Chapter 20: Mechanical waves

Transcription

1 Chapter 20: Mechanical waes! How do bats see in the dark?! How can you transmit energy without transmitting matter?! Why does the pitch o a train whistle change as it approaches or leaes a station? Make sure you know how to: 1. Graph sinusoidal unctions 2. Mathematically describe position-ersus-time or simple harmonic motion. 3. Determine the period and the amplitude o ibrations using a graph. CO: An article in the 2004 medical journal Rheumatology was titled From bats and ships to babies and hips (Kane, Grassi, Sturrock and Balint, 2004). The article described the history o the deelopment o medical techniques used to detect birth deects, breast tumors, joint problems, and other medical conditions. Interestingly, the article started by discussing bats. Lazzaro Spallanzani ( ), an Italian priest, studied them extensiely and was ridiculed or proposing that bats see with their ears. Spallanzani s bat problem, as it was termed, remained a scientiic mystery until 1938, when inally two young Harard students, Donald Griin and Robert Galambos, explained the mechanism o bat naigation. But een beore that in 1912 a Canadian, Reginald Fessenden, patented deices using a bat mechanism to locate sunken ships. The same idea was used by Paul Langein and Constantin Chilowsky to construct a bat-like generator that on April 23, 1916 during World War 1 was used to detect and then sink a German U-boat (UC-3). Finally, in the early 1950s Proessor Ian Donald o Glasgow suggested using bat signals to study sot abdominal tissues. He like Spallanzani was ridiculed or suggesting a totally unnecessary procedure or something that one could inestigate by a simple manual examination. As it turned out the irst obserations done by Spallanzani, the explanations proided by Griin and Calambos, and testing and application experiments conducted by many led to break though diagnostics not only in medicine, but also in metallurgy and astronomy. They all relate to the same simple phenomenon, whose name and real explanation you will learn later in this chapter. Lead: Vibrational motion occurs when a system that is disturbed rom its equilibrium position moes back toward this position but oershoots. It stops beyond the equilibrium position and returns only to oershoot again. This repetitie motion caused by a restoring mechanism leads to ibrational motion. In the last chapter, we looked only at the ibrational motion o one object but did not consider what happens in the enironment contacting the ibrating object. In this chapter we ocus on the eect that the ibrating object has on the medium that surrounds it. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

2 20.1 Obserations: pulses and wae motion Our eeryday experience indicates that the medium surrounding a ibrating object is also disturbed. For example, i you hit one prong o a tuning ork, it ibrates. You can see this ibration i you shine light on a small mirror glued on a prong o the tuning ork (see Fig. 20.1a). Simultaneously, you hear sound. I you dip the prongs o the ibrating tuning ork in a pan with water, you see the water splashing (Fig. 20.1b). The ibrating ork disturbed the water. When you touch the prongs and they stop ibrating, you stop hearing sound. The ibrating tuning ork must be disturbing the air adjacent to the ork. The disturbance is transmitted to our ear causing the sound we heard. It is diicult to ollow ibrating prongs as they moe ery quickly. Obserational experiments described below help us slow down the process. Figure 20.1 Vibrating tuning ork produces sound Obserational Experiment Table 20.1 Disturbances in dierent media. Obserational experiments 1 Tie a rope to a doorknob and then shake once the end that you hold. A disturbance traels along the rope. Analysis The disturbance moes along the rope but the rope ibers (see the little ribbon on the rope) remain in the same position ater the disturbance passes. 2. Place a beach ball in a swimming pool and Styrooam pieces on the surace. Push the ball up and down once. A circle spreads outward increasing in radius. The Styrooam pieces moe up and down. The Styrooam pieces moes up and down but do not trael across the pool. The circular disturbance produced by the ball s up and down motion spreads but the water does not moe across the pool. Pattern In the two experiments, we saw: (1) that a disturbance propagated in the medium at an obserable speed. (2) the particles o the medium did not trael across space. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

3 Waes and wae ronts The moing circular disturbance in the last experiment is called a wae ront (see the top iew in Fig. 20.2a). Eery point on the wae ront has the same displacement rom equilibrium. I instead o pushing the ball down and up once, you push it down and up in a regular pattern, you see a continuous group o humps and dips (waes consisting o many wae ronts) o circular shape moing outward on the surace o the water (Fig. 20.2b). You hae seen such repetitie waes many times or example, waes on the ocean or on a large lake. Similar phenomena happen i we continuously shake the end o a long rope up and down. Figure 20.2 Waeronts Wae motion inoles a disturbance produced by a ibrating object (a source). The disturbance propagates though a medium and causes points in the medium to ibrate. To explain this new type o motion, we repeat the beach ball experiment. When the ibrating beach ball moes down, water under the ball s bottom surace is pushed out to the side o the ball. We hae ormed a hump in the water at the sides o the ball. This hump alls back down towards its equilibrium leel and oershoots as the ball lits up out o the water. The alling hump pushes neighboring water upward. The hump is starting to propagate away rom the ball. A disturbance moes (propagates) outward, but the water at a particular location moes up and down and not in the direction the wae traels. Thus, the wae exists because o the interactions o the neighboring sections o the water and their coordinated ibrations. We can use the same idea to explain an experiment with a stretched slinky. According to our understanding o the wae motion, i we push or pull one coil seeral times, it will pull the slinky coil attached to it. This coil pulls the next coil, which in turn pulls the next coil, which so on and so orth. We see a pattern o propagating compressions and rareactions moing along the slinky (Fig. 20.3). This is called a longitudinal wae; the disturbance propagates along the slinky and each coil moes back and orth parallel to the direction o trael o that disturbance. We can disturb a slinky in a dierent way. I you ibrate the coil you are holding up and down perpendicular to the slinky, the coils moes up and down perpendicular to the orientation o Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

4 the slinky and to the direction that the disturbance traels (Fig. 20.4). This is called a transerse wae. Longitudinal and transerse waes In a longitudinal wae the ibrational disturbance in the medium is parallel to the direction o propagation o the disturbance. In a transerse wae the ibrational disturbance in the medium is perpendicular to the direction o propagation o the disturbance. The wae we obsered on the surace o water is more complicated; the water layers moe in an elliptical path so the wae is part transerse and part longitudinal. We obsere another important property o waes when watching waes on the surace o the water or on a slinky or on a rope. When a wae reaches the wall o the container or the end o the slinky or rope, it relects o the end and moes in the opposite direction (Figure 20.5 shows this relection or one pulse.) Thus, i we want to study simple wae motion and not worry about relected waes, we need to hae a ery long medium. Any boundary between dierent media causes a relected wae in the same medium. Figure 20.3 Longitudinal Wae Figure 20.4 Transerse Wae Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

5 Figure 20.5 Pulse relects o o ixed end Reiew Question 20.1 Can we say that particles in the medium moe parallel to the direction o wae propagation in longitudinal waes and they moe perpendicular to the direction o transerse wae propagation? Explain your answer Physical quantities and wae equation Waes are ubiquitous: water waes, string waes, sound waes, and waes in Earth initiated by earthquakes. To understand waes and predict their behaior, we need to inent some physical quantities that are useul or describing them and to ind relationships between the physical quantities. We then use these relationships or practical purposes and to predict new phenomena. Let us use a wae source that ibrates with simple harmonic motion perhaps a motor that can ibrate the end o the rope. We attach this ibration source to a ery long rope (Fig. 20.6a). With a really long rope, we aoid the complications o wae relections i we obsere the process or the time interal beore the relection occurs. The motor source moes the let end o the rope up and down perpendicular to the rope and thus produces a transerse wae. Figure 20.6 Wae produced by ibrating source At time zero, the source starts at y " # A. The source ibrates sinusoidally; thus the source s displacement rom equilibrium is described by a sinusoidal unction o time (Fig. 20.6b): Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

6 2$ y " Acos ( t) (20.1) T where y is the displacement o that end o the rope rom its equilibrium position, A is the amplitude o the ibration, and T is the period o ibration. The let end o the rope moes up and down with the ibrator. This end pulls the next part o the rope, which pulls its neighboring part, which pulls its neighboring part, and so on. The wae propagates along the rope at a particular speed. We now hae our quantities that describe the wae started by this ibrating source. Period T in seconds is the time interal or one complete ibration o a point in the medium at any point along the wae s path. Frequency in Hz ( s 1 ) is the number o ibrations o a point in the medium as the wae passes. Amplitude A is the maximum displacement o a point o the medium rom its equilibrium position as the wae passes. Speed in m/s is the distance a disturbance traels during a time interal diided by that time interal. Wae Equation We choose the positie x direction as the direction in which the disturbance traels at speed. The let end o the rope at x " 0 is attached to the ibrating source. According to Eq. (20.1), it oscillates up and down with a ertical displacement 2$ y(0, t) " Acos( t). T The shape o the rope at ie dierent times is shown in Fig I you look at any other position along the x -axis, the rope ibrates the same way as it does at x " 0 only in a dierent part o the ibration cycle. For example, at the irst marker to the right o x " 0, you see that at time zero the rope displacement is A ; at time then 0; and inally back to A. T t ", its y displacement is zero; then # A ; 4 Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

7 Figure 20.7 Shape o wae at ie consecutie times We can mathematically describe the disturbance yxt (, ) o the rope at some positie position x to the right o x " 0 at some arbitrary time t. Eery point to the right o x " 0 ibrates with the same requency, period, and amplitude as the rope at the x " 0. But there is a time delay x % t " or the disturbance at x " 0 to reach the location x. The ibration disturbance at position x at time t is the same as the disturbance at x " 0 was at the earlier time t x. Thus, the ibration o the rope at position x can be described as: 2$ x yxt (, ) " Acos[ ( t& )]. (20.2) T You see rom the snapshot graphs o the disturbance (see Fig. 20.7e) that the disturbance pattern repeats in a distance T. This distance separates neighboring locations on the rope that hae the same displacement y and the same slope or the locations o waeronts. This distance is called a waelength and is gien the Greek symbol ' (lambda): ' " T ", where is the speed o trael o the wae, T is the period o ibration o each part o the rope, and is the requency o ibration ( 1 T ). Waelength ' in m equals the distance between two nearest points on a wae that hae exactly the same displacement and shape (slope) or the distance between the nearest waeronts: ' " T " (20.3) We can substitute this expression or waelength into Eq. (20.2) and rearrange a little to get the so-called the wae equation: Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

8 2$ x t x t x yxt (, ) " Acos[ ( t& )] " Acos[2 $ ( & )] " Acos[2 $ ( & )]. T T T T ' Wae equation The wae equation indicates the displacement rom equilibrium y at location x at time t when a wae o waelength ' traels at speed toward the right through a medium. t x y Acos ( * + ". 2 $, & - ) T ' / (20.4) The ibration requency is the inerse o the ibration period ( 1 " ). T Tip! Notice how Eq. (20.4) is mathematically symmetrical with respect to the period o the wae and the waelength. The wae has two repetitie processes. I the location x is ixed, the equation shows that the point in the medium ibrates in time with period T. I the time t is ixed, the space has a waelike appearance with waelength '. ( t x) I we write equation 20.4 as y " Acos. 2$ & 2 $ 2 T ' / 3, the part 2 x $ ' shows how the displacement o a point at a distance x rom the origin o the wae is dierent rom the displacement o the point at the origin. The 2$ x is called a phase. Two points are said to be in ' phase when at the same time their displacements are the same, or they hae the same phase. By deinition o the waelength, two points, separated by the distance equal to a multiple o waelengths are always in phase. The wae equation describes the disturbance rom equilibrium o a medium through which a wae traels. When the medium is two dimensional, a line o points that hae the same disturbance at the same clock reading is the amiliar wae ront. As we already know, one example o a wae ront is a ripple in a water pond surrounding a ibrating object. All particles orming a wae ront on the ripple hae the same displacement rom their equilibrium position. Conceptual Exercise 20.1 Make sense o the wae equation A wae on an ininitely long t x slinky is described mathematically as y (0.1 m)cos ( * + ". 2 $, & - ) 0.7 s 1.8 m / you know about this wae and represent it graphically. Sketch and Translate Compare the gien equation with the wae equation [Eq. (20.4)]: t x y Acos ( * + ". 2 $, & - ) T ' / Say eerything. Thereore A " 0.1 m, T " 0.7 s, and ' = 1.8 m. We can also say that at t " 0, the starting point with the coordinate x " 0 had a displacement o Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

