Sound Longitudinal Waves Interference Pressure Graphs Standing Waves in a String: Two fixed ends Speed of Sound Wavefronts Frequency & Pitch (human range) Standing Waves in a Tube: One open end Two open ends
The Human Ear Musical Instruments (and other complex sounds) Sonar & Echolocation Beats Doppler Effect (and sonic booms) Intensity Sound Level (decibels) Longitudinal Waves As you learned in the unit on waves, in a longitudinal wave the
particles in a medium travel back & forth parallel to the wave itself. Sound waves are longitudinal and they can travel through most any medium, so molecules of air (or water, etc.) move back & forth in the direction of the wave creating high pressure zones (compressions) and low pressure zones (rarefactions). The molecules act just like the individual coils in the spring. The faster the molecules move back & forth, the greater the frequency of the wave, and the greater distance they move, the greater the waves amplitude. molecule wavelength, rarefaction Animation compression Review of Sound
Sound Waves: Molecular View When sound travels through a medium, there are alternating regions of high and low pressure. Compressions are high pressure regions where the molecules are crowded together. Rarefactions are low pressure regions where the molecules are more spread out. An individual molecule moves side to side with each compression. The speed at which a compression propagates through the medium is the wave speed, but this is different than the speed of the molecules themselves. wavelength, Pressure vs. Position The pressure at a given point in a medium fluctuates slightly as sound waves pass by. The wavelength is determined by the distance between consecutive compressions or consecutive rarefactions. At each compression the pressure is a tad bit higher than its normal pressure. At each rarefaction the pressure is a tad bit lower than normal. Lets call the equilibrium (normal) pressure P0 and the difference in pressure from
equilibrium P. P varies and is at a max at a compression or rarefaction. In a fluid like air or water, Pmax is typically very small compared to P0 but our ears are very sensitive to slight deviations in pressure. The bigger P is, the greater the amplitude of the sound wavelength, wave, and the louder the sound. Pressure vs. Position Graph P B A: P = 0; P = P0 B: P > 0; P = Pmax C: P < 0; P = Pmin x
A C Pressure vs. Time The pressure at a given point does not stay constant. If we only observed one position we would find the pressure there varies sinusoidally with time, ranging from: P0 to P0 + Pmax back to P0 then to P0 - Pmax and back to P0 The cycle can also be described as: equilibrium compression equilibrium rarefaction equilibrium The time it takes to go through this cycle is the period of the wave. The number of times this cycle happens per second is the frequency of the wave in Hertz. Therefore, the pressure in the medium is a function of both position and time!
P Pressure vs. Time Graph T t Rather than looking at a region of space at an instant in time, here were looking at just one point in space over an interval of time. At time zero, when the pressure readings began, the molecules were at their normal pressure. The pressure at this point in space fluctuates sinusoidally as the waves pass by: normal high normal low normal. The time needed for one cycle is the period. The higher the frequency, the shorter the period. The amplitude of the graph represents the maximum deviation from normal pressure (as it did on the pressure vs. position graph), and this corresponds to loudness.
Comparison of Pressure Graphs Pressure vs. Position: The graph is for a snapshot in time and displays pressure variation for over an interval of space. The distance between peaks on the graph is the wavelength of the wave. Pressure vs. Time: The graph displays pressure variation over an interval of time for only one point in space. The distance between peaks on the graph is the period of the wave. The reciprocal of the period is the frequency. Both Graphs: Sound waves are longitudinal even though these graphs look like transverse waves. Nothing in a sound wave is actually waving in the shape of these graphs! The amplitude of either graph corresponds to the loudness of the sound. The absolute pressure matters not. For loudness, all that matters is how much the pressure deviates from its norm, which doesnt have to be much. In real life the amplitude would diminish as the sound waves spread out. Speed of Sound As with all waves, the speed of sound depends on the medium
through which it is traveling. In the wave unit we learned that the speed of a wave traveling on a rope is given by: Rope: v = F F = tension in rope = mass per unit length of rope In a rope, waves travel faster when the rope is under more tension and slower if the rope is denser. The speed of a sound wave is given by: Sound:
v = B B = bulk modulus of medium = mass per unit volume (density) The bulk modulus, B, of a medium basically tells you how hard it is to compress it, just as the tension in a rope tells you how hard it is stretch it or displace a piece of it. (continued) Speed of Sound Rope: Sound: v =
v = F B (cont.) Notice that each equation is in the form v = elastic property inertial property The bulk modulus for air is tiny compared to that of water, since air is easily compressed and water nearly incompressible. So, even
though water is much denser than air, water is so much harder to compress that sound travels over 4 times faster in water. Steel is almost 8 times denser than water, but its over 70 times harder to compress. Consequently, sound waves propagate through steel about 3 times faster than in water, since (70 / 8) 0.5 3. Mach Numbers Depending on temp, sound travels around 750 mph (340 m/s), which would be Mach 1. Twice this speed would be Mach 2, which is about the max speed for the F-22 Raptor. Speed Racer drives a car called The Mach 5, which would imply it can go 5 times the speed of sound. Click below for video Short version here v = Temperature & the Speed of
Sound Because the speed of sound is inversely proportional B to the mediums density, the more dense the medium, the faster sound travels. The hotter a substance is, vibrate and the more room they take up. the faster its molecules/atoms This lowers the substances density, which is significant in a gas. So, in the summer, sound travels slightly faster outside than it does in the winter. To visualize this keep in mind that molecules must bump into each other in order to transmit a longitudinal wave. When molecules move quickly, they need less time to bump into their neighbors. The speed of sound in dry air is given by: v 331.4 + 0.60 T, where T is air temp inC. Here are speeds for sound: Air, 0 C: 331 m/s
