Vibrations & Sound Chapter 11 Waves are the practical applications o oscillations. Waves show up in nature in many orms that include physical waves in a medium like sound, and waves o varying electric and magnetic ield better known collectively as light. Water waves are perhaps the best known physical orm o waves as they are so visual and part o everyday lie. The image above shows a combination o waves in water as concentric circles. With this image, a story can be told o the past since something happened to create these waves, and the shapes o the waves are part o the clues to their origin. Understanding waves can unlock mysteries like this all across the universe.
Types o Waves and Wave Properties Lesson 1 Lesson Objectives escribe the dierence between transverse and longitudinal waves Understand wavelength, requency, and how they relate to wave speed Know the relationship between simple harmonic motion and waves Vocabulary Wave Transverse wave Longitudinal wave Check Your Understanding 1. A water wave has a shape that looks like a mathematical unction. What mathematical unction can describe the shape o a wave? 2. What does a wave have in common with a pendulum? Introduction In the study o pendulums, the position could be described with a sine or cosine unction. Sine and cosine look like waves. There are some deinite parallels between oscillating motion and waves. But waves are an abstract concept. A wave is not a material object you can quantiy. Waves cannot be captured and stored in a bottle, at least not in a conventional way. Waves are the act o energy being transerred between two places in space and time. Just like energy was an abstract concept, so are waves. The language and basic concepts o waves must be mastered to apply the knowledge o waves and their connection to energy. These are the concepts addressed in this lesson. Lesson Content Wave concept and types o waves What is a wave? The answer is not easy to describe initially. Tie a piece o string to a long spring or slinky ixed at one end, and compress a section o it. Release that compression and you will see it relax but the section next to it will compress and then relax. This will continue
rom the end being held to the other end. It will actually bounce back to a certain extent. The piece o string will barely have moved. What was seen traveling through the spring or slinky? The video below shows a slow motion compressional wave moving through a slinky type spring. http://www.youtube.com/watch?vubrlaccqk Slow motion compressional wave in a slinky Even though there is an obvious disturbance moving through the spring, no part o the spring moves more than a little bit to the let or right. This leads to the deinition o a wave as a disturbance or energy transerred rom one point in space to another. Mechanical waves carry the disturbance through matter, and these will be the ones addressed in these lessons on waves. Light is a wave that is a disturbance in interacting electric and magnetic ields, but these type o waves will be addressed later. Compressional waves are sometimes called longitudinal waves, as they travel along the length o the medium. Sound is a compressional wave. There are waves other than compressional waves, where the movement is perpendicular to the direction o the actual wave motion. These waves are called transverse and have the distinctive wave shape like waves seen in water. A computer simulation showing waves o both types can be seen at the link below. Wave properties http://www.acs.psu.edu/drussell/emos/waves/wavemotion.html Types o Waves shown by computer simulation Mechanical waves need three things to exist. They need an initial energy or the disturbance, they need a medium to carry the disturbance, and a physical connection to the medium. Once a wave is started, it can be characterized by three properties. The size o the disturbance is called the amplitude oten represented by the letter A. The distance over which the waves is spread is called the wavelength, represented by λ. The time it takes a wave to completely pass a static point in the medium is called the period, T. Figure 1 shows a transverse wave and longitudinal wave with wavelength and amplitude labeled.
Figure 1. Characteristics o waves (diagram taken rom http://stokescience.co.uk/waves.html -- to be replaced) Figure 1 shows the terminology used or the maximum and minimum displacement as the crest and trough respectively. The equivalent or longitudinal waves is also shown as compressions and rareactions. Even though the medium o the wave does not move much, the wave itsel is seen to propagate through the medium with a deinite speed. The speed o the wave is typically x λ constant and can be described as wavelength over period: v wave. t T Wave motion can be just a single pulse, but many times the wave motion is a repeated motion over some length o time. Ocean waves come one ater another. In the case o repeated wave motion instead o a single pulse, it is useul to talk about how requently waves pass a static point. Frequency is deined as how many waves per second travel through a static point. I 10 waves pass through a static point in a single second, the period o those waves is 1/10 th o a 1 second. The relationship between requency and period can be expressed as:. T The velocity o a wave can be written in terms o requency by replacing 1/T with. vwave λ Mathematical description o waves A transverse wave looks like a sine curve, so it is the easiest type to help explain the mathematic description o waves. However, all waves can be described as shown. A generic sine expression can be used to describe the vertical displacement o the medium o a y x, t Asin Bt + Cx +, where A,B, C, and transverse wave in space and time, so that ( ) ( ) are constants. The irst letter, A, is just the maximum displacement o the wave or amplitude.
