not polystyrene oam. At any rate when you bang on an object, it will vibrate at its natural requency. This principle is used in many musical instruments xylophones immediately come to mind. Forced Vibrations: Speakers have a small coil that is set to vibrating by the electrical output o an ampliier. Attached to the coil is a paper cone. The vibrating coil then orces 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 this cheap speaker has a very small area and does not do a very good job o transerring the sound energy into the air. So it sounds lousy and weak. When the big piece o cardboard was brought out and the speaker was pressed onto it, the speaker orced the cardboard to vibrate. Since it was mechanically connected to the cardboard, the vibrations were easily transerred. The cardboard had a very large area in contact with the air, so it was more eicient 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 orced vibration. Forced vibration The vibration o an object that is made to vibrate by another vibrating object in contact. A tuning ork makes a very weak sound - you can barely hear the thing. Place a vibrating tuning ork against a window or desktop, however, and the sound will become much louder. Another example o this that you experienced was the coat hanger on a string deal. When the clothes hanger was hung rom a piece o string and struck with something, you couldn t really hear anything (unless you placed your ear very close to the hanger). But i you pressed the string into your ear, you heard a deep resonant gong/bell type sound when the hanger was tapped. The string was orced to vibrate and conducted the sound to your ears. People do not recognize their own voice. Have you ever heard a tape recording o 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 dierent than the vibrations that travel through the air.

17 Forced vibration is very important in music. Many instruments have sounding boards which are orced to vibrate to make the instrument sound louder - pianos are a good example o this. Other instruments have bodies that act as sounding boards. Guitars, violins, banjos, mandolins, and ukuleles it into this category. The vibrating strings o these instruments produce a very pitiul weak sound, but place the same string on your average Martin guitar, and you get a whole dierent deal. Much o what makes up the quality o sound produced by an instrument depends on how well it can transer sound energy into the air. We've all heard o these abulous old violins rom 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 o sound that these violins produce. So a Stradivarius violin can sell (i anyone is willing to sell theirs) or millions o dollars. How these violins were made - the secret that gives them their rich sound is not known. All sorts o people are desperately trying to duplicate the eat, but so ar, no one has succeeded. At least according to the music experts. Resonance Discussed: Resonance is sometimes called sympathetic vibration. It means to "re sound" or "sound again". I two objects which have the same natural requency 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 irst instrument orces the air to vibrate at its natural requency. These sound waves travel to the other object and causes it to vibrate at that very requency. But this is also its natural requency. So the waves induce a vibration. Each compression arrives in phase with the vibrations o the object and adds 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 o resonance can be done with a book. The book is hung rom a bar. A person applies a pu o air to the book, causing it to swing a little. When it comes all the way back rom the swing, another pu is applied. It swings just a bit arther out. Apply yet another pu, it swings more. Eventually you get the book to really swing. What you are doing is applying energy at the resonant requency o the system. So the motion builds up and becomes greater and greater. For this to happen, however, the energy must be ed in at the resonant requency o the object. We can associate these resonant waves with standing waves in the object. I you blow randomly, this will not work. Musical instruments play things we call notes. A note is a speciic requency rom a thing called a scale, which is Tuning Fork a collection o eight notes called an octave. The notes are: A, B, C, D, E, F, and G. A C, or example, on some scales is 262 Hz. The interesting thing is that i you multiple the requency o a note, you get the same note, Flute 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 uller. The reason that dierent musical instruments sound dierent think piano and mandolin, even when Clarinet they are playing the same note, is that each instrument has its own set o harmonics. Some instruments only produce a undamental requency lutes oten do this, while other instruments produce a bunch o harmonics.

18 In the examples below, you see a pressure Vs. time graph or dierent things and instruments. A tuning ork, the irst example, produces a single requency, so its graph resembles a sine wave. The lute and clarinet do not appear to be sine waves. This is because o the presence o harmonics and the law o superposition. Tuning Fork Flute Clarinet Intensity Harmonics Harmonics Harmonics The last graph (above) shows the intensity o the dierent harmonics or the same instruments. The tuning ork only has the irst harmonic. The lute has a strong 2 nd and 4 th harmonic. These are stronger than the undamental requency. The clarinet has a strong 5 th and 1 st harmonic. This is why they each sound dierent to our ears.