9 0 0 y (0, 0) (0.1 m) cos ( * + ". 2 $, & - ) = 0.1 m 0.7 s 1.8 m /. We can write a mathematical equation t or the motion o the wae source at location x " 0 : y source " (0.1 m)cos 2 $ *, s 1. Notice that at x " 0, y is a unction o time only, as we are looking at the disturbance at only one location. Rearranging the equation ' " T, we get " ' / T " (1.8 m)/(0.70 s) = 2.6 m/s, a reasonable number. Simpliy and Diagram This is an idealized situation where the amplitude does not change with time or with the distance it propagates. Knowing the amplitude and the period, we can draw a disturbance-ersus-time graph or one particular point in the medium (we do it or x " 0 in Fig. 20.8a). Knowing the amplitude and the waelength, we can draw a graph that represents an instantaneous picture o the whole slinky at a particular time (done or t " 0 in Fig. 20.8b). The latter graph is like a photograph o the slinky. Figure 20.8 Wae disturbance at one (a) location and (b) one time Try It Yoursel: A slinky ibrates with amplitude 15 cm and period 0.50 s. The speed o the wae on the slinky is 4.0 m/s. Write the wae equation or the slinky. ( * t x + ) Answer: y " (0.15 m) cos. 2 $, s 2.0 m / Tip! As the wae equation is a unction o two ariables, when we represent waes graphically, we need two graphs one showing how one point o the medium changes its position with time ( y-s- t; x" const ) and the other one showing the displacements o multiple points o the medium at the same time a snapshot o the wae ( y-s- x; t " const ). A wae has been characterized using ie physical quantities: period, requency, speed, waelength, and amplitude. The dependence o these quantities on characteristics o the medium through which the wae traels is the subject o the next section. Reiew Question 20.2 One might say that there are two speeds to consider when talking about wae motion. What two speeds might these be? Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

10 20.3 Dynamics o wae motion: speed and the medium We deined the speed o a wae as the distance that a disturbance o the medium traels in a time interal diided by that time interal. This is the operational deinition o speed. It tells us how to determine the speed but does not explain why a particular wae has a certain speed. Our goal in this section is to inestigate what determines the wae speed. We will conduct experiments in which we record the changes in the speed o disturbances (pulses) on a slinky as one property o the slinky is changed while other properties are held constant. We assume that the speed o the pulse is the same as the speed o the wae or the same conditions and the pulse does not slow down or speed up as it traels. The experiment is shown in Fig In the table we only show the aerage results or multiple experiments and do not show the uncertainty or simplicity, so you can ocus on the patterns. Figure 20.9 Measuring wae speed on a slinky Obserational Experiment Table 20.2 What aects wae speed? Obserational Experiment Analysis Measure the distance between people holding the slinky The speed o pulse is: and the time interal or a pulse to trael that distance: " distance/time interal = d / % t Distance (m) Time interal (s) = (4.0 m)/(1.0 s) = 4.0 m/s. Eect o amplitude Change the amplitude o the pulses: Eect o amplitude on speed: Amplitude (m) Distance (m) Time interal (s) Amplitude (m) Speed (m/s) Eect o requency Change the requency o the pulses: Eect o requency on speed. Frequency (Hz) Distance (m) Time interal (s) Frequency (Hz) Speed (m/s) Eect o stretching slinky Change orce pulling on end o slinky (length o slinky also changes): Eect o orce pulling on end o slinky (and possibly on length o slinky): Force (N) Distance (m) Time Interal (s) Force (N) Speed (m/s) Patterns! The speed o a pulse does not depend on the amplitude or requency.! The speed does depend on how hard you pull the end. The patterns that we ound in the Obserational Experiment Table 20.2 are surprising the speed o the pulse does not depend on the amplitude o the pulse or how oten the pulses are Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

11 generated on the requency. In other words the source o the waes does not aect the speed o the waes on the slinky! Howeer, pulling harder on the end o the slinky exerting a larger orce leads to the increase in the speed. When the slinky is pulled harder, the coils spread arther apart thus not only the tension changes but also what is called the linear density o the slinky changes the slinky now has less mass per unit length. Thus, these properties o the slinky aect how ast a pulse or a wae propagates in it! I we do similar experiments with sti springs that stretch less than slinkies, we ind that the wae speed is proportional to the square root o the orce pulling on the end o the spring. This applies to other objects such as strings. 4 F. M on M The subscripts indicate the pulling orce o one part o the medium on a neighboring part. This result makes sense pulling on a spring makes it stier. The stier the spring the more its parts interact with each other and the aster they transer the disturbance. Other experiments indicate that the mass per unit length o ibrating particles (coils in the experiments in Obserational Experiment Table 20.2) aects the speed. The speed o a wae is inersely proportional to the square root o the mass per unit length o the media (or example the coils): m 4 1/ " 1/ 5. L Here the Greek letter 5 is used or the linear density (not or the coeicient o riction). The dependence on mass can be understood i we think o Newton s second law. The greater its mass, the less the acceleration o each particle (coil) o the ibrating medium, and the longer the time needed to moe the next coil. The pulse traels slowly i the mass o coils is large. We can combine these two actors into one equation: F M on M ". 5 Let us check whether the units are correct: N * kg m+* m + m " ", 2 -, - ". kg/m 0 s 10kg 1 s The units are correct. To make sure that the speed does in act depend on the aboe properties, we can perorm a testing experiment. Take a rope o a known mass and length, attach one end to a wall and pull on the other end exerting a known orce. Use the aboe expression or speed to predict the time a pulse traels rom one end to the wall and back again. As the pulse traels ery ast, to minimize experimental uncertainty we will count 10 trips. Table 20.3 Testing the expression or wae speed. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

12 Testing experiment Prediction Outcome Predict the time interal or a pulse The wae speed is: We conduct the to trael rom one end o the rope to experiment seeral FS on R the other end and back again. The " times and measure rope is pulled by a spring scale. m/ L the time to be = 20 N 0.60 kg/5.0 m = 12 m/s. The time interal should be: 2L % t " = 2(5.0 m)/(12 m/s) = 0.83 s. We predict that ten round trips will take 8.3 s s. The predicted outcome lies within this interal. Conclusion The outcome is consistent with the prediction. We did not disproe the relation describing how the speed o a wae depends on the properties o the medium. Wae speed The speed o a wae on a string or in some other medium depends on how hard the string or medium is pulled F M on M (one part o the string or medium pulling on the adjacent m part) and the mass per unit length 5 " o the string or medium: L F M on M " (20.5) 5 Tip: We hae two mathematical expressions or the speed o the pulse: d " and % t F M on M ". The irst expression is a deinition o speed. The second is a cause eect 5 relationship the speed depends on the elastic properties o the medium and on its mass per unit length. We just learned that or elastic waes the speed depends only on properties o the medium. Period and requency o a wae depend on the ibration source the medium ibrates at the same requency as the source (there is an important exception discussed later in the chapter, the Doppler eect). Thus the waelength depends on both the ibration requency o the source and on the speed o the wae through the medium, which depends on properties o the medium. I the ibration requency is low and the wae speed high, a crest o the wae has more time to trael beore the next crest leaes the source. Thus the waelength is large: / ' ". Howeer i the speed is low and/or the requency high, the waelength is small: ' " /. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

13 The amplitude depends on how big the source ibration is and on how big the ibrations in the medium are they are not necessarily the same as we learn next. The amplitude does not aect the speed o wae propagation or the waelength o the wae. Notice in the ollowing table that wae speed also depends on the type o wae. Transerse waes trael slower in the same medium than longitudinal waes. Table The speeds o dierent types o waes in dierent media Type o wae Medium Sample alues o the speed Sound in a gas Air 340 m/s Sound in a liquid Water 1500 m/s Compression (longitudinal) wae in solids Shear or transerse wae in solids Compression wae in a thin rod or bone Transerse wae on a string Clay Sandstone Limestone Granite Salt Bone Violin A-string Violin G-string 1000 m/s 2000 m/s 4000 m/s 5000 m/s 6000 m/s About 0.6 times the speed o compression waes gien aboe 3000 m/s 288 m/s 128 m/s Example 20.2 Waes on a slinky You place a slinky on the loor. Your riend holds the ar one end with a spring scale and you hold the other end and stretch the slinky (Fig. 20.9). Ater measuring with the spring scale the orce needed to hold her end, she then holds that end securely with her hand without changing the length o the stretched slinky. You then abruptly start a longitudinal impulse down the slinky. What inormation do you need to collect to be able to predict the time interal it takes the pulse to reach your riend? Ater you make the prediction, what experiment can you conduct to check the correctness o the prediction? Sketch and Translate The experimental setup is already sketched in Fig Your riend has recorded the magnitude o the orce she exerts on the end o the slinky F S on S. You can also measure the mass o the slinky m and the distance d between you and the riend. The latter two quantities are used to determine the slinky s mass per unit length. The aboe quantities are used in Eq. (20.5) to predict the speed. To determine the speed experimentally, you need to measure the time interal % t it takes the pulse to trael between you and the riend, leading to an independent measurement o the speed. Simpliy and Diagram The aboe reasoning assumes that there is no riction between the slinky and the loor. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

14 FS on S FS on S Represent Mathematically We know that " ". This equation allows you to 5 m/ d predict the speed. To check i the prediction matches the speed, we measure using a more direct d method based on the equation ". % t Sole and Ealuate Ater we collect the measurements or orce and mass per unit length and make a prediction, we measure the speed using the time interal and distance. I the assumptions are alid, the speed measured this way should be about the same as the predicted speed. To do the experiment correctly, the uncertainty o the prediction and o the measurement should be considered to be sure they oerlap. Try It Yoursel: Large slinkys can stretch long distances 5 or 10 m. Try to estimate the percent uncertainty in your ability to measure this distance and also the percent uncertainty in measuring the time interal or one pulse to trael that distance. Answer: The distance uncertainty depends on the instrument we use. I we use a tape measure that is calibrated in centimeters, then the uncertainty in the measurement o the length will be hal o one cm, which is m. This makes percent uncertainty about (0.005 m/5 m)(100 %) = 0.1 %. The time uncertainty is due to the use o the stopwatch or due to human reaction time which is about 0.2 s. Depending on the stopwatch, we choose the biggest uncertainty or the estimate. For example the smallest time measured by the stop watch is 0.01 s. Thus human reaction time contributes more to the uncertainty o the time measurement. I the total time o trael o the pulse is 2 s then percent time uncertainty o about (0.2 s/2 s)(100 %) = 10 %. Our measured alue will probably be uncertain by about 10 percent maybe more. Reiew Question 20.3 Why do you think the speeds o the waes on the A-string and the lower requency G-string o a guitar are dierent? Energy, power, and intensity o waes So ar the physical quantities that we used to describe waes were inented or a ery simple case: a slinky or rope with a simple harmonic motion ibrating source. I there is no damping o the ibrations as the wae propagates along this one-dimensional medium, then the amplitude o the ibration at any point in the medium is the same as that o the source. What happens to the amplitude o ibration i we make the model a little more complicated a twodimensional medium (like the surace o a small lake) or a three-dimensional medium (like the air surrounding a balloon that pops and produces sound)? We will analyze situations in which the source still undergoes simple harmonic motion and we neglect damping conersion o ibrational energy into thermal energy. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

15 Wae amplitude and energy in a two-dimensional medium To understand what happens to the wae amplitude in a two-dimensional medium, consider Fig A beach ball bobs up and down in water in simple harmonic motion thus producing circular waes that trael outward across the water surace in all directions. The shaded circles in Fig represent the peaks o the waes and are called wae crests. We obsere that the amplitudes o the crests decrease as the waes moe arther rom the source. Why? Figure Wae crests at one time on water surace Suppose that the wae source uses 10 J o energy to produce a small number o waes during a 1.0-s time interal and that all o that energy is transerred to these spreading waes. As the wae propagates, more and more particles ibrate since the circular wae has a longer circumerence as it gets arther rom the source. Howeer, the total energy o all ibrating particles cannot exceed 10 J. Since there are more ibrating particles as the wae moes outward, the energy per particle will hae to decrease. Now, let us do a quantitatie analysis. Surround the source with two pretend rings (the dashed lines in Fig ), one at a distance R rom the source and the other at distance 2 R. I the energy o the source per unit time is 10 J/s, then this energy irst moes through the circumerence C1 " 2$ R and then through the second larger circumerence C " 2 2$ R" 2C. The circumerence o the second ring is 2 times more than the irst, but the 2 1 same energy per unit time moes past the second ring. Consequently the energy per unit circumerence length ( energy/ C " energy/2$ r) passing the second ring is one-hal that passing the irst ring. For any arbitrary distance rom the source, the energy per unit time and per unit circumerence length is inersely proportional to the distance rom the source. Two-dimensional waes produced by a point source The energy per circumerence length and per unit time crossing a line perpendicular to the direction the wae traels decreases as 1/r, where r is the distance rom the point source o the wae. The power per unit length at a distance 2 R rom the source is hal the power per unit length at a distance R. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