Air, 20 C: 343 m/s Iron: 5130 m/s Glass (Pyrex): 5640 m/s Water, 25 C: 1493 m/s Diamond: 12 000 m/s Why can you hear campers across a lake? Wavefronts crest trough Some waves are one dimensional, like vibrations in a guitar string or sound waves traveling along a metal rod. Some waves are two dimensional, such as surface water waves or seismic waves traveling along the surface of the Earth. Some waves are 3-D, such as sound traveling in all directions from a bell, or light doing the same from a flashlight. To visualize 2-D and 3-D waves, we often draw
wavefronts. The red wavefronts below could represent the crest of water waves on a pond moving outward after a rock was dropped in the middle. They could also be used to represent high pressure zones in sound waves. When several waves overlap it is know as DIFFRACTION. Echoes & Reverberation An echo is simply a reflected sound wave. Echoes are more noticeable if you are out in the open except for a distant, large object. If you went out to the dessert and yelled, you might hear a distant canyon yell back at you. The time between your yell and hearing your echo depends on the speed of sound and on the distance to the to the canyon. In fact, if you know the speed of sound, you can easily calculate the distance just by timing the delay of your echo. Reverberation is the repeated reflection of sound at close quarters. If you were to yell while inside a narrow tunnel, your reflected
sound waves would bounce back to your ears so quickly that your brain wouldnt be able to distinguish between the original yell and its reflection. It would sound like a single yell of slightly longer duration. Animation Single echo Sonar SOund NAvigation and Ranging In addition to locating prey, bats and dolphins use sound waves for navigational purposes. Submarines do this too. The principle is to send out sound waves and listen for echoes. The longer it takes an echo to return, the farther away the object that reflected those waves. Sonar is used in commercial fishing boats to find schools of fish. Scientists use it to map the ocean floor. Special glasses that make use of sonar can help blind people by
producing sounds of different pitches depending on how close an obstacle is. If radio (low frequency light) waves are used instead of sound in an instrument, we call it radar (radio detection and ranging). Sonic Booms When a source of sound is moving at the speed of sound, the wavefronts pile up on top of each other. This makes their combined amplitude very large, resulting in a shock wave and a sonic boom. At supersonic speeds a Mach cone is formed. The faster the source compared to sound, the smaller the shock wave angle will be. Wave front Animations Another cool animation
Animation with sound (click on The Doppler Effect, then click on the button marked: Movie: F-18 Hornet breaking the sound barrier (click on MPEG movie) Doppler Effect A tone is not always heard at the same frequency at which it is emitted. When a train sounds its horn as it passes by, the pitch of the horn changes from high to low. Any time there is relative motion between the source of a sound and the receiver of it, there is a difference between the actual frequency and the observed frequency. This is called the Doppler effect. The Doppler effect applied to electomagnetic waves helps meteorologists to predict weather, allows astronomers to estimate distances to remote galaxies, and aids police officers catch you speeding. The Doppler effect applied to ultrasound is used by doctors to measure the speed of blood in blood vessels, just like a cops radar
gun. The faster the blood cell are moving toward the doc, the greater the reflected frequency. Animation (click on The Doppler Effect, then click on the button marked: The Doppler effect the change in pitch due to a moving wave source. 1) Objects moving toward you cause a higher pitched sound. 2) Objects moving away cause sound of lower pitch. 3) Used in radar by police and meteorologists and in Doppler Equation fL = fS
( v vL v vS f L = frequency as heard by a listener f S = frequency produced by the source ) v = speed of sound in the medium vL = speed of the listener v S = speed of the source This equation takes into account the speed of the source of the sound, as well as the listeners speed, relative to the air (or whatever the medium happens to be). The only tricky part is the
signs. First decide whether the motion will make the observed frequency higher or lower. (If the source is moving toward the listener, this will increase f L, but if the listener is moving away from the source, this will decrease f L.) Then choose the plus or minus as appropriate. A plus sign in the numerator will make f L bigger, but a plus in the denominator will make f L smaller. Examples are on the next slide. Doppler Set-ups fL = fS ( v vL v vS )
The horn is producing a pure 1000 Hz tone. Lets find the frequency as heard by the listener in various motion scenarios. The speed of sound in air at 20 C is 343 m/s. 343 f L = 1000 343 - 10 = 1030 Hz still 10 m/s ) ( f L = 1000 still 10 m/s
( 343 + 10 343 ) = 1029 Hz Note that these situation are not exactly symmetric. Also, in real life a horn does not produce a single tone. More examples on the next slide. Doppler Set-ups (cont.) fL = fS
( v vL v vS ) The horn is still producing a pure 1000 Hz tone. This time both the source and the listener are moving with respect to the air. ( 10 m/s 3 m/s 343 - 3
f L = 1000 343 - 10 = 1021 Hz f L = 1000 10 m/s 3 m/s ( 343 + 3 343 - 10 ) ) = 1039 Hz
Note the when theyre moving toward each other, the highest frequency possible for the given speeds is heard. Continued . . . Doppler Set-ups (cont.) fL = fS ( v vL v vS ) The horn is still producing a pure 1000 Hz tone. Here are the final two motion scenarios. f L = 1000
10 m/s 3 m/s 3 m/s ) 343 + 3 343 + 10 ) = 963 Hz f L = 1000 10 m/s
( 343 - 3 343 + 10 ( = 980 Hz Note the when theyre moving toward each other, the highest frequency possible for the given speeds is heard. Continued . . . Doppler Problem Mr. Magoo & Betty Boop are heading toward each other. Mr. Magoo drives at 21 m/s and toots his horn (just for fun; he doesnt actually see her). His horn sounds at 650 Hz. How fast should Betty drive so that she hears the horn at 750 Hz? Assume the speed o sound is 343 m/s.
