PH 221-3A Fall 2010 Waves - II Lectures 27-28 Chapter 17 (Halliday/Resnick/Walker, Fundaentals of Physics 8 th edition) 1
Chapter 17 Waves II In this chapter we will study sound waves and concentrate on the following topics: Speed of sound waves Relation between displaceent and pressure aplitude Interference of sound waves Sound intensity and sound level Beats The Doppler effect 2
Sound waves are echanical longitudinal waves that propagate in solids liquids and gases. Si Seisic i waves used dby oil explorers propagate in the earth s crust. Sound waves generated by a sonar syste propagate in the sea. An orchestra creates sound waves that propagate in the air. The locus of the points of a sound wave that has the sae displaceent is called a wavefront. Lines perpendicular to the wavefronts are called rays and they point along the direction which the sound wave propagates. An exaple of a point source of sound waves is given in the figure. We assue that the surrounding ediu is isotropic i.e. sound propagates with the sae speed for all directions. In this case the sound wave spreads outwards uniforly and the wavefronts are spheres centered e ed at the point source. The single arrows indicate the rays. The double arrows indicate the otion of the olecules of the ediu in 3 which sound propagates.
v Δp B Bulk odulus If we apply an overpressure p on an object of volue V, this results in a change of volue V as shown in the figure. The bulk odulus of the copressed aterial p is defined as: B SI unit: the Pascal V / V Note : The negative sign denotes the decrease in volue when p is positive. The speed of sound Using the above definition of the bulk odulus and cobining it with Newton's second law one can show that the speed of sound in a hoogeneous isotropic ediu with bulk odulus B and density is given by the equation: v B pv Note 1 : V Bulk odulus is saller for ore copressible B edia. Such edia exhibit lower speed of sound. Note 2 : Denser aterials (higher ) have lower speed of sound 4
The Speed of Sound 5
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Exaple: Thunder, lightning and a rule of thub There is a rule of thub for estiating how far away a thunderstor is. You can estiate your distance fro a bolt of lightning i by counting the seconds between seeing the flash and hearing the thunder and then dividing by 3 to obtain the distance in k. Why does this rule work? 7
The Speed of Sound Proble : Both krypton (Kr) and neon (Ne) can be approxiated at onatoic ideal gases. The atoic ass of krypton is 83.8 u, while that of neon is 20.2 u. A loudspeaker produces a sound whose wavelength in krypton is 1.25. If the loudspeaker were used to produce sound of the sae frequency in neon at the sae teperature, what would be the wavelength? 8
Traveling sound waves. Consider the tube filled with air shown in the figure. We generate a haronic sound wave traveling to the right along the axis of the tube. One siple ethod is to place a speaker at the left end of the tube and drive it at a particular frequency. Consider an air eleent of thickness x which is located at position x before the sound wave is generated. This is known as the "equlibriu position" of the eleent. Under these conditions the pressure inside the tube is constant In the presence of the sound wave the eleent oscillates about the equlibriu position. At the sae tie the pressure at the location of the eleent oscillates about its static value. The sound wave in the tube can be described using one of two paraeters: 9
p v s Traveling sound waves. One such paraeter is the distance s x, t of the eleent fro its equilibriu i position i, cos s x t s kx t. The constant s is the displaceent aplitude of the wave. The angular wavenuber k and the angular frequency have the sae eaning as in the case of the transverse waves studied tdidin chapter 16. The second possibility is to use the pressure variation p fro the static value. p x, t p sinkxt The constant p is the wave's pressure aplitude. The two aplitudes are connected by the equation: p v s Note : The displaceent and the pressure variation have a phase difference of 90. As a result when one paraeter has a axiu the other has a iniu and vice versa. 10
2 L Interference Consider two point sources of sound waves S 1 and S2 shown in the figure. The two sources are in phase and eit sound waves of the sae frequency. Waves fro both sources arrive at point P whose distance fro S 1 and S 2 is L1 and L2 respectively. The two waves interfere at point P. At tie t the phase of sound wave 1 arriving fro S at point P is kl t 1 1 1 At tie t the phase of sound wave 2 arriving fro S at point P is kl t In general the two waves at P have a phase difference 2 2 2 2 2 1 kl2 t kl1t k L2 L1 L2 L1 The quantity L L is known as the " path hlength hdifference" L 2 1 2 between the two waves. Thus L Here is the wavelength of the two waves. 11