16 Figure Wae crest amplitude decreases with distance rom source Three-dimensional waes Consider now a three-dimensional space. Suppose, or example, that a balloon pops and sound moes outward in all directions surrounding the popped balloon. Consider two imaginary spheres that surround the source (Fig ), one at distance R rom the source and the other at distance 2 R. I the energy o the source per unit time is 10 J/s, then this energy irst moes through the surace area A 2 1 " 4$ R and then through the second larger area A " 4 $ (2 R) " 4A. The area o the second sphere is 4 times more than the irst, but the same energy per unit time moes through it. Consequently, the energy per unit area through the second sphere is one-ourth that through the irst sphere. For any arbitrary distance rom the source, the energy per unit time and per unit area will be inersely proportional to the distance squared. Three-dimensional waes The energy per area and per unit time passing across a surace perpendicular to the direction the wae traels decreases as 2 1/r, where r is the distance rom the point source o the wae. The power per unit area at a distance 2 R rom the source is 1/4 o the power per unit area at a distance R. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

17 Figure Sound decreases as get arther rom source Tip! The letter symbol or the area and the amplitude o ibrations is the same - A. Make sure that you think o the meaning o this symbol in the context o the situation to use the appropriate quantity. Wae power and wae intensity We deined the power P o a process in Chapter 6 as energy change per unit time interal ( P"% U / % t ). The intensity o a wae is deined as the energy per unit area and per unit time interal that crosses perpendicular to an area in a medium in which a wae traels. Intensity Energy % U P " " I " " time area % t A A 2 The units o intensity I are energy (joule) diided by the product area ( m ) and time (s), or 2 2 J/m s. The unit J/s is called a watt; thus, the unit o intensity is watt/ m ( 2 W/m ). We ound that the intensity o a wae propagating in a three-dimensional medium 2 decreases in proportion to the inerse o the distance squared rom the source ( I 4 1/ r ). The energy o a ibrating system depends on the amplitude o the ibrating medium. For example or an object attached to a spring: U s max Tip! In the equation aboe the letter A reers to the amplitude o ibrations. " 1 2 ka 2 (20.6) Thus, i the energy (and intensity) on the let side o the aboe equation decreases by one-orth (1/4) when the distance doubles, the amplitude on the right side decreases by 1/2 (since 1/2 squared is 1/4). This is true or waes in three-dimensional space (sound or light), whose amplitudes o ibration decrease in proportion to 1/ r, where r is the distance rom the point source o the wae. What happens to the wae amplitude o two-dimensional waes? Example 20.3 Water wae in pool You do a cannonball die o the high board in a pool. The wae amplitude is 50 cm at your riend s location 3.0 m rom where you entered the water. What is the wae amplitude at a second riend s location 5.0 m rom where you entered the water? Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

18 Sketch and Translate Visualize the person jumping into the pool and the circular wae crests moing outward. We consider the water wae to be a two-dimensional wae. The amplitude o the impulse at 3.0 m rom the source is 50 cm. We wish to ind the amplitude o the impulse at a distance r = 5.0 m rom the source. Simpliy and Diagram Assume that there is no transer o mechanical energy into internal energy and the changing amplitude is only due to increased distance. Represent Mathematically The energy per circumerence length and per unit time passing perpendicular to the direction o wae trael decreases as 1/r, where r is the distance rom the source. Thus, Ur 2 (1 / r2) r1 " ". U (1 / r) r r1 1 2 We also know that the energy o a ibration is proportional to the square o the amplitude o the 2 ibration: U 4 A. Thus: Thereore Sole and Ealuate Thus, A r2 U U r2 r2 1 2 r1 A r 2 r1 A 1/2 1/2 * r + * 3.0 m + ", - A ", m 1 2 A r " ". * r + ", /2 A r2 (0.50 m) = (0.77)(0.50 m) = 0.39 m. This decreased amplitude arther rom the source seems reasonable. Try It Yoursel: You set up longitudinal ibrations on a really long slinky. I there is no riction or other orm o energy transer rom mechanical energy, how does the amplitude at a distance o 2.0 m rom the source compare to the amplitude 4.0 m rom the source? Answer: They are the same this is a linear medium and the same ibrational energy passes eery position on the slinky. Reiew Question 20.4 What happens to the intensity o a wae as it propagates in a three-dimensional medium? Explain your answer Relection and impedance matching Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

19 Until now, we hae considered waes moing through a homogeneous medium, a medium with the same properties eerywhere. Howeer, when we irst obsered the pulses on a string or a slinky, we saw that they came back ater reaching the end attached to a door or being held by a person. What happens to a wae when it reaches the end o a medium or when there is an abrupt change rom one medium to another? Consider the ollowing obseration. You hold one end o a rope with the other end ixed (Fig ). You shake the end once and obsere a traeling pulse. When the pulse reaches the ixed end, the pulse bounces back in the opposite direction (relection). The amount o relection at an interace between dierent medium (the ixed end is a dierent medium or the wae compared to the rope) and the nature o the relected wae can proide considerable inormation about dierences o the two media. Geologists, or instance, hae determined that the Antarctic ice sheet rests on earth rather than water because radio waes relected rom the bottom o the ice sheet are characteristic o an ice-earth interace rather than an ice-water interace. Consider more experiments. Figure Snapshots o pulses on a string A thin rope (small mass per unit length) on the let is woen to a thicker rope with a larger mass per unit length on the right. Send an upward transerse pulse to the right in the thin rope (Fig ). A partially relected inerted pulse returns to our hand and a partially transmitted upright pulse traels in the thicker rope. Figure 20.14, Snapshot o relection and transmission at interace With the ropes connected in the opposite order (the thick rope on the let and thin rope on the right), send a transerse upright pulse along the thick rope toward the right (Fig ). Now, Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

20 a partially relected upright pulse returns to our hand and a partially transmitted upright pulse traels in the thin rope. How do we explain these obserations? In the irst experiment, because o its large mass per unit length, the second thick rope is harder to accelerate upward than i a thin rope was on the right. A small-amplitude upward pulse is initiated in the thick rope. As the thick rope does not "gie," it exerts a downward orce starting an inerted, relected pulse back toward the let in the thin rope. In the second experiment when the upright pulse in the let thick rope reaches the interace, the small rope restrains the thick rope less than usual. The thick rope initiates a large upright transmitted pulse in the thin rope and because o the less resistance rom the small rope oershoots and starts a small upright relected pulse back in the thick rope. Why is it important to understand the patterns that occur when waes trael rom one medium to another? You saw or example that in the irst experiment a signiicant amount o the pulse energy traels back thus less energy is transmitted orward. I an important wae signal is traeling through the air and it all comes back when reaching an interace into another medium, little inormation is transmitted into the second medium. For example, your hard skull relects most o the sound energy traeling through the air rom another person s oice or rom a musical eent. Howeer, your ear has been built so that much o the energy o sound waes in the air can reach the inner ear and be detected. On the other hand, i we want to aoid detecting sound waes, we can create an interace that relects most o the energy back. For example, you may wear earplugs while on a long airplane ride or i trying to sleep in a noisy enironment. Impedance We can inent a physical quantity that characterizes the degree to which waes are relected and transmitted at the boundary between dierent media. In physics this quantity is called impedance. The impedance Z o a medium is a measure o the diiculty a wae has in distorting the medium and depends both on the elasticity and the density o the medium the medium s elastic and inertial properties. Elastic property is related to the interactions between the particles in the medium (the stronger the interactions, the greater the property). The inertial property characterizes the density o the medium. Impedance is deined as o the square root o the product o the elastic and inertial properties o the medium: Impedance = Z " (Elastic property)(inertial property) (20.7) I two media connected together at an interace hae dierent elastic and inertial properties, or example the two dierent thickness (density) ropes woen together, a wae traeling in one medium is partially relected and partially transmitted at the boundary between the media. I both the elastic property and the inertial property o two media are the same, a wae moes rom one medium to the other as i there is no change. I the impedances o two media are ery dierent (or example a wae traeling rom a medium with small impedance to a medium with high impedance (like a pulse on a rope attached to a wall), then most o the wae energy is relected back into the irst medium and does not trael into the second medium. An example o this eect Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

21 is the use o ultrasound to detect anomalies in internal organs. Ultrasound relects o organs because o their dierent densities. The shape o a baby inside mother s womb can be determined by looking at relected waes. Matching impedances Ultrasound is poorly transmitted rom air outside the body to tissue inside the body. The impedance o the body is much greater than that o air and most o the ultrasound energy is relected at the air-body interace and does not get inside to look at objects inside the body. To aoid this problem, the ultrasound emitter is held directly against the body, which is coered with a gel that "matches" the impedance between and the emitter and the body surace. This matching allows the ultrasound wae to propagate inside the body. Scientists and engineers oten build special detectors that make a better impedance match with a sound source or example, when detecting seismic waes in Earth caused by earthquakes. One part o the seismometer actually becomes part o Earth and ibrates with it during an earthquake. They are so sensitie that they can detect a man jumping on the ground one km rom the seismometer. Animals hae also adapted to detect seismic signals. Elephants hae sti cartilage and dense at in their eet. This special tissue makes an impedance match with the earth s surace. Elephants at times lean orward on their ront eet seemingly to detect ibrations in the earth, or example prior to a new herd approaching a water hole. Reports describe Asian elephants as igorously responding to earthquakes, een trumpeting at the approach o an earthquake. So ar we ocused on the pulses and waes that relect at the boundary between two media. In addition, some absorption o a wae's energy may take place at the boundary between two media or in any part o a medium through which the wae traels. The coordinated ibrations o the atoms in the medium is turned into random kinetic energy, that is, into thermal energy. The rate o this conersion aries greatly. The slower the conersion, the longer the wae lies. Reiew Question 20.5 Why is it impossible to create only one traeling wae on a slinky? 20.6 Superposition principle and problem soling strategy In past sections, we studied the behaior o a single wae traeling in a medium and initiated by a simple harmonic oscillator. Most periodic or repetitie disturbances o a medium are combinations o two or more waes o the same or dierent requencies traeling through the same medium at the same time. The sound coming rom a iolin, or instance, may be a combination o almost 20 sinusoidal waes, each o dierent requency and waelength. To understand what happens in a medium through which two or more waes simultaneously pass, consider seeral simple experiments. Obserational Experiment Table 20.5 Adding two pulses passing through a medium. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

22 Obserational experiment An upright pulse traeling right rom the let side o a rope meets an inerted pulse traeling let rom the right side o the rope. The rope looks undisturbed as the pulses pass in the middle. Ater passing, the pulses continue moing in the same direction with the same amplitude as i they had neer met. Two upright pulses trael rom each end o the rope toward the center. When the pulses pass the center, the disturbance is twice that i only one pulse was present. Ater passing the center o the string, the pulses continue moing in the same direction with the same amplitude as though they had neer met. Pattern When pulses meet, the resultant pulse is the sum o the two o them (direction is important). Analysis The part o a string where the pulses met is pulled up by the pulse coming rom the let, and down by the pulse coming rom the right causing a zero disturbance. The part o a string where the pulses met is pulled up by the pulse coming rom the let, and up by the pulse coming rom the right producing a double amplitude pulse. It appears that each pulse pulls the string separately and the inal displacement o the string at a particular time and location is just a combination o the two pulses at that time and location adding them together: ynet " y1 # y2 This looks like superposition we encountered beore. For example, we added the orces that other objects exerted on an object o interest to ind the sum o the orces each orce produces an eect on an object independently o other orces and the net eect is a result o adding the orces. We encountered superposition when we studied electric and magnetic ields too. Figure Superposition o two waes We can test i the superposition principle works or the waes. Imagine we hae two ibrating sources in water or example, you push beach balls A and B up and down at the same requency, one in each hand (Fig a). They send out sinusoidal waes, so that each particle o the surace ibrates sinusoidally. I our understanding o the principle o superposition is correct, then there will be some locations in water where the waes coming rom two sources cause the net displacement o the surace to be twice the displacement i there was only one wae source. Consider point C in Fig a. I the sources A and B ibrate with the same amplitude Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