fL = fS ( v vL v vS ) 750 = 650 vL = 28.5 m/s 21 m/s vL (
343 + v L 343 - 21 ) Blue shift vs. red shift. http://library.thinkquest.org/19537/?tqskip1=1&tqtime=0208 Interference As we saw in the wave presentation, waves can passes through each other and combine via superposition. Sound is no exception. The pic shows two sets of wavefronts, each from a point source of sound. (The frequencies are the same here, but this is not required for interference.) Wherever constructive interference happens, a listener will here a louder sound. Loudness is diminished where destructive
interference occurs. A A: 2 crests meet; B constructive interference B: 2 troughs meet; constructive interference C: Crest meets trough; destructive interference C Interference: Distance in Wavelengths Weve got two point sources emitting the same wavelength. If the difference in distances from the listener to the point sources is a multiple of the wavelength, constructive interference will occur. Examples: Point A is 3 from the red center and 4 from the green center, a difference of 1 . For B, the difference is zero. Since 1 and
0 are whole numbers, constructive interference happens at these points. If the difference in distance is an odd multiple of half the wavelength, destructive interference occurs. Example: A Point C is 3.5 from the green B center and 2 from the red center. The difference is 1.5 , so destructive interference occurs there. Animation C animation 2 Constructive or destructive? Interference: Sound Demo
Using the link below you can play the same tone from each of your two computer speakers. If they were visible, the wavefronts would look just as it did on the last slide, except they would be spheres instead of circles. You can experience the interference by leaning side to side from various places in the room. If you do this, you should hear the loudness fluctuate. This is because your head is moving through points of constructive interference (loud spots) and destructive interference (quiet regions, or dead spots). Turning one speaker off will eliminate this effect, since there will be no interference. Listen to a pure tone (up to 1000 Hz) Interference: Noise Reduction The concept of interference is used to reduce noise. For example, some pilots where special headphones that analyze
engine noise and produce the inverse of those sounds. This waves produced by the headphones interfere destructively with the sound waves coming from the engine. As a result, the noise is reduced, but other sounds can still be heard, since the engine noise has a distinctive wave pattern, and only those waves are being cancelled out. Noise reduction graphic (Scroll down to Noise Cancellation under the Applications of Sound heading.) Acoustics Acoustics sometimes refers to the science of sound. It can also refer to how well sounds traveling in enclosed spaces can be heard. The Great Hall in the Krannert Center is an example of excellent acoustics. Chicago Symphony Orchestra has even recorded there. Note how the walls and ceiling are beveled to get sound waves reflect in different directions.
This minimizes the odds of there being a dead spot somewhere in the audience. scroll down to zoom in on the Great Hall pic. Can you hear it ? Click above for a news story and HERE for extended version. Beats are pulsing variations of loudness caused by interference of sounds of slightly different frequencies. Sound BEATS
Two overlapping waves with slightly different frequencies gives rise to the phenomena of beat frequencies (beats). The beat frequency is the difference between the two sound frequencies (shown below to the right). For example, the beat frequency of the two tuning forks shown below to the left is: 440 Hz 438 Hz = 2 Hz. Beats Example Mickey Mouse and Goofy are playing an E note. Mickeys guitar is right on at 330 Hz, but Goofy is slightly out of tune at 332 Hz. 1. What frequency will the audience hear? 331 Hz, the average of the frequencies of the two guitars. 2. How often will the audience hear the sound getting louder and softer? They will hear it go from loud to soft
twice each second. (The beat frequency is 2 Hz, since the two guitars differ in frequency by that amount.) Beats Weve seen how many frequencies can combine to produce a complicated waveform. If two frequencies that are nearly the same combine, a phenomenon called beats occurs. The resulting waveform increases and decreases in amplitude in a periodic way, i.e., the sound gets louder and softer in a regular pattern. Hear Beats When two waves differ slightly in frequency, they are alternately in phase and out of phase. Suppose the two original waves have frequencies f1 and f2. Then their superposition (below) will have their average frequency and will get louder and softer with a frequency f beat =of| f|1 f-1 -f2f2| |.