Constructive intereference. The wave at P resulting fro the interference of the two waves that t arrive fro S and ds has a axiu 1 2 aplitude when the phase difference 2 2 012 0,1, 2,.... L 2 L L 0,, 2,... Destructive intereference. The wave at P resulting fro the interference of the two waves that arrive fro S and S has a iniiu aplitude when the phase difference 1 2 2 21 0,1, 2,.... L 21 1 L L /2, 3 /2, 5 /2,... 2 Δ L equal to an integral ultiple of λ constructive L equal to a half-integral ultiple of λ interference Δ destructive interference 12
At an open-air concert on a hot day (T c = 25 C, V s = 346.5 /s), a person sits at a location that 7.0 and 9.1 respectively fro speakers at each side of the stage. A usician, waring up, plays a single 494 Hz tone. What does the spectator hear? 13
Constructive and Destructive Interference of Sound Waves Assue that two loudspeakers in the figure are vibrating out of phase instead of in phase. (see exaple 2 fro the text) The speed of sound is 343 /s. What is the sallest frequency that will produce destructive interference at point C? 14
Constructive and Destructive Interference of Sound Waves Speakers A and B are vibrating in phase. They are directly facing each other, are 7.80 apart, and are each playing a 73.0 Hz tone. The speed of sound is 343 /s. On the line between the speakers there are three points where constructive interference occurs. What are the distances of these three points fro speaker A? 15
Intensity of a sound wave Consider a wave that is incident norally on a surface of area A. The wave transports energy. As a result power P (energy per unit tie) passes through A. We define the wave i ntensity I as the rat i o P/ A I P A SI units: W/ The intensity of a haronic wave with displaceent aplitude 2 v 2 1 2. In ters of the pressure aplitude 2 s is given by: I s I p 2 2v v Consider a point source S eitting a power P in the for of sound waves of a particular frequency. The surrounding ediu is isotropic so the waves spread uniforly. The corresponding wavefronts are spheres that have S as P their center. The sound intensity at a distance r fro S is: I 2 4 r 1 The intensity of a sound wave for a point sources is proportional to r 2 16
The decibel The auditory sensation in huans is proportional to the logarith of the sound intensity I. This allows the ear to percieve a wide range of sound intensities. The threshold of hearing I is defined as the lowest sound intensity that can be detected by the huan ear. I 10 W/ o 0 12 2 The sound level is defined in such a way as to iic the response I of the huan ear. 10log is expressed in decibels (db) Io We can invert the equation above and express I in ters of as: I I o Note 1 : 10 /10 For I I we have: 10log1 0 o Note2 : increases by 10 decibels every tie I increases by a factor of 10 4 For exaple 40 db corresponds to I 10 I o 17
Exaple 1: Express the threshold of hearing (2.5 x 10-12 W/ 2 ) and the threshold of pain (1 W/ 2 ) in decibels. Exaple 2: At a distance of 60 fro a jet airline the intensity is 1 W/ 2. I 180 =? 18
The intensity of sound near a loud rock band is 120dB. What is the intensity of sound near two such rock bands playing together? 19
Proble (Decibels): Two identical rifles are shot at the sae tie and the sound intensity level is 800dB. What would be the sound intensity level if only one rifle were shot? (hint: the answer is not 400db) 20
Proble : Two sound sources each eit sound power uniforly in all directions. There are no reflections. Both sources are located on the x axis, one at the origin and the other at x = +123. The source of the origin eits four ties ore power than the other source. Where on the x axis is the intensity of each sound equal? Note there are 2 answers. 21
Sound standing waves in pipes Consider a pipe filled with air that is open at both ends. Sound waves that have walengths that satisfy a particular relation with the length L of the pipe setup standing waves that have sustained aplitudes. The siplest pattern can be set up in a pipe that is open at both ends as shown in fig.a. In such a pipe standing waves have a antinode (axiu) in the dispaceent aplitude The aplitude of the standing wave is plotted as function of distance in fig.b. The pattern has an node at the pipe center since two adjacent antinodes are separated by a node (iniu). The distance between two adjacent antinodes is /2. v v Thus L /2 2 L Its frequency f 2L The standing wave of fig.b is known as the " fundaental ode" or " first haronic" of the tube. Note : Antinodes in the displaceent aplitude correspond to nodes in the pressure aplitude. This is because s and p are 90 out of phase. 22