The wave will repeat itsel when the quantity B( nt ) n( 2π ) since sine will repeat itsel when the parameter inside increases by a actor o 2π in a time equal to the period or subsequent 2π intervals o the period. Thereore, the coeicient B can be written as: B 2π. Here is T a parallel with circular motion and simple harmonic motion. The angular requency term described the rate o revolution in time so that one ull revolution in a period T was: 2π ω 2π. The coeicient B can be written as angular requency, ω. In space or along T the x-axis, the wave will have repeated itsel every wavelength o distance. For this to represented in the sine unction, the x-contribution should equal 2π when x equals some integer 2π amount o wavelengths: Cx kx x. The C coeicient is renamed and is called the wave λ 2π number, k. Finally, the coeicient describes how the sine curve is not perectly aligned λ with the origin. The curve may start to the let or right, which describes the phase o the curve, designated by the phase constant, φ. The generic equation or the vertical displacement o the transverse wave can be written as: ( ) sin ( ω φ) dy d ( sin ( kx + ωt + φ) ) y t A kx + t +. Calculus can be used to ind the rising or vertical velocity o the medium. to ( v y ) max ( ) vy A ωacos kx + ωt + φ with a maximum velocity equal dt dt ω A. The acceleration is the derivative o velocity, so that: ( cos( + ω + φ) ) dv d kx t ( + + ) with an acceleration that is dt dt y a 2 sin 2 y ωa ω A kx ωt φ ω y opposite in direction to the velocity. This description o the wave matches the components o circular motion and also the descriptions o simple harmonic motion. Lesson Summary A wave is a disturbance that transers energy rom one place to another. A mechanical wave travels through a medium made o mass in some state, with examples o water waves in water and sound waves in air. Longitudinal waves move along the direction o travel o the wave itsel. Transverse waves move perpendicular to the direction o travel o the wave. All waves have wavelengths and requencies that deine the wave speed, v wave Waves can be described with the same ormalism as simple harmonic motion. λ.
Review Questions Review Problems Further Reading / Supplemental Links Points to Consider 1. How is sound a mechanical wave? 2. oes the speed o a wave traveling through a taught string depend on the tension o the string?
Sound and other common waves Lesson 2 Lesson Objectives Calculate wave speed on a stretched string istinguish between standing and traveling waves escribe sound as a wave Be able to calculate sound intensity in decibels Check Your Understanding 1. It is harder to make a wave pulse in a heavy metal chain compared to a thin rope. Why? 2. Why does sound travel at dierent speeds in dierent media? 3. oes a wave made in water stop waves coming rom dierent directions when they meet each other? Introduction Understanding the basics o waves leads to applying that knowledge to some common types o waves. The application o waves to practical phenomenon is truly an eye opening experience. It was stated earlier that waves are abstract concepts. To apply the concept o waves to explain natural events is one o those light bulb moments rom learning physics. This lesson will address waves in ropes, strings, and chains and also waves in matter that are collectively called sound. The concepts covered in this lesson will set up the very popular phenomenon called music. Lesson Content Speed o a wave in a cord A wave pulse created in a cord that is held at both ends will travel down the length o the string. The cord can be a string, rope, chain, or equivalent lexible linear wave carrier. In this case, let the cord be a string. Its speed on the string depends on how tight the string is pulled and the density o the string itsel. A consideration o orces will show the exact relationship.