Beats Animation (click on start simulation) f combo = ( f1 + f2 ) / 2 soft loud f. Resonance - the ability of an object to vibrate by absorbing energy at its natural frequency. The place were something likes to The dramatic move. increase in a wave's amplitude due to a forced
vibration at the object's natural frequency. Tuning Forks & Resonance Tuning forks produce sound when struck because, as the tines vibrate back and forth, they bump into neighboring air molecules. (A speaker works in the same way.) Animation Touch a vibrating tuning fork to the surface of some water, and youll see the splashing. The more frequently the tines vibrate, the higher the frequency of the sound. The harmonics pics would look just like those for a tube with one open end. Smaller tuning forks make a high pitch sound, since a shorter length means a shorter wavelength. If a vibrating fork (A) is brought near one that is not vibrating (B), A will cause B to vibrate only if they made to produce the same frequency. This is an example of resonance. If the driving force (A) matches the natural frequency of B, then A can cause the amplitude of
A B B to increase. (If you want to push someone on a swing higher and higher, you must push at the natural frequency of the swing.) Resonance: Shattering a Can sound waves really shatterGlass a wine glass? Yes, if the frequency of the sound matches the natural frequency of the glass, and if the amplitude is sufficient. The glasss natural frequency can be determined by flicking the glass with your finger and listening to the tone it makes. If the glass is being bombarded by sound waves of this frequency, the amplitude of
the vibrating glass with grow and grow until the glass shatters. What structural difference do you see between acoustic and electric guitars? Electric guitars usually have a solid body while acoustics have a hollow body and at least one sound hole. We know that the vibrating string makes the sound, but why is the acoustic so much louder than the electric if the electric is not plugged into an amp? The electric gets louder from the electronic amplification of the amp. The acoustic uses the air inside the hollow body as an amplifier. The chunk of air inside the body resonates http://www.archive.org/detail
s/SF121 Resonance is the place were thing like to move or vibrate. https://youtu.be/nFzu6 CNtqec http://en.wikipedia.org/ wiki/Tacoma_Narrows _Bridge http://encarta.msn.com /media_461550807/col lapse_of_the_tacoma_ narrows_bridge.html Can you hear it???? WE will check your hearing in a
few minutes!! Frequency & Pitch Just as the amplitude of a sound wave relates to its loudness, the frequency of the wave relates to its pitch. The higher the pitch, the higher the frequency. The frequency you hear is just the number of wavefronts that hit your eardrums in a unit of time. Wavelength doesnt necessarily correspond to pitch because, even if wavefronts are very close together, if the wave is slow moving, not many wavefronts will hit you each second. Even in a fast moving wave with a small wavelength, the receiver or source could be moving, which would change the frequency, hence the pitch. Frequency Pitch Amplitude Loudness Listen to a pure tone (up to 1000 Hz) Listen to 2 simultaneous tones (scroll down)
What am I actually hearing? The Human Ear The exterior part of the ear (the auricle, or pinna) is made of cartilage and helps funnel sound waves into the auditory canal, which has wax fibers to protect the ear from dirt. At the end of the auditory canal lies the eardrum (tympanic membrane), which vibrates with the incoming sound waves and transmits these vibrations along three tiny bones (ossicles) called the hammer, anvil, and stirrup (malleus, incus, and stapes). The little stapes bone is attached to the oval window, a membrane of the cochlea. The cochlea is a coil that converts the vibrations it receives into electrical impulses and sends them to the brain via the auditory nerve. Delicate hairs (stereocilia) in the cochlea are responsible for this signal conversion. These hairs are easily damaged by loud noises, a major cause of hearing loss! The semicircular canals help maintain balance, but do not aid hearing.
Animation Ear Anatomy Range of Human Hearing The maximum range of frequencies for most people is from about 20 to 20 thousand hertz. This means if the number of high pressure fronts (wavefronts) hitting our eardrums each second is from 20 to 20 000, then the sound may be detectable. If you listen to loud music often, youll probably find that your range (bandwidth) will be diminished. Some animals, like dogs and some fish, can hear frequencies that are higher than what humans can hear (ultrasound). Bats and dolphins use ultrasound to locate prey (echolocation). Doctors make use of ultrasound for imaging fetuses and breaking up kidney stones. Elephants and some whales can communicate over vast distances with sound waves too low in pitch for us to hear (infrasound).