23 and the same period, then they each send a wae toward point C. The source A wae will cause C to ibrate as: The source B wae will cause C to ibrate as 2$ a ya " Acos[ ( t& )] T 2$ b yb " Acos[ ( t& )] T I the distances a and b are equal, then the two ibrations arrie in the same phase at point C ( 2 $ a b " 2 T $ ). I one wae causes C to moe up, the other one will also cause it to moe T up. I one causes C to moe down, the other one will also cause C to moe down (Fig b shows the displacements rom the A and B waes to C at one instant). I the idea o superposition is correct and the point C is at the same distance rom both sources, then the amplitude o ibration at C will be twice the amplitude caused by one wae source. There are many points at the same distance rom both sources, and the ibrations at these points will hae twice the amplitude caused by one wae. We should get the same eect or other points that are one waelength arther rom source A than rom B. This will also be true or points that are any integral multiple o waelengths arther rom A than rom B (i a& b" n' or n = 0, 1, 2, ). At any o these points, the resultant ibration amplitude due to both waes will be about two times bigger than the ibration amplitude i only one wae was present. We can reason urther. Suppose now that we choose a point D located so that the wae rom A arries there at a maximum displacement and the wae rom B arries at a minimum displacement (see Fig a). Then the A wae pushes point D up while B wae pushes it down the same distance. The net displacement is zero. This situation happens i one wae has to trael a distance that is hal a waelength longer than the other see Fig b. We conclude that at locations where 1 a& b" n ( ') with n an odd integer ( n = 1, 3, 5, ), there will be no ibration 2 because the two waes will always cancel each other. Figure Waes cancel at D Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

24 I we know the source positions and the waelengths o the waes, we can predict the locations o these two types o places: where the waes either reinorce each other or cancel each other. For this testing experiment we can use two speakers at dierent locations and predict places where our ears will hear no sound and places that are extra loud. We try this experiment next. A procedure or soling wae intererence problems is described on the let side and applied to the problem in Example 20.6 on the right side. Example 20.4 Where is there no sound? Two sound speakers separated by 100 m ace each other and ibrate in unison at a requency o 85 Hz. Determine three places between the speakers where you cannot hear any sound. Sketch and Translate! Construct a sketch o the situation. Label all known quantities. Where do you hear no sound? Simpliy and Diagram! Decide what simpliications you are making.! Draw displacement-ersus-time or displacement-ersus-position graphs to represent waes i necessary. Represent Mathematically! Represent the situation described in the problem using the relationships between physical quantities. Sole and Ealuate! Sole the equations or an unknown quantity.! Ealuate the results to see i they are reasonable (the magnitude o the answer, its units, how the solution changes in limiting cases, and so orth). Assume that the amplitude o the sound does not change as the wae traels away rom the source (this is a reasonable assumption i the two distances are not too dierent). The person should hear no sound at places where the distance rom the let speaker to the person is some odd multiple o a hal waelength arther than the distance rom the right speaker. Should hear no sound i: 1 a& b " n ( '). 2 Here a is x and b is 100 m - x : ' 3' 5' x (100 m x) ",, where sound 340 m/s 340 m/s ' " " " " 4.0 m! 1 85 Hz 85 s Sole the equation in the last step or places where we hear no sound: ' 3' 5' x (100 m x) ",, ' 3' 5' 7 2x & 100 m =,, ' 3' 5' 7 2x " 100 m +,, ( ') ( 3' ) ( 5' ) 7 x ". 50 m#, 50 m +, 50 m / / / 3 Using ' = 4.0 m, we expect no sound at: x " 51 m, 53 m, 55 m,.... Note that the irst place is 51 m rom the let speaker and 49 Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

25 m rom the right (one hal waelength dierent). The next place is 53 m 47 m = 6 m or 1.5 waelengths. The answers seem reasonable. When we conduct the experiment we ind that at the predicted locations we hear no sound. Try it yoursel: For the last situation, where should the person stand to hear sound with twice the amplitude? Answer: The condition or these places is that the ear should be the same distance or one or more ull waelengths dierent distances rom the sound sources: [ x (100 m x )" 0, ',2',...]!"This condition is satisied at positions 50 m, 52 m, 54 m,. We can now assert the superposition principle with greater conidence. Superposition principle When two waes pass through the same medium at the same time, the net displacement at a particular time and location in the medium is the sum o the displacements that would be caused by each wae i alone in the medium at that time. Mathematically this statement can be written as: y " y # y #...# y (20.8) net 1 2 n The process when two or more waes o the same requency trael in the same medium and add together to orm a smaller or larger wae is called intererence. Locations where the waes add to create a larger disturbance are called locations o constructie intererence. Places where two waes add to produce a smaller disturbance (to cancel each other) are called locations o destructie intererence. Huygens Principle The superposition principle allows us to explain another phenomenon that we obsered many times. Imagine a beach ball ibrating with simple harmonic motion in water. Circular waes moe outward rom the ball. How can we explain their ormation using the superposition principle? C. Huygens ( ) proided the answer. He was selected by King Louis XIV to direct all scientiic research in France, just as Aristotle was selected by Alexander the Great to be his court scientist. During his tenure with the king, Huygens deeloped the best telescopes o his time and inented a pendulum clock so accurate that it easily detected small dierences in ree all acceleration at dierent locations on the earth. Huygens idea or how waes were created in an orderly way as they moed out rom a source is shown in Fig You see a wae ront (like the crest o a wae) represented by line AB moing at a speed away rom a point source S. To determine the location o the wae ront a short time t ater it is at cured line AB, one needs to draw a large number o circular arcs, each centered on a dierent point o wae ront AB. Each arc represents a waelet that moes away rom a point on the original wae ront. The radius o a waelet r depends on the speed o the wae ront at the point where the waelet originates and on the time t that the waelet has traeled away rom the point. These three quantities are related by the equation r " t. The addition o these waelets produces a new wae ront whose shape can be determined i one adds Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

26 the disturbances due to all waelets. The new waeront is determined by drawing a tangent to the ront edges o the waelets line CD in Fig To test Huygens idea, consider two new experiments. Figure Creating new waeront Testing Experiment Table 20.6 Testing Huygens principle Testing experiment Prediction Outcome The waelets produced by each pin when added should produce a straight wae ront. Place a ibrating board with many closely placed pointed pins in water so the pins moe up and down each creating a waelet in the water. We obsere the predicted straight wae ront waes. Place in a tray o water a barrier with a narrow slit opening in the path o a plane wae. The ibrations o water in the narrow slit should produce a circular waelet beyond the slit. We obsere a circular waelet spreading behind the narrow slit. Conclusion Predictions based on Huygens principle match the outcomes o the testing experiments. These experiments do not disproe the principle and gie us more conidence in the principle. Reiew Question 20.6 Your riend says that it is impossible or two waes to arrie at the same point and still hae no motion at this point. How can you conince him that this is possible? 20.7 Sound waes We used sound in some o our examples in this chapter. Howeer, we hae not discussed the nature o a sound wae. As we know, or a wae to propagate, there needs to be a ibrating Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

27 source and a medium o interacting particles. When some o the particles are disturbed rom an equilibrium position, they push and pull nearby particles and so on. Early in this chapter, we attached a tiny mirror to one prong o a tuning ork (Fig. 20.1a). When light rom a laser pointer shined on the mirror, we obsered the relected spot o light on the wall ibrating up and down making a line. I we touched the tuning ork so the sound stopped, the ibration stopped and there was just a dot o light on the wall. I we hit the ork harder, the sound was louder and the line on the wall was longer. I you touch the ibrating tuning ork, the sound dies. I you hit the tuning ork and then put the ends o the prongs down on the surace o water in a pan (Fig. 20.1b), we saw waes on the water surace produced by the ibrating prongs. We conclude that sound sources are ibrating objects. What happens so that our ears hear a ibrating tuning ork or any other sound? Let s use a tuning ork and a microphone to get a better eel or the nature o a sound wae in air. Remember that a microphone has a magnet and a membrane with a coil attached. Varying air pressure causes the membrane to ibrate and the ibrating coil in a magnetic ield causes a corresponding ibrating electric signal (electromagnetic induction rom Chapter 18). Eidently, the sound wae is a arying pressure in the air. The ibrating prongs o the tuning ork causes air pressure changes and these pressure changes trael through the air rom one place to another. We also know that sound propagates in water and een in solid objects you can easily hear a ibrating tuning ork with the handle touching a table and your ear on the table. Does sound propagate in a acuum? To answer this question we could theoretically perorm the irst experiment under a acuum jar, but it is diicult to eacuate the jar completely. In addition, the tuning ork sitting on the bottom o the jar would ibrate the jar, which will produce sound. Thus we hae to answer the question by reasoning. I a tuning ork ibrates in a acuum, there will be no medium or the tuning ork prongs to ibrate against and consequently no way or a sound wae to originate and no medium or it to trael rom one place to another. Sound needs a medium in order to trael rom a source to a sound receier, such as the ear. The measurements o the speed o sound in dierent media show that sound speed increases in solids compared to liquids and air, which supports the idea that the sound has the same nature as waes on a slinky but instead o ibrating coils interacting with each other, there must be the particles o matter doing the same thing. In summary, sound inoles the transmission o a disturbance between interacting particles that ibrate. The disturbance propagates in all directions as a wae. The analysis o these compressions shows that sound is a longitudinal wae. How do humans hear sound? Your body is a dierent medium or the sound than the air through which the sound traels. When a wae reaches a dierent medium, it is partially relected and partially transmitted into it. We perceie these ibrations as sound. We can detect loud sounds with many parts o our body een our hands placed on an inside car door when the radio is playing. But we can t distinguish with our hands much about the sound is it a ballad, a Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

28 symphony, or a country western singer? To distinguish the details about the sound, we rely on our ears. The ear and hearing A simpliied sketch o the human ear is shown in Fig The part o the ear at the side o our head, called the pinna, gathers and guides sound energy into the auditory canal. Because o the pinna, the pressure ariation at the eardrum due to a sound wae is about two times the pressure ariation that would occur without the pinna and auditory canal. The pressure ariation at the eardrum causes it to ibrate. A large raction o the ibrational energy is transmitted through three small bones in the middle ear known collectiely as the ossicles and indiidually as the hammer, anil, and stirrup. These three bones constitute a leer system that increases the pressure that can be exerted on a small membrane o the inner ear known as the oal window. The pressure increase is possible or two reasons. (1) The hammer eels a small pressure ariation oer the large area o its contact with the eardrum, whereas the stirrup exerts a large pressure ariation oer the small area o its contact with the oal window. This dierence in areas causes the pressure to be increased by a actor o about 15 to 30. (2) The three bones act as a leer that causes the orce that the stirrup exerts on the oal window to increase, although the displacement caused by the orce is less than i the bones were not present. Because o all these eects, pressure luctuation o the stirrup against the oal window can be 180 times greater than the pressure luctuation o a sound wae in air beore it reaches the ear (a actor o 2 due to the pinna and auditory canal, a actor o up to 30 because o the area change rom the eardrum to the oal window, and a actor o 3 rom the leer action o the ossicles). Figure Parts o the ear The increased pressure luctuation against the oal window causes a luctuating pressure in the luid inside the cochlea o the inner ear. The luctuating pressure is sensed by nere cells along the basilar membrane. Neres nearest the oal window respond to high-requency sounds whereas neres arther rom the window respond to low-requency sounds. Thus, our ability to distinguish the requency o a sound depends on the ariation in sensitiity o dierent cells to dierent requencies along the basilar membrane. This ability is remarkable in that the basilar membrane is only about three centimeter long, and yet a normal ear can distinguish sounds that dier in requency by about 0.3 percent (a requency dierence o 3 Hz can be distinguished or two sounds near 1000 Hz). This ability to distinguish sounds o slightly dierent requency spans Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