Figure 2. Forces on a string with a wave pulse Figure 2 shows a wave pulse on a string with a mass per unit length o µ with the top portion labeled with the tension orces on either side. The top section o the pulse is trying to be pulled back down to its normal position shown by the dotted line by a downward radial orce. At this instant in time, the top portion o the wave pulse is experiencing a circular motion type orce pulling it towards the center o the arc shaped pulse. The sum o orces on this top segment must be equal to a net centripetal orce. Fy mac The radial orce F r is equal to the two vertical components o the tension on either side, so that 2F sinθ ma 2F sinθ ma T C T C 2 vt 2FT sinθ m. R The central angle o the arc ormed by the arc segment is double the angle θ, since it breaks down into two right triangles each with angle θ. The mass per unit length µ is equal to total mass divided by total length, or the mass o a small piece divided by its small length or a uniormly distributed mass. m m µ s R ( 2θ ) solving or mass, m µ R( 2θ). Also, or small angles, sinθ θ. This is true since the portion o the arc at the top o the wave pulse is chosen to be a small piece. Putting these all together, the tangential velocity o the arc can be ound. This tangential velocity is the horizontal wave speed along the length o string. 2 vt 2FT sinθ m R 2 vt 2 T 2 R Fθ µ R( θ) v wave FT µ
Example: A rope o mass 0.50kg and length 2.5m is used to set up a hanging mass m 5.0kg. The physical set up is similar to that shown in igure 2 without the standing wave established yet. a) Find the minimum tension in the rope (ignore the mass o the rope itsel). b) I the rope is plucked like a guitar string, a wave will be created. Find the speed o that wave. Solution: Part a. The only orces on the hanging mass are tension upward and weight downward, and the mass is stationary. Once the rope is plucked, it will experience a small acceleration upward and downward as the wave propagates. These will be small compared and ignored or simplicity. F y 0 F T mg 0 F 39.2N F 5.0 ( 9.8 2 ) m T mg kg 0.50 Part b. The mass per unit length o the rope can be ound using µ m kg 0.20 s 2.5m FT 39.2N The speed o the wave in the rope can be ound using vwave, and µ 0.20 v 196 m wave s Standing Waves and Traveling Waves In the previous example, a rope was plucked like a guitar string. I a device was installed that repeated the plucking so that the wave in the rope was repeated, a constant traveling wave would be generated. These waves would travel down the length o the rope until they ran into the pulley and then would relect backwards. The combination o waves would create a random jiggling unless the requency allowed those rebounding waves to combine in a constructive wave to create a standing wave. The distinction between a standing wave and traveling wave is challenging to visualize. The video below shows the dierence visually. http://www.youtube.com/watch?v03curkjqde0&eaturerelated Standing wave versus traveling wave A standing wave seems to stay stationary and show its wave orm in one place. epending on the input requency, the wave may be dierent sizes. This standing waves will have stationary points called nodes, and points where the maximum displacement occurs where the crests and troughs are located called anti-nodes. Figure 2 shows the irst three standing waves possible in a cord held taught by a hanging mass. The nodes and anti-nodes are shown with N s and A s. s T kg m kg m.
Figure 2. Standing waves in a cord held in tension The distance between pulley and wall must hold the varying hal intervals o wavelength. A trend is apparent where the number o antinodes corresponds to number o hal wavelengths squeezed on to the cord. L n λ n 2 or 2L λ n n The phenomenon o the standing wave can be combined with the two deinitions o wave speed to relate required requency to get certain numbers o nodes and anti-nodes to show up. Example: Let igure 2 represent the same rope in the previous example. Find the requency o vibration required to achieve the three cases o standing waves shown. Solution: The two deinitions o wave speed applicable are v F µ v T wave and wave λ. Setting these n FT two equal to each other and solving or requency: n. The irst three requencies 2L µ are 1 3.5Hz, 2 7.0Hz, and 3 10.5Hz. It makes sense that the rope would have to be wiggled aster to get more standing waves to show up. A standing wave can be created in more than just cords. The video below was taken at the US Naval Academy at their wave lab. They can control the requency o the waves with a set barrier to make standing waves in water very well. http://www.youtube.com/watch?vnpeevou4z8&eaturerelated Standing waves in water, USNA Wave Lab
Power dissipated in a harmonic wave These standing waves can be described with the trigonometric unctions introduced in lesson 1. These waves can be described as harmonic waves and they can be described by the same ormalism used with simple harmonic motion. The power transmitted by a harmonic wave can be expressed using the time derivative o the energy o the harmonic wave. The wave has 1 2 1 2 2 E KE mv µ x ω A. some kinetic energy associated with it: 2 2 ( ) 1 2 2 ( µ ( ω )) de d x A dx P dt dt dt ( A ) 2 1 2 2 2 µ ω The time derivative o horizontal position with respect to time corresponds to the wave velocity. Since the mass itsel is not moving sideways, the kinetic energy corresponded to the localized velocity or upward velocity in the case o a transverse wave. Power dissipated by a harmonic wave, P µω Av 1 2 2 2 wave Sound as a Wave Sound is a wave in a physical medium that can be heard by the human ear. In this sense, sound is just waves we can hear. All o the applications discussed so ar can apply to sound waves. The speed o the wave in a cord can be generalized so it can apply to sound. Speed o mechanical waves elastic property inertial property The inertial property describes how resistant the medium is to transmission o a vibration, and elastic property describes how much the medium wants to return to its original position. Generally speaking, the inertial property can be described by mass density and the elastic P property by the bulk modulus. The bulk modulus, B, can be expressed as: B. A V V more detailed derivation o this relationship with calculus is listed in the supplemental links section o this lesson. v wave B ρ Sound will travel in more than just air. Using the properties o several materials, table 1 below can be constructed to show the speed o sound waves in dierent media.