Hear the full range of audible frequencies (scroll down to speaker buttons) Intensity The amount of energy in a wave. The amplitude of the wave determines how loud something is. The units used for intensity are decibels. Intensity All waves carry energy. In a typical sound wave the pressure doesnt vary much from the normal pressure of the medium. Consequently, sound waves dont transmit a whole lot of energy. The more energy a sound wave transmits through a given area in a given amount of time, the more intensity it has, and the louder it will sound. That is, intensity is power
per unit area: P I = A Suppose that in one second the green wavefronts carry one joule of sound energy through the one square meter opening. Then the intensity at the red rectangle is 1 W / m 2. (1 Watt = 1 J / s.) wavefronts 1 m2 Intensity Example If you place your alarm clock 3 times closer to your bed, how
many times greater will the intensity be the next morning? answer: Since the wavefronts are approximately spherical, and the area of a sphere is proportional to the square of its radius (A = 4 r 2), the intensity is inversely propotional to the square of the distance (since I = P / A). So, cutting the distance by a factor of 3 will make the intensity of its ring about nine times greater. However, our ears do not work on a linear scale. The clock will sound less than twice as loud. Threshold Intensity The more intense a sound is, the louder it will be. Normal sounds carry small amounts of energy, but our ears are very sensitive. In fact, we can
hear sounds with intensities as low as 10-12 W / m 2 ! This is called the threshold intensity, I 0. I 0 = 10 -12 W / m 2 This means that if we had enormous ears like Dumbos, say a full square meter in area, we could hear a sound delivering to this area an energy of only one trillionth of a joule each second! Since our ears are thousands of times smaller, the energy our ears receive in a second is thousands of times less. Sound Level in Decibels
The greater the intensity of a sound at a certain place, the louder it will sound. But doubling the intensity will not make it seem twice as loud. Experiments show that the intensity must increase by about a factor of 10 before the sound will seem twice as loud to us. A sound with a 100 times greater intensity will sound about 4 times louder. Therefore, we measure sound level (loudness) based on a logarithmic scale. The sound level in decibels (dB) is given by: = 10 log I I0 (in decibels) Ex: At a certain distance from a siren, the intensity of the sound waves might be 10 5 W / m 2 . The sound level at this location would be: 10 log (10 5 / 10 12) = 10 log (10 7 ) = 70 dB
Note: According to this definition, a sound at the intensity level registers zero decibels: 10 log (10 12 / 10 12) = 10 log (1 ) = 0 dB The Decibel Scale The chart below lists the approximate sound levels of various sounds. The loudness of a given sound depends, of course, on the power of the source of the sound as well as the distance from the source. Note: Listening to loud music will gradually damage your hearing! Source Anything on the verge of being audible Whisper Normal Conversation Busy Traffic Niagara Falls Train Construction Noise Rock Concert
Machine Gun Jet Takeoff Rocket Takeoff Decibels 0 30 60 70 90 100 110 120 130 150 180 }
Constant exposure leads to permanent hearing loss. Pain Damage Intensity of Sound demo ! Some call it selective hearing Which has the greater intensity? A
B Intensity & Sound I Level = 10 log I0 Every time the intensity of a sound is increased by a factor of 10, the sound level goes up by 10 dB (and the sound seems to us to be about twice as loud). Lets compare a 90 dB shout to a 30 dB whisper. The shout is 60 dB louder, which means its intensity is 10 to the 6th power (a million) times greater. Proof: 60 = 1 - 2 = 10 log (I 1 / I 0 ) - 10 log(I 2 / I 0 ) = 10 log I 1 / I 0 I2 / I0 60 = 10 log (I 1 / I 2 ) 6 = log (I 1 / I 2 )
10 6 = I 1 / I 2 Compare intensities: 100 dB vs. 75 dB answers: factor of 100 factor of 316 (10 2.5 = 316) Compare sound levels: 4.2 10 4 W / m 2 vs. 4.2 10 7 W / m 2 differ by 30 dB ( Is differ by 3 powers of 10 ) Compare intensities: 80 dB vs. 60 dB Complex Sounds Real sounds are rarely as simple as the individual standing wave
patterns weve seen on a string or in a tube. Why is it that two different instruments can play the exact same note at the same volume, yet still sound so different? This is because many different harmonics can exist at the same time in an instrument, and the wave patterns can be very complex. If only fundamental frequencies could be heard, instruments would sound more alike. The relative strengths of different harmonics is known as timbre (tam-ber). In other words, most sounds, including voices, are complex mixtures of frequencies. The sound made by a flute flute is predominately due to the first & second harmonics, so its waveform is fairly simple. The sounds of other instruments are more piano complicated due to the presence of additional harmonics. Combine Harmonics
Create a Complex Sound violin e. Standing wave - a wave pattern that occurs when two waves equal in wavelength and frequency meet from opposite directions and continuously interfere with each other. Need help wi th harmonics ? node antinode
How many nodes and antinodes are in a standing wave that has 3 wavelengths? Natural Frequency Standing Waves occur at what are called Natural Frequencies or Harmonics. All Natural Frequencies are multiples of the Fundamental The first harmonic is known as the fundamental. A. What is the 5th Harmonic if the fundamental is 12.7 Hz?