29 the human hearing range rom about 20 Hz to 20,000 Hz. What about high requency sounds greater than 20,000 Hz? Ultrasound Ultrasound waes are pressure waes at a requency higher than the upper limit o human hearing. We cannot hear such high requencies because o limitations o the ear. On the other hand, many animals such as bats, dolphins and whales, dogs, and some types o ish communicate using ultrasound signals. Ultrasound waes are widely used or diagnostics in medicine. Pregnant women hae ultrasound scans to determine whether the unborn baby has certain health deects, and sometimes to determine the sex o the baby. The ultrasound relects dierently o arious organs, thus producing an interpretable picture or the doctor (Fig ). Many applications o ultrasound are based on the short waelength o the waes, which are smaller than internal body parts and help distinguish the details o those parts. Sound waes o longer waelength do not allows us to obtain the same inormation. Howeer, some applications o ultrasound are not related to the human body. Figure Ultrasound photo o child in womb Caitation When low-intensity high-requency sound traels through non-elastic media such as water and most liquids, there is a small uniorm luctuation o the pressure in the luids about their normal pressure (see Fig ). I the amplitude and intensity o the ultrasound wae increases, the rareaction part o the wae may hae such low pressure that the liquid pulls apart or ractures leaing a small caity in the liquid, a phenomenon known as caitation. When a caity is ormed, the liquid eaporates into it orming a low-pressure bubble in the liquid, which begins to collapse due to the high pressure surrounding liquid. As the bubble collapses, the pressure and temperature o the apor inside increases. The bubble eentually shrinks to a tiny raction o its original size, at which point the gas inside can hae a rather iolent explosion, which releases a signiicant amount o energy in the orm o a shockwae and as isible light. The temperature o this apor inside the collapsed bubble may be up to 5000 K and the pressure up to 1000 atm. Figure Depiction o pressure ariation due to sound wae Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

30 Ultrasonic deices use caitation bubbles to clean suraces. When used or cleaning purposes, ultrasound can remoe harmul chemicals. Caitation is oten used to homogenize or mix and break down suspended particles in a colloidal liquid compound, such as paint mixtures or milk. Many industrial mixing machines are based upon this design principle. Ultrasound and caitation is used to break kidney stones. Future applications are planned or joint and muscle treatment, or corrections o blood clots, and or high intensity ocused ultrasound non-inasie treatment o cancer. Reiew Question 20.7 How can you decide i sound is indeed produced by ibrating objects? 20.8 Loudness and Sound Intensity In eery day lie we describe dierent sounds using the words loudness, pitch, and timbre (the quality o sound). In this and the next section we learn how these words correspond to arious physical quantities. Loudness We already know rom preious experiments with a tuning ork and laser pointer that two sounds o dierent loudness hae dierent amplitudes. Is the loudness o sound determined only by the amplitude o ibrations? The answer would be yes i our ears were equally sensitie to all requencies o sound. Howeer, they are most sensitie to requencies around 500 Hz; sounds o less amplitude at this requency are perceied as loud as sounds o greater amplitude at a requency o 2000 Hz. Sounds o ery high amplitude at requency o 20 Hz or 20,000 Hz are perceied as not loud our ears can barely hear them i at all. This leads to dierent measures o the apparent loudness o a sound the physical quantities intensity and intensity leel. Sound Intensity As noted, our impression o the loudness o a sound depends partly on the requency o the sound and on the amplitude o the ibrations that the sound induces in our eardrums, which in turn depends on the magnitude o the pressure ariation caused by the sound wae. Normal atmospheric pressure is x 10 N/m. The pressure luctuation aboe and below atmospheric pressure o a barely audible sound is called the threshold o audibility. For the normal ear o a young person, the threshold pressure is about atmospheric pressure. A sound that is harmul to the ear is about about x 10 N/m, or less than one-billionth 6 10 times greater in pressure, or 2 20 N/m. Instruments built to measure the loudness o a sound do not measure the sound's pressure amplitude but instead measure two other quantities called intensity and intensity leel. The intensity I o a wae o any type was deined beore [see Eq. (20.6)] as the wae energy that crosses an area A perpendicular to the wae's direction in a time interal % t diided by the area Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

31 and the time interal. Intensity can also be deined in terms o power P since power is the energy that lows per unit time interal. Quantitatie Exercise 20.5 Sound energy absorbed at a construction site The intensity o sound at a construction site is W/m. I the area o the eardrum is sound energy is absorbed by one ear in an 8-h workday? cm, how much Represent Mathematically The intensity I o the sound, the area A o the eardrum, and the time interal % t o the exposure hae been gien. We wish to determine the sound energy % U absorbed. Equation (20.7) relates all o these quantities: Energy % U Intensity " I " " time area % t A To ind the energy, multiply both sides by the time interal and the area: % U " I % t A Sole and Ealuate In using the aboe, we need to conert the time interal into seconds and the area into 2 m ( * 3600 s + )( 2 * 10 m + )., 2 -/ % U "(0.10 W/m ).(8.0 h), - (0.20 cm ) " J 1 h / cm 13 This looks like a small number. Howeer, our perception o sound energy depends on the power per unit area, not the total energy. Our enironment includes sounds that dier in intensity rom about W/m (barely audible) to nearly 1 W/m (a painul sound). The noise in a classroom is about Intensity leel W/m. Because o the wide ariation in the range o sound intensities, a dierent quantity or measuring sound intensities, called intensity leel, has been deeloped. Intensity leel is not a measure o a sound's intensity but is a comparison o the intensity o one sound and the intensity o a reerence sound. Intensity leel is deined on a logarithmic scale as ollows: I 8 " 10log10 (20.9) I where I 0 is a reerence intensity to which other intensities I are compared (or sound, 12 2 I 0 " 10 W/m ). The log in Eq. (20.10) is the logarithm to the base ten o the ratio I / I 0. The logarithms to the base o 10 are oten used in engineering and acoustics; they are called common logarithms 1. For simplicity common logarithms assume the base o 10 but it is not written. As I and I 0 in Eq. (20.10) hae the same units, intensity leel is a unit-less quantity because the units cancel. Neertheless, intensity leel has a dimensionless unit called the decibel (db). The unit 0 1 I x y " x. For common logarithms b " 10, thus or y " log( x) ; x " 10 y. y " b, then log b( ) Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

32 seres as a reminder o what we are calculating, much like the radian is a dimensionless reminder o one way o speciying angles. The intensity leels 8 o seeral amiliar sounds are listed in Table Table 20.6 Intensities and Intensity Leels o Common Sounds Source o Sound Intensity Intensity leel Description 2 ( W/m ) (db) Large rocket engine (nearby) Jet takeo (nearby) Pneumatic rieter; machine gun Rock concert with ampliiers (2 m); jet Pain threshold takeo (60 m) Construction noise (3 m) Subway train Heay truck (15 m) Constant exposure endangers Niagara Falls hearing Noisy oice with machines; aerage actory Busy traic Normal conersation (1 m) Quiet oice Quiet Library Sot whisper (5 m) Rustling leaes Barely audible Normal breathing Hearing threshold Quantitatie Exercise 20.6 Noisy classroom The sound in an aerage classroom has an intensity o W/m. (a) Determine the intensity leel o that sound. (b) I the sound intensity is doubled, what is the new sound intensity leel? Represent Mathematically To calculate the intensity leel 8, we need to compare the gien sound intensity I with the reerence intensity or sound the log o that ratio: * I + 8 " 10log, -. I I 0 " 10 W/m and then take 10 times Sole and Ealuate (a) The ratio o the sound intensity I and the reerence intensity I 0 is: & 7 2 I 10 W/m " " & 12 2 I 10 W/m Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

33 Since 5 log 10 is 5 2, the sound intensity leel is: 5 8 " 10log10 " 10(5) " 50 db. (b) I I is doubled, the ratio o I and I 0 becomes: The sound intensity leel is, then, The classroom is pretty quiet. I I 2910 Try It Yoursel: The intensity at a concert is Answer: 110 db. W/m & " " & W/m 5 L " 10log(2910 ) " 10(5.3) " 53 db J/s m. What is the intensity leel? Tip! We can calculate the intensity leels listed in Table 20.6 easily by considering the power o 10 by which the sound's intensity exceeds the reerence intensity. For example, the sound intensity in a noisy oice is about reerence intensity o intensity leel is 10(8) = 80 db W/m which is 10 times more intense than the W/m. Since the noisy oice is 8 powers o 10 greater than I 0, the The intensity scales are used in ields other than acoustics. An example is a amiliar Richter scale or seismic actiity. The scale uses a single number to represent the amount o seismic energy released by an earthquake. Sometimes the Richter scale is known as local magnitude scale. To make Richter scale scientists calculate a base -10 logarithm o the combined horizontal amplitude o the largest displacement rom zero as recorded by a speciic instrument (Wood- Anderson Torsion seismometer). Thus an earthquake that is 6.0 on a Richter scale has a 10 times bigger amplitude than the one that measures 5.0. The problem with the scale is that it saturates at high amplitudes thus geologists use a dierent scale called moment magnitude scale. Determining Intensity rom Intensity Leel Sometimes in our problem soling we need to calculate the intensity o a sound wae (or some other wae) rom its known intensity leel. To do this, we use another general property o logarithms [see Eq. (A.4) in Appendix A.3]. I 8 " log( I / I0), 10 7 I / I " /10 8 /10 7 I " 10 I0 (20.11) For example i a shout produces an intensity leel o 75 db, the intensity o the sound is: 2 We need to ind 5 y " log 10. From the knowledge o logs: 5 10 " 10 y " x, thus in this case y " 5. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

34 I " 10 I " 10 I "(10 )(10 ) I " I " (10 W/m ) " W/m 75/ & 12 2 & Reiew Question 20.8 How is the loudness o sound related to the physical quantities characterizing wae motion? 20.9 Pitch, requency, and complex sounds I you take seeral tuning orks o dierent sizes and hit each ork with a mallet, they produce sounds o approximately the same intensity but you hear them as dierent pitch sounds. The smaller ork makes a higher pitch sound and the bigger ork a lower pitch sound. The smaller the ork has a higher ibration requency, and the bigger the ork has a lower ibration requency. Thus, pitch is related to the requency o sound. Pitch is not a physical quantity (there are no units or pitch) but is a physiological characteristic o sound. It is used to indicate a person s subjectie impression about the requency o a sound. A normal ear is sensitie to sounds in the audio-requency range rom about 20 Hz to 20,000 Hz. As a person ages, the limits o this hearing range shrink. Complex sounds, waeorms and requency spectra There is more to sound than the sensations o its loudness and pitch. For example, an oboe and a iolin playing concert A equally loud at 440 Hz sound ery dierent. What other property o sound causes this dierent sensation to our ears? Let s use a microphone and a computer to look at a pressure-ersus-time graph o a sound wae produced by playing one key on a piano (see Fig notice that the graph has negatie alues, thus it must be representing the gauge pressure). There is a characteristic requency, but the wae is not a pure sinusoidal like wae. What causes this shape? Figure Pressure-ersus-time o sound rom one piano note To help answer this question, lets use two tuning orks with the second ork ibrating at twice the requency o the irst. A microphone can detect the sound when either or both orks ibrate. A computer records the electric signal produced by the microphone. When we strike each ork separately, the computer shows sinusoidal waes (Fig a and b). When we hit the two orks simultaneously, the signal looks dierent (Fig c) it is more like that recorded when playing the piano. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

35 Figure Pressure-ersus-time graphs caused by one or two tuning orks To understand the shape o the wae in Fig c, recall the superposition principle. Instead o one wae arriing at the microphone, the sound rom two tuning orks simultaneously arries at the microphone. We can use the superposition principle to explain the shape o the pressure-ersus-time graph or the sound produced by both tuning orks (Fig c). To construct the graph you add the pressure ariations caused by each wae at each instant in time. The result is the signal shown in Fig 20.23c. The luctuating pressure pattern is called a waeorm. Sometimes we wish to determine the requencies and amplitudes o the sinusoidal pure sounds that produced a complex wae. This is how it is done. A microphone detects a sound wae and conerts the pressure ariation into ariable electric current which changes with time exactly the same way as the pressure near the microphone. This electric signal passes through a deice called a spectrum analyzer. This deice electronically analyzes the complex waeorm and plots on a screen the amplitude o waes at dierent requencies that were added together to orm the complex wae. The amplitude-ersus-requency graph is called a requency spectrum o the complex wae. A requency spectrum o the complex waeorm in Fig c is shown in Fig The height o the line at each requency is proportional to the amplitude o the wae at that requency. Since the amplitude o wae 2 is twice that o wae 1, its ertical height is twice the height o wae. Wae 2 is also twice the requency o wae 1 (in this example only). Figure Frequency spectrum produced by two tuning orks Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