2L n n Standing waves in tubes open at both ends The next three standing wave patterns are 2 shown in fig.a. The wavelength where known as the Standing waves in tubes open at one end and closed at the other The first four standing wave patterns are shown in fig.a. They have an antinode at the open end and a node at the closed end. 4L L The wavelengt h n n n 1, 2, 3,... The integer n is haronic nuber The corresponding frequencies 4L n n f n n L n nv 2L L for n=1,3,5,7,... 23
Longitudinal Standing Waves Proble A person hus into the top of a well and finds that standing waves are established at frequencies of 42, 70.0 and 98 Hz. The frequency of 42 Hz is not necessarily the fundaental frequency. The speed of sound is 343 /s. How deep is the well? 24
Beats. If we listen to two sound waves of equal aplitude and frequencies 1 2 1 2 1 2 f and f f f and f f we perceive the as a sound of frequency f1 f2 fav. in addition we also perceive "beats" which are variations in the 2 intensity of the sound with frequency f f f. The displaceents of the beat 1 2 two sound waves are given by the equations: s s cos t, and s s cos t. 1 1 2 2 These are plotted in fig.a and fig.b. Using the principle of superposition we can deterine the resultant displaceent as: 1 2 1 2 s s1s2 scos1tcos2t2scos t cos t 2 2 1 2 1 2 s 2 s costcos t where and 2 2 Since 25 1 2
T beat f f f beat 1 2 T' 1 2 1 2 s 2 s cost cos t where and 2 2 The displaceent s is plotted as function of tie in the figure. We can regard it as a cosine function whose aplitude is equal to 2 s cos t. The aplitude is tie dependent but varies slowly with tie. The aplitude exhibits a axiu whenever cos t is equal to either +1 or -1 which happens twice within one period of the cos t function. Thus the angual frequency of the beats 2 2 2 The frequency of the beats f 2 2 2 f f beat beat 1 2 beat 12 1 2 1 2 26
The Doppler effect Consider the source and the detector of sound waves shown in the figure. We assue that the frequency of the source is equal to f. We take as the reference frae that surrounding air through which the sound waves propagate. If there is relative otion between the source and the detector then the detector perceives the frequency of the sound as f f. If the source or the detector ove towards to each other f f. if on the other hand the source or the detector ove away fro each other f f. This is known as the " Doppler " v vd effect. The frequecy f is given by the equation: f f. Here vs and v v vs are the speeds of the source and detector with respect to air, respsctively. When the otion of the detector or source is towards each other the sign of the speed ust give an upward shift in frequency. If on the other hand the otion is away fro each other the sign of the speed ust give a downward shift in frequency. The four possible cobinantions are illustrated in the next page. D 27
v S v D D v v f f f f v v S v S v D v v f f D f f v v S v S v D v v f f D v v v v f f v v v S v D D S v v f f v v 28 S D S
A train approaching a siren. The train encounters ore wave fronts per unit tie than when stationary. f = f + additional # of condensations additional # of condensations in a tie t= (V r t)/λ The Doppler Effect A train receding fro the siren. The train encounters fewer wave fronts per unit tie than when stationary. in 1 sec additional # = V r /λ -> f = f + V r /λ = f(1 + V r /fλ) = f(1 + V r /V) A receiver on the train will detect a higher frequency when approaching a siren, and a lower frequency when receding. 29
The source (train) is in otion, the receiver is stationary A stationary receiver will detect a higher frequency when it is front of the train and a lower frequency when behind the train. The wavelength ahead of the train is shorter λ = λ V E /f and behind the train is longer λ = λ + V E /f when the train is stationary. 30
Proble: The trucks travel at the sae speed. They are far apart on adjacent lanes and approach each other essentially head-on. One driver hears the horn of the other truck at a frequency that is 1.20 ties the frequency he hears when the truck is stationary. The speed of sound is 343 /s. At what speed dis each htruck oving? 31