B. B. What is the wavelength of the 3rd harmonic if the length of string is 1.2m? https ://youtu.be/X8qZO6g_X5Q Standing Waves: 2 Fixed Ends When a guitar string of length L is plucked, only certain frequencies can be produced, because only certain wavelengths can sustain themselves. Only standing waves persist. Many harmonics can exist at the same time, but the fundamental (n = 1) usually dominates. As we saw in the wave presentation, a standing wave occurs when a wave reflects off a boundary and interferes with itself in such a way as to produce nodes and antinodes. Destructive interference always occurs at a node. Both types occur at an antinode; they alternate. n = 1 (fundamental)
n=2 Node Animation: Harmonics 1, 2, & 3 Antinode Wavelength Formula: 2 Fixed Ends =2L (string of length L)
=L n=1 n=2 Notice the pattern is of the form: 2 L = n where n = 1, 2, 3, . =2L 3
n=3 Thus, only certain wavelengths can exists. To obtain tones corresponding 1 = L to other wavelengths, one 2 n=4 must press on the string to change its length. Sound vibrating string standing wave patterns A standing wave (transverse or longitudinal) is a wave that vibrates between two fixed points (a fixed length that the wave vibrates in). Strings or columns of air can vibrate to form standing waves.
Sound vibrating string standing wave patterns When a string or column of air vibrates at its natural frequency, resonance occurs. When resonance occurs, the waves amplitude increases. With a sound wave, this means an increase in the sounds loudness. Sound vibrating string standing wave patterns When a string or a column of air vibrates at a frequency other than its natural frequency, standing waves will not occur and the string or column of air will not resonate and the waves amplitude (loudness) will not increase. Sound vibrating strings v = wave velocity
fn = frequency (nth harmonic) v n = harmonic number fnn = n 2 L (n = 1, 2 . . . ) L = length of string n=1 n=2 n=3 Sound vibrating strings n = 1 (fund. frequency), 1 = 2 L, L= 1 1st harmonic lowest possible standing wave frequency n = 2 (2nd harmonic), 2 = L, L = 2
2nd harmonic integral multiple of the fund. frequency (f2 = 2 f1) n = 3 (3rd harmonic), 3 = 2/3 L, L = 3/2 3 3rd harmonic integral multiple of the fund. frequency (f3 = 3 f1) n = 4 (4th harmonic), 4 = L, L = 2 4 4th harmonic integral multiple of the fund. frequency (f = 4 f ) Sound A longitudinal standing wave pattern on a slinky 1. Suppose that a string is 1.2 meters long and vibrates in the first, second and third harmonic standing wave patterns. Determine the wavelength of the waves for each of the three patterns. 2. The string below is 1.5 meters long and is vibrating as the first harmonic. The string vibrates up and down with
33 complete vibrational cycles in 10 seconds. Determine the frequency, period, wavelength and speed for this wave. And the answer is 1. For the first harmonic, the length of the string is equivalent to one-half of a wavelength. If the string is 1.2 meters long, then onehalf of a wavelength is 1.2 meters long. The full wavelength is 2.4 meters long. Second harmonic: Third harmonic: 1.2 m 0.8 m 2. The frequency refers to how often a point on the medium undergoes back-and-forth vibrations; it is measured as the number of cycles per unit of time. In this case, it is f = (33 cycles) / (10 seconds) = 3.3 Hz The period is the reciprocal of the frequency.
T = 1 / (3.3 Hz) = 0.303 seconds avelength of the wave is related to the length of the rope. For the first harmonic as pictured in this problem, wavelength ngth of the rope is equivalent to one-half of a wavelength. That is, L = 0.5 W where W is the wavelength. : = 2 L = 2 (1.5 m) = 3.0 m The speed of a wave can be calculated from its wavelength and frequency using the wave equation: v = f W = (3.3 Hz) (3. 0 m) = 9.9 m/s Waves on strings are similar t o pipes F = nv / 2L S=f Determine the length of guitar string required to produce a fundamental frequency
(1st harmonic) of 256 Hz. The speed of waves in a particular guitar string is known to be 405 m/s. Given: v = 405 m/s Find: L = ?? f1 = 256 Hz speed = frequency wavelength great problems wavelength = speed / frequency wavelength = (405 m/s) / (256 Hz) wavelength = 1.58 m Diagram:
Vibrating String Example Billy decides to play his own ukulele. He plays a D on his guitar which has a fundamental frequency of 296 Hz. The wave travels down the string with a speed of 386 m/s. What is the length of the string? F= nv 2L L = 1 (386) 2 (296) .65m Hear what a ukulele sounds like. (Scroll down.) Pitch
Indicates how high or low a note is. How fast the molecules of a medium vibrate or its frequency determine pitch. A B Which set of waves is a higher note? Octaves & Ratios Some mixtures of frequencies are pleasing to the ear; others are not. Typically, a harmonious combo of sounds is one in which the frequencies are in some simple ratio. If a fundamental frequency is combined with the 2nd
harmonic, the ratio will be 1 : 2. (Each is the same musical note, but the 2nd harmonic is one octave higher. In other words, going up an octave means doubling the frequency.) Another simple (and therefore harmonious) ratio is 2 : 3. This can be produced by playing a C note (262 Hz) with a G note (392 Hz). The Musical Scale Mathematical Relationships in the form of Ratios 1 9/8 5/4 4/3 3/2 5/3 15/8 2
DO RE MI FA SO LA TI DO 264 297 330 352 396 440 495 528 C D E F G A B C Note Name Key Color Frequency C 528
B 495 B flat A A flat 475 440 422 G 396 G flat
380 F 352 E 330 E flat D D flat C 317 297 285
264 Standings Waves in a Tube: 2 Open Ends Like waves traveling on a string, sound waves traveling in a tube reflect back when they reach the end of the tube. (Much of the sound energy will exit the open tube, but some will reflect back.) If the wavelength is right, the reflected waves will combine with the original to create a standing wave. For a tube with two open ends, there will be an antinode at each end, rather than a node. (A closed end would correspond to a node, since it blocks the air from moving.) The pic shows the fundamental. Note: the air does not move like a guitar string moves; the curve represents the amount of vibration. Maximum vibration occurs at the antinodes. In the middle is a node where the air molecules dont vibrate at all. Harmonics animation 1st, 2nd, and 3rd Harmonics n = 1 (fundamental)
Wavelength Formula: 2 Open Ends (tube of length L) As with the string, the pattern is: 2L = n where n = 1, 2, 3, . Thus, only certain wavelengths will reinforce each other (resonate). To obtain tones corresponding to other wavelengths, one must change the tubes length.