36 Most sounds we hear are complex waes consisting o sinusoidal waes o many dierent requencies. The sound we call noise includes waes o a broad range o requencies, and we seldom associate a pitch with the noise. Howeer, i the complex sound is the result o the addition o the wae o high amplitude at some low requency (called undamental) and waes at smaller amplitudes but at higher requencies that are requencies are multiples o the undamental requency (called harmonics), then we usually identiy the pitch o the sound as being that o the undamental requency. This is true een when the amplitudes o the higher harmonics are greater than that o the undamental. The waeorms and requency spectra o complex waes rom a piano and rom a iolin playing concert A at 440 Hz are shown in Fig Notice that the iolin A has many requency components, which are all multiples o 440 Hz (or example, 880 Hz, 1320 Hz, 1760 Hz, 2200 Hz, and een higher requencies). It is a complex wae. Figure Piano and iolin concert A waeorms and requency spectra Seeral physiological aspects o our hearing system can cause ariations in the pitch we hear een though the requency o the sound is unchanged. For example, a loud, high-requency sound may seem to hae a higher pitch than the same requency sound played sotly. Under some conditions these ariations may be considerable. Fortunately, these conditions do not oten arise in music, and the pitch change can usually be ignored. Quality o Sound A iolin while playing concert A at 440 Hz sounds dierent rom an oboe or French horn playing the same note. We say that the sounds o the instruments hae a dierent quality or timbre, the characteristic that distinguishes one sound rom others o equal pitch and loudness. The quality o a sound at a particular pitch depends on the requency spectrum o the sound on the number o higher harmonics that are part o the sound, the relatie amplitudes o these harmonics, and on other more complex characteristics o the sounds. Beat requencies The quality o sound is also aected when seeral instruments all play the same music (or example, the irst iolin section o an orchestra). Although they try to play identical requencies, there can be slight dierences. One iolinist may be playing concert A at 439 Hz and another at 441 Hz. What happens when these two iolinists play together? Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

37 Example 20.7 Beats Two sound sources are equidistant rom a microphone that is connected to a computer that records the arying sound pressure as a unction o time due to the two sound sources (Fig a). Source 1 ibrates at 10 Hz and source 2 at 12 Hz. During 1.0 s, the ariation o pressure at the microphone due to wae 1 is shown in Fig b and that due to wae 2 is shown in c. These are pressure ariations due to only one wae source playing at a time. Determine the pressure-ersus-time wae pattern that would be seen i both sound sources played at the same time. Figure 20.26(a) Sketch and Translate We need to add the two waes shown in Fig b and c to see the pattern produced when both sounds play simultaneously (Fig d). Note that the pressures shown in the Fig b and c graphs are the pressures aboe or below atmospheric pressure caused by each sound i that was the only sound present. Simpliy and Diagram Note in Fig d that at time 0.00 s the positie pressure caused by wae 1 cancels the negatie pressure caused by wae 2. The net disturbance at time 0.00 s is zero. At 0.25 s, both waes hae a maximum negatie pressure ariation and the net disturbance is twice that negatie pressure. At 0.50 s, the two wae pressure ariations again cancel each other causing a zero pressure disturbance. Figure 20.26(b)(c)(d) Beats produced by two sounds Represent Mathematically The pattern shown in Fig d rom 0.00 s to 0.50 s is called a beat. It has a requency o about 11 Hz (the aerage o the two wae requencies) but its amplitude is modulated rom zero to twice the amplitude o the indiidual waes and back to zero again at 0.50s. A second beat is produced rom 0.50 s to 1.00 s. The beat requency is the number o beats Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

38 per second in this case 2 beats/s. This is just the dierence in the two wae requencies (12 Hz 10 Hz) = 2 Hz. In general, the number o beats produced per unit time, the beat requency beat, equals the magnitude o the dierence in the requencies o the two waes causing the beat pattern: " beat 1 2 Sole and Ealuate As noted aboe, the beat requency in this case is beat " 10 Hz 12 Hz " 2 Hz. Note that we took the absolute alue o the requency dierence. Try It Yoursel: What would be the period o the amplitude modulation on the graph in Fig d i one source ibrates at 67 Hz and the other one at 69 Hz. Answer: The beat requency would be the same (2 Hz) and the period o the beat requency pattern would be T " 1/ " 0.5 s. We can now answer the question posed beore the aboe example. When the two iolins play concert A simultaneously, the waes that they make add together and produce ibrations whose amplitude changes. The amplitude changes with the requency that equals the dierence o the two iolin requencies 2 Hz in that example. " (20.10) beat 1 2 These ibrations o changing amplitude are beats and the requency o the amplitude change is called the beat requency. With many instruments playing together, the amplitudes o the sounds produced are modulated at beat requencies. This produces a pleasing so-called chorus eect. Beats are useul in precise requency measurements. For example, a piano tuner can easily set the requency o the middle C string on the piano to 262 Hz by obsering the beat requency produced when the piano string and a 262-Hz tuning ork are sounded simultaneously. I a beat requency o 3 Hz is obsered, then the piano string must be ibrating at either 259 Hz or 265 Hz. The string is then tightened or loosened until the beat requency is reduced to zero. Reiew Question 20.9 What happens when two sinusoidal sounds, one at twice the requency o the other, are simultaneously produced? How do we know? Applications: Standing waes on strings The superposition o waes (adding waes) is the basis or much that is needed to understand how musical instruments work. Consider string instruments irst. Start by obsering the ibration o a rope with one end attached to a ixed support (a wall) while the other end is Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

39 held in your hand. You start shaking that end up and down and obsere an interesting eect. Depending on the requency with which you shake your hand, the rope has either ery small amplitude and irregular ibrations or large amplitude sine-shaped ibrations. The latter are not traeling waes but are instead coordinated ibrations with parts the rope ibrating up and down with large amplitude and other parts not ibrating at all. Graphs o these large amplitude ibrations are shown in Fig and are called standing waes. How is the irst standing wae ibration produced? Figure Standing waes on a string Tip! Notice that graphs in Fig represent seeral snapshots o the same wae. The times at or which the solid line and the dashed lines represent the positions o dierent points are dierent. Obserational Experiment Table 20.7 Producing the irst standing wae ibration. Obserational experiment You shake the rope at a relatiely low requency and obsere the large amplitude ibration shown below. Analysis A series o sketches o the disturbances caused by your shaking at requency is shown at the right. A pulse starts moing to the right (b-d). When it reaches the ixed end, the upward pulse is relected and inerted to a downward pulse traeling back toward your hand (e-g). When it reaches your hand, it once 1 Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

40 again relects and inerts to an upward pulse (part h). To make this pulse bigger, you gie your hand another upward shake, making sure to time the shake so that it adds to the preious upward pulse that has just been relected (part h). Pattern You must shake the rope upward each time a preious pulse returns. The amplitude o the disturbance The traeling subscript along 1 the means rope grows. the lowest You need standing one shake wae during requency. the time The interal word ibration 2L that it takes is not or a unit the pulse and can be remoed rom the aboe equation. Thus, this lowest requency ibration o the rope is one to trael down the string and back: 2L 1 ibration shake per time interal : " or 1 " undamental requency = " 2L 2L 1 1 " 1 : " 2L " 2L (20.13) This requency is traditionally called the undamental requency. I the shaking requency is less than this, each new pulse will interere destructiely with preious pulses, and the rope will ibrate little. Should we raise and lower our hand at a requency 1 ", each new pulse adds 2L to the last and creates a large-amplitude ibration. The waelength o this undamental requency ibration is: ' 1 " " " 2L 1 2L The length o the rope equals one hal o the waelength o the wae that has this undamental requency. Let s test this idea. Testing Experiment Table 20.8 Fundamental ibration requency Testing Experiment Prediction Outcome Predict the The speed o a wae in the string is: When we pluck a undamental 1/2 1/2 banjo D string ( ibration requency F ) ( Peg on String (80.0 N) ) ". / ". / " 807 m/s. and compare it to o a banjo D string..( 3 2 mstring / Lstring )/ 3.(0.179 x 10 kg)/(0.690 m) / 2 3 the requency o The mass o the Consequently, we predict that the undamental requency should a tuning ork, we string is g, its be: ind = 587 1D length is m, 807 m/s 1 and the wooden peg " " " 585 s " 585 Hz Hz. 1 2L 2(0.690 m) pulls on its end exerting an 80.0-N Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

41 orce. Conclusion The agreement is excellent. We did not disproe the requency expression, but instead built conidence in it. Now consider the ibration that produces a standing wae on the rope at twice the requency o the undamental ibration (see Fig ). You ibrate the rope so that your hand has two up and down shakes during the time interal that is needed or a pulse to make a round trip down the rope and back (Fig ). Figure Producing second standing wae ibration The new pulse adds to a pulse started earlier and that has completed a round trip back to your hand. This earlier pulse relects rom your hand and joins the new pulse becoming a larger amplitude pulse. Thus, the requency 2 o this second standing wae ibration is: We can see a pattern now. 2 2 ibrations " second standing wae ibration = " 2. 2L 2L Standing wae requencies on a string The requencies n that a string with ixed ends can ibrate are: n " n or n" 1, 2,3,... (20.11) 2L where is the speed o the wae on the string and L is the string length. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

42 Since ' " ibrations:, we can write an expression or the waelengths o these allowed standing wae 2L ' n " " " or n " 1, 2,3,,,. (20.12) n n( ) n 2L Note that a rope or string will ibrate with large amplitude at any requency or which a pulse produced by the wae source (or example, the ibrating hand) is reinorced by a new pulse ater completing a round-trip up the string and back again. The wae patterns that are produced on the rope or these dierent alues o n do not appear to trael or moe. They appear to be stationary ibrations, called standing waes. Standing waes are caused by the superposition o waes moing in opposite directions. How are standing waes dierent rom amiliar to us traeling waes? When a traeling wae moes along the string each point on the string ibrates between the maximum and minimum disturbances (between y " 6 A). Howeer dierent points reach the maximum displacements at dierent times, in other words dierent points hae dierent phases at the same time. In a standing wae dierent points on the string moe in dierent patterns in some places they ibrate with large amplitude ibration (called antinodes) and in other places with no ibration at all (called nodes). Howeer, dierent points reach their maximum displacements at the same time they ibrate in phase. When one plays a musical instrument, she excites more than one o its standing wae ibrations (see complex waes in the preious section). For example, when you bow a iolin string, you excite simultaneously 10 or 20 standing-wae ibrations. The pitch o the sound is determined by the standing wae o the lowest requency present in the complex wae. The quality o the sound depends on the number and relatie amplitudes o the other standing-wae requencies that make up the complex sound wae and on the way that this spectrum o tones changes with time. Example 20.8 A Playing a banjo A banjo D-string is 0.69 m long and has a undamental requency o 587 Hz. The string ibrates at a higher undamental requency i its length is shorter. To shorten the string you can hold the string down oer a ret. Where should you place the ret to play the note F-sharp, which has a undamental requency o 734 Hz? Sketch and Translate We can isualize the situation by picturing a string o dierent lengths. We know that when the length o the string is L " 0.69 m, the undamental requency is 1 " 587 Hz. What is the new length o the string L ' or which the undamental requency is 1 ' " 734 Hz? Simpliy and Diagram Assume that the speed o the wae on the string is independent o where the inger pushes down on the string. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