Gases Liquids Solids Air at 0 o C 331 Water 1493 Aluminum 5100 Air at 20 o C 343 Methyl alcohol 1143 Copper 3560 Hydrogen at 0 o C 1286 Sea water 1533 Iron 5130 Oxygen at 0 o C 317 Lead 1322 Helium at 0 o C 972 Vulcanized rubber 54 Table 1. Speed o sound in meters/second in dierent media The human ear is a very dynamic sensor o sound that is capable o sensing very low amplitude sounds and high amplitude sounds. To make all these sound amplitudes it on one scale, a logarithmic unction is used so all sounds can be compared to the lowest amplitude that can be heard called the threshold o hearing. The scale is measured in decibels and can be expressed as: 10log I 12 W β where I0 10 2. Figure 3 shows a list o sound amplitudes measured m I 0 in decibels. Figure 3. Graphical view o the logarithmic scale o sound amplitudes (taken rom http://www.soundadvice.ino/thewholestory/san1.htm -- need to recreate)
Sound needs a medium in which to transmit energy rom particle to particle so that it could be detected by the combination o bones, membranes, and nerves in the human ear. In this description, there must be no sound in space. This is not necessarily true. There are a ew particles in space although they are very spread out. It would not be worth trying to talk to someone in space without a radio. However, many dramatic events occur in space like the births and deaths o stars and even the Big Bang event itsel that would move these scarce particles around. A recording has been made at the University o Virginia that uses real data to recreate the sound that represents the ormation o the universe. The link below goes to an instructive web-site that plays this sound ile as well as discuss the basics o sound. Lesson Summary http://www.astro.virginia.edu/~dmw8/bba_web/index_rames.html The sound o the universe The velocity o a wave in a cord can be expressed as v wave F µ T. A traveling wave is created by repeating the vibration that created the initial wave over and over so that the disturbance is recreated over and over again. A standing wave is the phenomenon o a traveling wave standing still although it is the combination o waves at just the right requencies. Power dissipated by a harmonic wave is P µω Av. 1 2 2 2 wave The velocity o any mechanical wave can be expressed as vwave B. ρ Sound is a type o mechanical wave that can be detected by the human ear. Sound intensities or amplitudes are measured in decibels using I. 12 W 0 10 2 m Review Questions 10log I β I 0 where
Review Problems Further Reading / Supplemental Links Waves in physical medium. etailed derivations and an online calculator to test concepts. http://hyperphysics.phy-astr.gsu.edu/hbase/waves/wavsol.html Points to Consider 1. Will sound waves cancel each other out i they run into each other? 2. o all vibrating strings create sound? 3. How is music made by a wind instrument?