=2L n=1 =L n=2 2 = L 3 n=3 =1L 2 n=4 Sound vibrating air columns
Tube open at both ends (open pipe) (shown below with antinodes at both ends) (all harmonics present n = 1, 2, 3. . .). Since the distance from one node to the next ( ) = the distance from one antinode to the next, the standing wave equation for a pipe open at both ends is the same as that of a vibrating string. v fnn = n 2 L (n = 1, 2 . . . ) Sound open pipes vibrating air columns n = 1 (fund. frequency), 1 = 2 L,
L = 1 1st harmonic lowest possible standing wave frequency n = 2 (2nd harmonic), 2 = L, L = 2 2nd harmonic integral multiple of fund. frequency (f2 = 2 f1) n = 3 (3rd harmonic), 3 = 2/3 L, L= 3/2 3 3rd harmonic integral multiple of Fig 11.45, p.403 Slide 83 Standings Waves in a Tube: 1 Open End If a tube has one open and one closed end, the open end is a region of maximum vibration of air moleculesan antinode. The closed end is where no vibration occursa node. At the closed end, only
a small amount of the sound energy will be transmitted; most will be reflected. At the open end, of course, much more sound energy is transmitted, but a little is reflected. Only certain wavelengths of sound will resonate in this tube, which depends on it length. Harmonics animation 1st, 3rd, and 5thHarmonics animations (scroll down) n = 1 (fundamental) Wavelength Formula: 1 Open End (tube of length L) This time the pattern is
different: or, n L = 4 4L = n where n = 1, 3, 5, 7, . Note: only odd harmonics exist when only one end is open. L= 1
4 n=1 L= 3 4 n=3 5 L= 4 n=5 L= 7 4 n=7
Sample problems here Extra notes Fig 11.44, p.403 Slide 89 http://www.phys.unsw.edu.au/jw/fl utes.v.clarinets.html Organizing the formula Harm. # # of
Waves in Air Column # of Nodes # of Antinodes LengthWavelength Relationship 1 1/2 1
2 Wavelength = (2/1)*L 2 1 or 2/2 2 3 Wavelength = (2/2)*L 3 3/2
3 4 Wavelength = (2/3)*L 4 2 or 4/2 4 5 Wavelength = (2/4)*L 5
5/2 5 6 Wavelength = (2/5)*L Example Example Problem Problem #1 #1 The speed of sound waves in air is found to be 340 m/s. Determine the fundamental frequency (1st harmonic) of an open-end air column that has a length of 67.5 cm. Given: v = 340 m/s L = 67.5 cm = 0.675 m
Find: f1 = ?? Wavelength = 2 Length because Wavelength = 2 0.675 m Wavelength = 1.35 m speed = frequency wavelength frequency = speed / wavelength frequency = (340 m/s) / (1.35 m) frequency = 252 Hz Diagram: =2L Nice graphics http://dev.physicslab.org/Document.aspx?doctype= 3&filename=WavesSound_ResonancePipes.xml
https://youtu.be/EsLTVrMSaZg Watch for homework https://youtu.be/pydXBbC70tY Practice problem If you have a closed end resonator vibrating at 789 Hz in a space of 1.8 m space, what is the Find the A. of the fundamental B. Fundamental frequency C. of the 5th harmonic D. Frequency of the 11th harmonic A. 5/4 L =
= 1.44 (1.8) = B. F1 = 789/5= 157.8 Hz C. = 5 x (1.44) = 7.2 m D. F = 11 X 157.8 = 1725.8 Hz What is the fundamental frequency of a 18 cm tall bottle .18 = = .72 m S=f 343 = f (.72) f = 476.4 Hz How long is the open pipe that resnates at a fundamental frequency of 600 Hz? S = f 343 = (600) = .57 m L= L = .5 (.57) = .29 m
The size of the musical interval is determined by the ratio of the frequencies of the two tones which comprise the interval. This sequence of frequencies is the natural scale. Intervals having a frequency of 9/8 or 10/9 are called whole tones and those characterized by the Palm are Pipes 16/15 ratio half tones. Length of CPVC pipe for 180 Frequency No.