43 Represent Mathematically The undamental requency o ibration 1 relates to the string length L by the equation 1 ", or 2L L ". We cannot sole immediately or the unknown length 2 1 when the string is ibrating at 734 Hz since we do not know the speed o the wae on the string. But we can use this idea to write an expression or the speed o the wae or both cases (the speed is the same or both cases): Sole and Ealuate " (2 L) " '(2 L'). 1 1 Diiding both sides o the expression or the speed sides by ' 1 and multiplying by 2, we get an expression or L ' : L' L * + L " ", - 1' 1' * + 1 * 587 Hz + L' ", - L", -(0.69 m) = 0.55 m 0 1 ' Hz 1 The answer is reasonable as it is less than 0.69 m. To ind the ret location, we need to ind the dierence between the original length L and the new length L ' : L L '" 0.69 m 0.55 m = 0.14 m. The ret should be 0.14 m rom the end o the string. Try It Yoursel: I tuned correctly, the undamental requency o the A string o a iolin is 440 Hz. List the requencies o seeral other standing-wae ibrations o the A-string. Answer: The other standing wae requencies are integral multiples o the undamental and include: 880 Hz, 1320 Hz, 1760 Hz, and so orth. Reiew Question A horizontal string o length L has one end passing oer a pulley with a block hanging at the end. A motor is attached to the other end o the string and exerts a constant horizontal orce on the string. When the motor is turned on, it ibrates the end o the string up and down at requencies that change slowly rom zero to some high requency. What do you obsere? Application: Standing Waes in Air Columns So ar we discussed how standing waes are ormed on strings that are ixed at both ends. In this section we learn how musical instruments made o pipes or tubes such as organs, clarinets, lutes, and trumpets work. Standing waes in open pipes Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

44 Suppose you hae pipes o dierent length and blow across their ends. They produce sounds o dierent pitch. Or you take a bottle, ill it with water, and blow across the top o the bottle. You again hear sound whose pitch depends on the amount o water in the bottle. Your can een play music by haing seeral bottles illed with dierent careully measured amounts o water and dierent people blowing into them. Each bottle can be one note in a song. How can we explain these phenomena? When we blow in a pipe, or lute, or through a ibrating reed, such as in a clarinet, we initiate pressure pulses at that end o the pipe (or example, by a ibrating reed). Alternating high and low pressure pulses leae the reed and moe in the air column squishing the air in the air column in a repetitie ashion. Consider the ate o one o these pressure pulses while it is moing in an air column with both ends open. Such pipe is called an open pipe. The pressure pulse moes along the pipe as shown in Fig a-c. At the open end, a high-pressure pulse can suddenly expand into open space causing a high-pressure pulse outside o the tube and leaing behind a low pressure relected pulse that traels back into the tube. This change rom high to low pressure is similar to a phase change o a pulse on a string relected o a ixed end. Unlike the phase o the pulse due to the displacement o air molecules (density pulse) the pressure pulse changes its phase when reaching an open end and does not change phase when it reaches a closed end. Figure Producing standing wae in open pipe This pulse o decreased pressure moes back toward the other end o the pipe where it started (parts d-) and is relected at the open end on the let (part g). Ater relection rom both open ends, the pulse again has increased pressure and is moing toward the right. I a new pressure pulse is initiated at this time, it intereres constructiely with the relected pulse, and its amplitude increases. A large-amplitude standing-wae ibration begins to orm in the open pipe. I, howeer, the new air pressure pulse is too early or too late, it intereres destructiely with the relected pulse, and the amplitude decreases. The requency o standing-wae ibrations in pipes depends on the time interal needed or a pulse to trael down the pipe and back again. I the sound pulse traels at a speed along a 2L pipe o length L, then the time interal T needed to trael a distance 2L is T ". To reinorce the ibration o air in the pipe, pressure pulses should occur once each time period T or 1 at a requency " T " 2L. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

45 This is the same equation as or the undamental requency o ibration o a string. Just as we can make a string ibrate at multiples o the undamental requency, the pipe can also ibrate with large amplitude i excited by air pressure pulses at integral multiples o the undamental requency. Howeer, in or pipes open on both ends, the speed o a pulse is independent o the pipe as all pulses propagate through the air inside the pipe (about 340 m/s). Standing wae ibration requencies in open pipes The resonant requencies n o air in an open pipe o length L in which sound traels at speed are: n " n( ) or n" 1,2,3,... (20.13) 2L Musical instruments that unction as open pipes include the lute, recorder, oboe, and bassoon. Tip! Be sure that you use the speed o sound in the medium that ills the pipe in the aboe expression or the standing wae requencies. Standing waes in one-end closed pipes Next, consider a pipe, which is closed at one end and open at the other end called a closed pipe. Musical instruments that behae somewhat like closed pipes include the clarinet, trumpet, trombone, and human oice. The closed end o these instruments is at the reed, mouthpiece, and ocal cords, respectiely. It is more complicated to determine the requencies o standing waes in closed pipes than in open pipes. Suppose that you initiate a pulse o increased pressure by a reed near the open end o the pipe (Fig a). When the pulse is relected at the closed end, its phase is unchanged (part d). I the pressure in the pulse is aboe normal pressure, the pressure in the relectie pulse is also aboe normal. When the relected pulse traels back to the open end, it is once again partially relected. Howeer, this time there is a change in phase; a high-pressure pulse traels into open space leaing behind a low-pressure relected pulse. There is now decreased pressure pulse traeling back to the right (part g). I a new pulse o increased pressure is initiated, it cancels the eect o the relected pressure pulse. To add to the relected pulse in a constructie way, we must wait or it to make an additional trip down the pipe and back again. During each round-trip, its phase is inerted once. Ater two round-trips it is again a pulse o increased pressure (part m), and a new pulse adds constructiely to increase its amplitude. For a large-amplitude standing wae to be produced, then, the time interal T between pulses must be 4L. The requency o the undamental is 1 " 1 T " 4L. The one-end closed pipe is unique in that so-called oertone ibrations (higher requency standing waes) do not occur at all requencies that are integral multiples o the undamental requency. For example, at twice the Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

46 requency o the undamental, a pulse makes only one trip up and down the pipe beore another pulse is started. The relected pulse, which is inerted, intereres destructiely with the new pulse. The pipe cannot ibrate at twice the undamental requency. A closed pipe can, howeer, ibrate at odd integral multiples o the undamental requency. Figure Producing standing waes in a closed pipe Testing the requency equation or closed pipes I our reasoning that sound in pipe-like instruments is due to standing waes in them is correct, then changing the medium inside a pipe should change the undamental requency produced by the pipe without changing its size. Our ocal cord ibrations initiate sounds by our oices. The mouth is like a chamber with one end closed at the ocal cord and the other end open at the mouth a closed pipe. I we hae a dierent gas in the throat and mouth, the speed o a wae in the throat and mouth will change, as should the standing wae requencies o sounds produced. Conceptual exercise 20.9 Funny talk What should happen to sound rom a oice ater a person inhales helium and sings a note? Sketch and Translate Assume that a person s ocal chamber is a resonant caity like a one endclosed pipe with the ocal cords at the closed end and the mouth at the open end. The speed o sound in helium gas is much greater than in air. Thus, the undamental requency ( 1 " ) o 4L sound rom our oice should be greater when illed with helium than when illed with air. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

47 Simpliy and Diagram. The length o the chamber does not change during the experiment. The speed o sound in air is 340 m/s and in helium it is almost three times higher 930 m/s. Thus the undamental requencies o the sound we produce should increase almost three times a dierence easily heard. When we inhale helium and speak or sing with it in our mouth, a ery high pitch sound is heard as predicted. Try It Yoursel: In our testing experiment with helium, we used gas with a greater speed o sound. What happens i we inhale a gas that has a lower sound speed than in air? Answer: An example o such gas is CO 2. Inhale it (as this gas is dangerous, only trained perormers can do it) and say a ew words. The pitch o your oice will be much lower. Another testing experiment might inole the receiing end o sound our ear. We know already that the auditory canal o the outer ear is a cm -long pipe closed at the end by the eardrum. I we ill it with water (being underwater or example), then the speed o sound will be signiicantly dierent (1500 m/s as opposed to 340 m/s in air). Thus, we should hear high requency sounds better under water than when in air. Quantitatie Exercise Listening under water Calculate the undamental requency o the auditory canal when (a) in air and (b) in water. How might this resonance o the auditory canal aect our ability to hear sounds at these requencies? Represent Mathematically Visualize the canal as a one end-closed pipe open to the air and closed at the eardrum. The length o the canal does not change but the speed o sound in the canal increases signiicantly i illed with water compared to air. Thus the undamental standing wae requency 1 " should be at higher requency in the water illed canal. 4L Sole and Ealuate (a) The undamental requency 1 or a closed pipe o length L " 3.0 cm = 0.03 m illed with air through which sound traels at a speed " 340 m/s is: 1 * + * 340 m/s + " 1, -" 1, -" 2800 Hz. 0 4L 1 0 4(0.030 m) 1 The ear is, in act, sensitie to sound in this requency range. (b) Sound traels at speed o the closed pipe auditory canal illed with water is: " 1500 m/s in water (see Table 20.4). The undamental requency 1 * 1500 m/s + " 1, -" 13,000 Hz. 0 4(0.030 m) 1 Our inner ears are not ery sensitie at this ery high requency. Howeer, the resonance o the auditory canal when underwater helps us hear high-requency sounds better than when the canal is illed with air. Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

48 Try It Yoursel: You sing middle C (256 Hz) aboe a graduated cylinder partially illed with water. What should be the height o the air space at the top o the cylinder (water is at the bottom) so you hear a resonance o your oice with the cylinder? Answer: 33 cm. We can now assert with conidence: Standing wae ibration requencies in one-end closed pipes The resonant requencies n o air in a pipe o length L closed on one end in which sound traels at speed are: n " n( ) or n" 1,3,5,7,... (20.14) 4L Standing-wae ibrations occur in many objects besides strings and pipes the swaying o tall buildings and the twisting ibrations o bridges are examples. Reiew Question When we studied traeling waes, we decided that the waelength o a wae depends on the requency o ibration and the speed o the wae in the medium. Is this statement true or standing waes in pipes? Putting it all together: Doppler Eect Until now, we hae been studying situations in which the source o a wae and the obserer were both stationary with respect to the earth and with respect to each other and the medium in which the wae propagated was at rest with respect to both the source and the obserer. In this case, the requency o a wae as it leaes its source is the same as the requency o the wae when detected by the obserer. In real lie objects emitting and detecting waes can be moing. What happens then to the requency o emitted or receied waes? To answer this question we start with some obserational experiments. Suppose you stand near a racetrack as ast-moing race cars pass. While a car approaches, you hear a high-pitched, whirring sound; as the car passes, the requency or pitch o the sound you hear suddenly and noticeably drops. A similar phenomenon occurs when you listen to the sound rom the horn o a passing car or the whistle o a passing train. The change rom a higher pitch as the sound source approaches to a lower pitch as it moes away is an example o the Doppler eect. The Doppler eect occurs when a source o sound and an obserer moe with respect to each other and with respect to the medium in which the sound traels. Is there a pattern in these obsered requency shits? Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

49 Consider more phenomena to see i there is a pattern in the requency change. We connect a generator to an electronic whistle that makes a sound at a source requency and a S microphone and spectrum analyzer that records the requency O o the sound detected by an obserer (the microphone). The source whistle is attached to a cart that can moe in one direction toward or away rom another obserer cart that can moe independently toward or away rom the source cart. Consider what happens to the requency o the obsered sound when either the sound source or sound obserer is moing with respect to the ground. In all experiments the air in which the sound propagates is at rest with respect to earth. Obserational Experiment Table 20.9 Eects o motion on obsered requency. Obserational experiment (a) Whistle source stationary with respect to the ground. Obserer moing toward source. Analysis The requency spectrum or the signal emitted by the source whistle ( S ) and heard by the obserer ( O ): S ; O (b) Whistle moing toward obserer. Obserer stationary with respect to the ground. The requency spectrum S ; O (c) Whistle stationary with respect to the ground. Obserer moing away rom source. The requency spectrum S < O (d) Whistle moing away rom obserer. Obserer stationary. The requency spectrum: < S O (e) Whistle moing with respect to the ground. Obserer moing at the same speed in the same direction. The requency spectrum: " S O Pattern Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