Resonance: Strings and Pipes Lesson 3 Lesson Objectives escribe intererence o waves and resonance Identiy wave phenomenon with sound like beats Understand the basics o sound physics with stringed instruments Understand the basics o sound physics with wind instruments Vocabulary Constructive intererence estructive intererence Check Your Understanding 1. How can a standing wave be made in water? 2. How does requency aect the appearance o a standing wave in a cord? B 3. Explain the equation: v wave ρ. Introduction The standing waves described in lesson 2 occur because waves can add and subtract rom each other when they encounter each other. This phenomenon is the basis o how musical instruments can create sounds in a controlled repeatable way. Sound waves are perhaps the most relevant way to demonstrate the principles o wave phenomenon. This lesson will explore how waves combine, and culminate with an exploration o stringed and wind musical instruments. Lesson Content Wave Intererence Waves can interere with each other either to combine or subtract rom each other. The video below shows two waves that pass though each other. As they do, they make a larger wave at one point. Constructive intererence occurs when waves interere with each other to add their amplitudes. estructive intererence occurs when the waves try to cancel each other out.
http://www.youtube.com/watch?vyvitr5th8jw Video example o constructive intererence I you set up two speakers and play the same requency sound, it is possible to get those sound waves to constructively interere so that certain places along a back wall will have a doubly loud sound. There will also be places where the sound cancels out. Figure 4 shows this arrangement. Figure 4. Sound intererence by two speakers A loud sound will be heard at a point along the wall i the path length dierence is equal to some whole number o waves. This means that the r nλ or constructive intererence and n 1 ( ) r n+ λ or destructive intererence. The integer n measures the number o the maxima 2 or minima that occur. At n 0, there is no path dierence which occurs directly in the middle between the two speakers. Example: For the case o the two speakers shown in igure 4, ind the irst three positions or constructive intererence and the irst three positions o destructive intererence. The distance between the speakers is 6.00m, the distance to the wall is 22.0m, and the requency o the sound is a middle C note ( 264 Hz). Solution: For constructive intererence, the irst three path dierences will be r λ, 1 r2 2λ, and r3 3λ. The speed o sound at room temperature is 343 m/s and v wave λ. Thereore, the n
wavelength o a middle C note is 343 m λ ( 261 1 ) s λ 1.31m and r1 1.31m, r2 2.62m, and r3 3.94m. The small triangle shown in igure 4 shares the same angle as the larger r y triangle, where sinθ and tanθ. Solving or the angle in the irst equation and L plugging it into the second allows y to be ound. For destructive intererence, the procedure is the same but with path lengths adjusted by a hal wavelength: r1 0.66m, r2 1.97m, and r3 3.29m. The results or the positions o the irst three minima and maxima are shown in the table below. s Maxima Minima n 1 y 4.94m y 2.42m n 2 y 10.72m y 7.65m n 3 y 19.18m y 14.40m Beats Combining two waves, especially sound waves, will create an interesting phenomenon. I the waves have two dierent requencies, they will alternate between constructive and destructive intererence to create a new wave pattern. In sound, this pattern becomes a new sound which pulsates at a new requency. Let two waves o dierent requency pass through the same point in time. The vertical displacement caused by the two waves can be described mathematically as: ( ) cos( ω ) and y ( t) A cos( ω t) y t A t 1 0 1 where ω1 2π 1 and ω2 2π 2. 2 0 2 The amplitudes o these two waves will combine so that a new displacement will be the sum o the these two: total ( cos( 2π ) cos( 2π )) y y + y A t + t. There is a trigonometric identity that will help 1 2 0 1 2 a b a+ b uncover the beats phenomenon. cos a+ cosb 2 cos cos 2 2 The total y-displacement equation now becomes: 1 2 1 2 ytotal 2A0 cos 2π t cos 2π + t 2 2
Figure 5. Beat requencies. The green is the combination o the red and blue. Stringed Instruments http://www.youtube.com/watch?vzwkiwaim3pw Resonance emonstration and Explanation A violin string can be adjusted to have just the right tension or a given length to recreate a particular note. I dierent mass density strings are also added to this instrument, it can create a range o requencies so that a skilled player could put together a variety o sounds across a particular musical scale. In music, the pitch o a musical note is related to the requency o the sound. However, each musical instrument creates its own tone quality. A violin or instance does not create a perectly smooth wave like a tuning ork. Figure 6 shows the amplitude versus time and the requency spectrum o a middle C note. The waves are airly regularly spaced indicating a predominant requency, but the amplitudes appear inconsistent. This shows the tone quality that makes the sound o a violin so distinctive. Also shown in the requency spectrum o igure 6 are the harmonic responses. Harmonics are the integer multiples o the undamental requency and they appear as the equally space spikes in the requency plot. The human ear hears these as one sound and part o the reason or the irregular waves seen in the amplitude plot.