Note #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15
F G A B flat C D E F G A B flat C D E F water 23.60 cm
21.00 cm 18.75 cm 17.50 cm 15.80 cm 14.00 cm 12.50 cm 11.80 cm 10.50 cm 9.40 cm 9.20 cm 7.90 cm 7.00 cm 6.25 cm 5.90 cm 349 Hertz 392 Hertz 440 Hertz
446 Hertz 523 Hertz 587 Hertz 659 Hertz 698 Hertz 748 Hertz 880 Hertz 892 Hertz 1049 Hertz 1174 Hertz 1318 Hertz 1397 Hertz Open End Pipes F = nv where n = 1,2,3 etc 2L
Palm Pipes Check to see the note of your palm pipe Follow along with the music and play your palm pipe when your note occurs. How could pipes that have the same note be different lengths? Can you tell what key each song is in? HAPPY BIRTHDAY Melody C C D C F E Harmony A A Bb Melody C C C A F E D Harmony
F C Bb C C D CGF Bb Bb A Bb Bb A F G F C A TWINKLE, TWINKLE LITTLE STAR F F C C D D C
Bb Bb A A G G F Harmony C C A A Bb Bb A G G F F E E C C C Bb Bb A A G C C Bb Bb A A G Harmony A A G G F F C A A G G F F C F F C C D D C Bb Bb A A G G F
Harmony C C A A Bb Bb A G G F F E E C Melody Melody Melody GOD BLESS AMERICA Melody F E D E D C GF G A Harmony A C
E D E F Melody Harmony E Bb A C F G A C G A Bb D D G F G F E F F E
C Melody E F G C F GA C G A Bb E A Bb C Harmony C D E E F
Melody D C Bb A G F Bb A G F Harmony F F E CE D F F E C G C G The fundamental (first harmonic) for an open
end pipe needs to be an antinode at both ends, since the air can move at both ends. This looks different than the wavelength that I showed you in Figure 3, but it is still half of a full wavelength. That means the length of the tube and frequency formula are L= Example 1: An open ended organ pipe is 3.6m long. a) What is the wavelength of the fundamental played by this pipe? b) What is the frequency of this note if the speed of sound is 346m/s? (Calculate it using the formulas youve just learned, although if you wanted you could use v = f ) c) What note could be played as the third harmonic on that pipe? d) If we made the pipe longer, what would happen to the fundamental note would it be higher or lower frequency?
a) L = = 2L = 2(3.6m) = 7.2 m b) c) Notice that the third harmonic is three times bigger than the first harmonic. d) If we made the pipe longer, the wavelength would be bigger (just look at the formula in part "a" of this example), and since wavelength and frequency are inversely related, that means the frequency would be smaller. Standing Waves: Musical Instruments As we saw with Schmedricks ukulele, string instruments make use of vibrations on strings where each end is a vibrational node. The strings themselves dont move much air. So, either an electrical
pickup and amplifier are needed, or the strings must transmit vibrations to the body of the instrument in which sound waves can resonate. Other instruments make use of standing waves in tubes. A flute for example can be approximated as cylindrical tubes with two open ends. A clarinet has just one open end. (The musicians mouth blocks air in a clarinet, forming a closed end, but a flutist blows air over a hole without blocking the movement of air in and out.) Other instruments, like drums, produce sounds via standing waves on a surface, or membrane. Hear and See a Transverse Flute Hear a Clarinet, etc. (scroll down) Standing Waves on a Drum Animation Credits
F-22 Raptor http://members.home.net/john-higgins/index2.htm Sonar Vision http://www.elender.hu/~tal-mec/html/abc.htm Krannert Center (acoustics) http://www.krannertcenter.com/center/ venues/foellinger.php Ukulele: http://www.glass-artist.co.uk/music/instruments/ukepics.html Tuning Forks: http://www.physics.brown.edu/Studies/Demo/waves/demo/3b7010.htm Waveforms: http://www.ec.vanderbilt.edu/computermusic/musc216site/what.is.sound.html http://www.physicsweb.org/article/world/13/04/8 Piano : http://www.mathsyear2000.org/numberland/88/88.html Credits Mickey Mouse: http://store.yahoo.com/rnrdist/micmousgoofp.html Dumbo: http://www.phil-sears.com/Folder%202/Dumbo%20sericel.JPG Sound Levels: http://library.thinkquest.org/19537/Physics8.html?
tqskip1=1&tqtime=0224 Angus Young: http://kevpa.topcities.com/acdcpics2.html Wine Glass: http://www.artglassw.com/ewu.htm Opera Singer: http://www.ljphotography.com/photos/bw-opera-singer.jpg