50 (a) & (b) Source and/or obserer are moing toward each other: the obsered requency is greater than the emitted source requency. (c) & (d) Source and/or obserer are moing away rom each other: the obsered requency is less than the source requency. (e) There is no relatie motion: the obsered and emitted requencies are the same. Note careully the obsered patterns in Table We try next to deelop a quantitatie explanation or these patterns. Doppler eect or a moing source To explain the Doppler eect, examine the waes created by a water beetle bobbing up and down at a constant requency on the water. I the beetle bobs up and down in the same place, the pattern o wae crests appears as shown in Fig The crests moe away rom the source at a constant speed. The large circle represents a crest produced by the beetle some time ago, and the small circle is a crest produced more recently. The distance between adjacent crests is one waelength and is the same toward obserer A as toward obserer B. The requency o waes reaching both obserers is the same. Figure Bobbin beetle creates wae crests Figure Crests created as beetle hops to right Now suppose the bobbing beetle moes to the right with respect to the water at a slower speed than the speed at which the waes trael. Each new wae produced by the beetle originates rom a point arther to the right (Fig ). Wae crest 1, the irst to be created, is a large circle centered at position 1 where the beetle started bobbing. Wae crest 2 is a smaller circle centered a step to the right at position 2, and so orth. Wae 1, which started an equal distance rom obserers A and B, reaches them at the same time. Howeer, wae 4 reaches B somewhat sooner than it reaches A because the wae originated to the right o center, closer to B than to A. Thus, B obseres our waes in a shorter time interal than does A. In general, i a wae source moes toward an obserer, the obserer detects a higher requency than the requency emitted by the source (see Experiment (b) in Table 20.9). I the source moes away rom the obserer, the obserer detects a lower requency than the requency emitted by the source (see Experiment (d) in Table 20.9). To derie an equation or calculating this shit, irst calculate the separation o crests in ront o the beetle. Consider the separation o wae crest 4 and the wae initiated by the beetle's ith step (wae crest 5 shown in Fig ). I the beetle bobs up and down at requency, the S Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

51 1 time between the start o wae 4 and the start o wae 5 is T ". During this time interal, the beetle moes a distance T S to the right, where S is the speed o the wae "source", in this case the beetle. Wae 4 has moed a distance T during that same time interal, where is the speed o the wae through the water. S Figure Waelength reduced in ront o hopping beetle We see in Fig that the crests o waes 4 and 5 arriing at obserer B are separated by a distance ' ' B (a waelength) gien by ' '" T T B Similar reasoning could be used or the waelengths o waes reaching obserer A. The beetle is moing away rom A (see Fig ) and the waelengths o the waes reaching A are: S ' A' " T # ST The waes are incident on obserer B at a requency OB and on A at requency OA wae speed 1 " " " ". waelength ' B ' T ( ) T( ) s s wae speed 1 " " " " waelength ' A ' T ( + ) T( + ) s s The inerse o the time interal T between the beetle s bobs is the beetle bobbing requency S 1 " (the beetle is the wae source). Substituting or 1 T T in the preceding equations, we hae an expression or determining the obsered requency either in ront or in back o the source: O ". (20.15) S (! ) The minus sign applies or an obserer B toward which the source is moing and the plus sign to an obserer A or which the source is moing away. Note that with the minus sign and the source moing toward the obserer, the obsered requency will be greater than the source requency in agreement with the qualitatie experiments. With the plus sign and the source moing away rom s Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

52 the obserer, the obsered requency is less than the source requency, also in agreement with the qualitatie experiments. Doppler eect or a moing obserer The obsered requency also diers rom the source requency i the obserer moes with respect to the ground and the source is stationary. Assume, or instance, that a riend remains stationary in the water o a swimming pool and creates waes by pushing a beach ball up and down. I you swim toward the source, you encounter the waes more requently than i you remain stationary (consistent with Experiment (a) in Table 20.9). I you swim away rom the source, you encounter the waes less requently than i you remain stationary (consistent with Experiment (c) in Table 20.9). I the obserer moes toward the source at speed O, then the waes appear to moe past the obserer at speed #. Since the waelength ' o the waes rom a stationary source is O uniorm in all directions, the obserer will encounter waes at a requency speed o obserer relatie to waes waelength + ' O O " ". I the obserer moes away rom the source, then the waes appear to moe past the obserer at speed and the obsered requency will be: O O speed o obserer relatie to waes " " waelength ' The waelength ' "relatie to the source requency S and the speed o the wae in the medium is ' " / S. Thus, O 6 6 * 6 + " " ", - ' 0 1. (20.16) O O O S / S This equation can be used to calculate the requency O detected by an obserer moing at speed O toward (plus sign) or away rom (negatie sign) a stationary source emitting waes at requency. S General equation or Doppler eect or sound (20.16): We can get a general equation or the Doppler eect by combining Eqs. (20.15) and O Doppler eect or sound When a sound source or sound obserer moes relatie to each other, the requency o the obsered (detected) sound diers rom the requency O S o the sound source: * 6 +, (20.17) 0 1 O O " S, -! S Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

53 where is the speed o the wae through the medium in which it traels; O is the speed o the obserer relatie to the medium (use the plus sign in Eq. (20.17) i the obserer moes toward the source and a negatie sign i the obserer moes away rom the source); and S is the speed o the source relatie to the medium (use the negatie sign in Eq. (20.17) i the source moes toward the obserer and a positie sign i the source moes away rom the obserer). Tip! I the signs look conusing, use the sign that makes a higher requency i the source is moing toward the obserer or the obserer is moing toward the source; similarly, choose signs the make the obsered requency lower i the source or obserer is moing away rom the other. Example Listen to train whistle A train traeling at 40 m/s with respect to the ground has a horn that ibrates at a requency o 200 Hz. Determine the requency o the horn s sound heard by a bicycle rider traeling at 10 m/s with respect to the ground in the same direction as the train when the cyclist is (a) ahead o the train and (b) when behind the train ater the train has passed. Sketch and Translate A sketch o the situations is shown in Fig a and b. Figure Bicycle rider hears train whistle Simpliy and Diagram Both speeds are assumed constant; the speed o sound is 340 m/s. Represent Mathematically (a) The train source whistle is moing toward the obserer, which causes an increase in the obsered requency. This increase occurs i you use the minus sign in Eq. (20.17) in ront o the source speed S. The obserer on bicycle is moing away rom the source causing a decrease in the obsered requency. This decrease occurs i you use a minus sign in Eq. (20.17) in ront o the obserer s speed O. (b) The train source whistle is now moing away rom the obserer, which causes a decrease in the obsered requency. This decrease occurs i you use the plus sign in Eq. (20.17) in ront o the source speed S. The obserer on bicycle is now moing toward the source causing an increase in the obsered requency. This increase occurs i you use a plus sign in Eq, (20.17) in ront o the obserer s speed O. Sole and Ealuate (a) The requency heard by the obserer (the cyclist) when moing ahead o the train is: Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

54 * 6 + ( 340 m/s 10 m/s ) (200 Hz) 220 Hz O " S, -" "!. S 2340 m/s 40 m/s/ 3 (b) The requency heard by the cyclist ater the train has passed is: O ( 340 m/s # 10 m/s ) "(200 Hz). " 184 Hz 2340 m/s + 40 m/s /. 3 The requencies o sound heard by the cyclist make sense. In (a), the train was moing toward the cyclist aster than he was moing away; they were moing closer and the sound heard by the cyclist was at a higher requency than that emitted by the train. In (b) ater the train had passed, they were moing apart and the requency o sound heard by the cyclist was less than that emitted by the train whistle. It is good to do qualitatie checks on your quantitatie answers to see i they make sense. Try it yoursel: What would the bicyclist hear i he were biking toward the on-coming train? Answer: 233 Hz. Measuring the speed o blood The Doppler eect is the basis o a technique used to measure the speed o blood low. High-requency sound waes ( < 20,000 Hz ) called ultrasound are directed into an artery, as shown in Fig The waes are relected by red blood cells back to a receier. The requency detected at the receier relatie to that emitted by the source R S indicates the cell s speed and the speed o the blood. (A similar arrangement is used to measure the speed o cars, but microwaes are used instead o ultrasound). To derie an equation or the requency shit, we consider the process in two parts: (1) The waes leae the source at requency S and strike the cell at requency '. (2) The waes relected rom the cell at requency where they are detected at requency. R ' return to the receier Figure Use Doppler eect to measure blood low speed The requency o the sound emitted rom the source by the equation ' striking a red blood cell is related to the requency S Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

55 ' * # 0 + " S, - & b 0 1 (20.18) where is the speed o sound in blood and b is the speed o the blood (a positie number as the blood is moing toward the source). The blood red cells relect this sound to a receier, which detects the sound. The cells act as a source at requency receier R is related to ' Substituting or by the equation b R " ', - '. The requency detected at the * # + 0 & 0 1 ' rom Eq. (20.18), we ind the requency detected at the receier R relatie to the requency emitted by the sound source : S * # 0 + * # b + # b R " S, -, -" S( ). (20.19) 0& b 10 # 0 1 & b This result allows us to measure the speed o blood using the Doppler eect. Example Buzzer moes in circle A riend with a ball attached to a string stands on the loor and swings it in a horizontal circle. The ball has a 400-Hz buzzer in it. When the ball moes toward you on one side, you measure a requency o 412 Hz. When the ball moes away on the other side o the circle, you obsere a 389 Hz requency. Determine the speed o the ball. Sketch and Translate Draw a sketch o the situation (Fig ). You detect a higher requency when the ball moes toward you and lower requency when it moes away, consistent with our understanding o the Doppler eect. Simpliy and Diagram The speed o sound in air is constant and equal to 340 m/s. The air is stationary with respect to the ground. Figure Using Doppler eect to ind speed during circular motion Represent Mathematically We can use Eq. (20.17) to ind an expression or the ratio o the requency when the source (the whistle) is moing toward you (the obserer) and when moing Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

56 away. This ratio can then be rearranged to get an expression or the speed o the source (the whistle): # 0 # " " # 0 S moing away rom O S # S S moing toward O S S where is the speed o sound in air and S is the speed o the source (the whistle).. Sole and Ealuate Rearranging the aboe, we get an expression that can be used to determine the source speed: * S moing toward O S moing away rom O S ", S moing toward O # - S moing away rom O To sole or the speed o the source S in terms o the obsered requencies (412 Hz when source is moing toward obserer and 389 Hz when moing away), we get: S * 412 Hz 389 Hz + ", -(340 m/s) = 9.8 m/s Hz Hz 1 The ball was moing at 9.8 m/s the correct unit and a reasonable magnitude. Try It Yoursel: What requency do you hear when the ball is moing perpendicular to your line o sight when in ront o you? Answer: 400 Hz. At that instant, the ball is not moing toward or away rom you it moes in the perpendicular direction no Doppler shit. S S, Note you can determine how ast the ball was moing in a circle by just comparing the sound requency when moing toward you and the requency when moing away. The same conceptual idea is used to determine the speed o a star moing in a circle in a galaxy (the quantitatie procedure is a little dierent as light is used instead o sound). The speed measurement can be used to determine the mass pulling inward on the star by other objects in the galaxy and is one source o a new mystery there seems to be an excess mass pulling in on the star. This mass is unseen and unaccounted or you will learn more later about this interesting dark matter problem. Reiew Question The siren on an ambulance sounds continuously as the ambulance irst approaches you and then moes away. How does the sound rom the siren change and why? Summary Word description Sketch and/or diagram Mathematical description Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter

57 Speed o wae depends on properties o the medium (or example, a orce like quantity F MonM (medium on medium) and a mass/length quantity 5 ). The waelength ' depends on requency and wae speed. Traeling Wae equation The traeling wae equation allows us to determine the displacement rom equilibrium y o a point in the medium at location x at time t when a wae o waelength ' traels at speed. Intensity I o a wae is the energy per unit area and time interal that crosses perpendicular to an area in a medium in which the wae traels. Intensity leel 8 "o a sound is the common log o the ratio o its intensity I and a reerence intensity I 0. Superposition principle When two or more waes pass through the same medium at the same time, the net displacement o any point in the medium is the sum o the displacements that would be caused by each wae i alone in the medium at that time. ' " T " " F MonM / 5 * t x+ y " A cos 2 $, - 0T ' 1 where % U P I " " % ta A * 8 10 log I + ", I I 0 " 10 J/m s. y " y # y y 1 2 # 3 #... Huygens principle Eery ibrating point o a medium produces a waelet with a circular ront surrounding it. The new wae ront o the wae is the superposition o all waelets due to ibrations on all points o preious wae ront. Standing waes on strings o length L on which waes at speed orm inphase ibrations o requencies n. The amplitudes o those ibrations are dierent or dierent points o the medium. Standing waes in open pipes o length L in which sound traels at speed are discrete large amplitude ibrations o requencies n. n " n( ) 2L or n " 1, 2, 3,... n " n( ) 2L or n " 1, 2, 3,... Etkina/Gentile/Van Heuelen Process Physics 1/e, Chapter