Figure 6. Violin playing a middle C note. The strings on a violin, viola, cello, and guitar are all ixed on both ends. This case was presented in lesson 2, and shown that the undamental wavelength and successive harmonics 2L were described by λ n n. The violin and other bowed instruments can create dierent requencies by the player using a inger to pinch the string to eectively change the length on a given string. A piano has one string per note or a total o 88 strings. Wind Instruments Sound waves in an open or closed pipe will resonate and have undamental requency waves just like the strings. An open pipe wind instrument acts like a ixed string since the waves coming out o each end o the pipe need to be in phase with each other so that a continuous adding sound or constructive intererence takes place. Figure 7 shows the analogous string and wind instruments with the violin and the lute. Both ollow the general rules or undamental wavelengths and ensuing harmonics described by the relationships below or all integers.
2L λ n 1 n v 2L n 1 The wind instruments can achieve dierent requencies by changing their eective lengths with n strategically placed holes that are alternately covered and uncovered by the player. Figure 8 shows the open and closed tube pipes and the undamental requency and 1 st two harmonics produced. Figure 7. Stringed and Wind Musical Instruments: the violin and the clarinet Figure 8. Open and Closed Tube Wind Instruments
A closed tube wind instrument can use its length to produce a relected wave and eectively hal the length necessary to produce constructive intererence. The harmonics only occur at the odd integers as shown in igure 8. The closed tube wind instruments like the clarinet ollow the general rule below as shown in the trend illustrated in igure 8 only or odd integers: 4L λ n 1 n Lesson Summary v 4L n 1 n Waves through a medium will interere with each other and either add their amplitudes or subtract amplitudes depending on the waves alignment. Two waves o the same requency will add or subtract depending on whether the path dierence is an even number o wavelengths or something in between. Two waves o dierent requency will combine to yield a pulsation with a requency equal to the dierence between the two requencies in a phenomenon called beats. Stringed instrument create undamental requencies and harmonics o those requencies 2L based on λ n n Wind instruments can either be open at both ends or closed at one end, and they each ollow a set o rules or the requencies they can produce. The open end instruments like the clarinet and some organs ollow the same rule as stringed instruments ixed at each end. The instruments with one end closed, like the lute and many brass 4L instruments, will produce undamental wavelengths based on λ n n Review Questions Review Problems Further Reading / Supplemental Links Points to Consider 1. The mechanical universe cannot just be described in terms o a bunch o static point masses. eend this statement with the concept o waves as your argument.
The oppler Eect Lesson 4 Lesson Objectives eine the oppler Eect Apply the oppler Eect to describe the motion o a system Understand the applications o the oppler Eect to Astronomy and terrestrial systems Vocabulary oppler Eect Check Your Understanding 1. What is the requency o a wave that has a period o 1/10 th o a second? 2. escribe how the equation that ollows describes waves. v λ 3. How does the amplitude o a wave aect its velocity? Introduction The oppler Eect is applied through many applications and has been at the center o many societal upgrades in terms o making lives better in practical ways and helping understand the Universe. While the oppler Eect gets the credit, it is really the culmination o the realization about the importance o understanding waves. Lesson Content The classic example illustrating the oppler Eect is the noticeable change in pitch you hear as a ire truck approaches you and them passes you. As the ire truck approaches, the sound waves are orced closer together and have a higher pitch since the truck is moving as it emits the sound waves. The ire truck passes by and then carries the wave source urther and urther away making the pitch seem lower. Figure 9 shows the oppler Eect graphically or a moving ire truck and stationary observer. Moving Source and Stationary etector Understanding the oppler Eect oers access to some very useul applications. The simplest case to study the eect is or a moving wave source and a stationary detector. The moving wave source could be a ire truck with its siren blaring, and the stationary detector could be the
ears on your head as you watch the approaching ire truck. Figure 10 shows the wave source at two successive positions over one period, T. The distance the source moves equals the source velocity times the period, vt. S Two successive waves are shown ahead o the source. Figure 9. oppler Eect rom a moving ire truck as heard by a stationary source Figure 10. oppler Eect or Moving Source and Stationary etector
These successive waves were transmitted some time, t, prior to the source reaching the irst dot. The crest o the irst wave shown is located a distance equal to the velocity o the wave times total time, v wave t. The second wave is located on period s worth o movement behind, or vwave ( t T) equal to ( ). The total distance between the initial source position and the urthest wave is vwavet vt S + vwave t T + λ v wave t v S T + v wave t v wave T + λ 0 vt S vwavet + λ. The equation can be solved or the squished wavelength that would be 1 vwave vs sensed by the detector: λ ( vwave vs ) T combined with T λ. From S s the point o view o the detector, they see the same speed o the source wave so that the wave vwave velocity equation gives: v wave λ and wavelength yields λ. This expression or the vwave vwave vs squished wavelength can be set equal to the previous one: which can be rearranged in terms o the detector requency, v wave s vwave vs s. This equation says that you can predict the requency heard by a detector i you know the speed o the wave and o the moving source. It also says that i you know the detector requency and source requency with the velocity o the wave, you can calculate the velocity o the source. This is the oppler Eect. In the case o sound waves, you can measure the shited pitch o a known sound source and calculate its velocity. Stationary Source and Moving etector I the detector is moving toward the detector, the equation above changes: v wave v wave s s. Solving or the detector requency and adjusting vwave vs vwave v or the two cases o an approaching detector or detector moving away gives: v v wave S vwave or the detector moving away rom the source, and v + v wave S vwave or the detector moving toward rom the source. http://www.youtube.com/watch?vm3mkzjlacai r. Hewitt explains oppler Eect
Moving Source and Moving etector I the detector is also moving, the requency o the source will seem shited according to the above equation so that v + v wave vwave v + v wave s vwave vs Example: v wave s vwave vs and putting these two together, and the requency at the approaching detector is v v + v wave wave s vwave vs vwave An ambulance emits a siren with a constant requency o 1000Hz. The ambulance is moving at 30.0 m/s toward an observer in a car approaching the ambulance with a velocity o 20.0 m/s. a) What requency does the observer hear? b) As the ambulance passes the car, the requency goes rom being higher than normal to now being lower than normal. What is the overall change in requency heard as the ambulance goes rom approaching to going away rom the observer? Solution: a) b) m m vwave + v 330 s + 20 s s 1000Hz 1167Hz m m vwave vs 330 s 30 s m m vwave + v 330 s 20 s s 1000Hz 861Hz m m where the sign reverses on vwave vs 330 s + 30 s the detector and source velocity lip signs since the derived equation assumed the two approached each other. The dierence in requency is equal to: 306Hz. The human ear can detect changes in requency equal to less than 1 Hz so this would be a dramatic change in requency. Applications o the oppler eect The oppler eect is used in ground based radar systems to determine the movements o weather systems. oppler is also used by astronomers to detect the radial speed o distant galaxies. Astronomers will look at the spectra o a galaxy and look or common absorption lines. The lines are due to clouds o gases like hydrogen in the galaxy absorbing certain requencies o light and leaving dark lines in the visible light spectrum. When these lines are
compared to samples o the same gases, they are ound to be shited slightly out o position. This shit is the oppler eect or light, and can be used to calculate the velocity o the galaxy either coming toward us or going away in space. An approaching object will have it spectra compressed or shited toward the blue, which is called a blue shit. A receding object will display a red shit. The oppler eect allowed or the discovery that almost all galaxies are red shited and thereore moving away rom us, and that we are in an expanding universe. Lesson Summary A moving source approaching a stationary detector will have the ollowing relationship v wave between the source requency and detector requency: s. vwave vs A detector moving toward a stationary source will ollow the equation: vwave + v S. vwave A detector moving away rom a stationary source will ollow the equation: vwave + v S. vwave A detector approaching a moving source is described by v + v. wave s vwave vs The oppler Eect is used to measure velocities o weather activity as well as the radial velocities o distance galaxies. Review Questions Review Problems Further Reading / Supplemental Links Points to Consider 1. What are some other uses o the oppler Eect?
2. What are some other examples o oppler Eect besides an approaching ire truck?