17 - E i 525 fiHAFTEH 1'." THE PRINCIPLE ClF...

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Unformatted text preview: E i 525 fiHAFTEH 1'." THE PRINCIPLE ClF LINEAR SUPERF‘DEITIDN AND INTERFERENCE PHENDMENA Section 11rd Longitudinal Standing Waves 14. a longitudinal standing wave is es- ls— o.eo m ——-l tablished in a tube that is open at both ends [see the drawing}. The length of the tube is dill} or. What is the wave— length of the waves that make up the standing wave'iI (nlflflflm (h) {140m {c} ilflil m {d} LEE! m {a} Lot] to Iii. A longitudinal standing wave is established in a tube open at only one end (see the drawing}. The frequency of the standing wave is fifil} Hz, and the speed of sound in air is 343 this. What is the length of the tobe‘?I (a) it. 1 3 m {b} are m (chitii? m {d} £3.52 or {e} {Ltd or . 3' Note to Instructors: Most of tire hemewdri: prohienes in this chapter are nvniinhiefor assignment via an oniine homework management progralf ;" as WileyPLLlS or Wehdssign. and those merited with the icon @ one presented in WileyPLUS using a guided rutorirttfomror that provides entranced interactivity See Preface for additional details. m Solution is in the Student Solutions Manual. “I Solution is available oniine at wunv.wiley.comtcollegetcutnell Section 111 The Principle of Linear Superposition, Section 13.1 Constructive and Destructive Interference of Sound Waves 1. In Figure l'i'fi', suppose that the separation between speakers A and E is 5130 rn and the speakers are vibrating in phase. Theyr are playing identical lES-Hs tones, and the speed of sound is 343 nits. What is the largest possible distance between speaker 13 and the ob- server at C. such that he observes destructive interference? 1. Two speakers, one directly behind the other, are each generating a Edfi-Ha sound wave. What is the smallest separation distance between the speakers that will produce destructive interference at a listener standing in front of them? The speed of sound is 343 rnts. 3. l'lil'l Concept Simulation 1?.1 at www.wiiey.eonttcollegetcutnell illustrates the concept that is pertinent to this problem. The drawing graphs a string on which twn pulses {half up and han down] are trav- eling at a constant speed of l crnr's at r = D s. Using the principle of linear superposition, draw the shape of the string’s pulses at t = l s, 1 s. 3 s. and 4 s. ll 4 E El 1+3 12 Distance, cm 1 cnu's 1 units 4. Two waves are traveling in opposite directions on the same string. The displacements caused by the individual waves are given by y, = [24.13 mm) sin(9.flfln't — [.25‘3'11‘} and y; = [3513 mmisinfflhflm + {1.4mm}. Note that the phase angles [9.00m — 1.35 stir] and (2.33m + {1.4% var} are in radians, t is in seconds, ands: is in meters. stir = dflfl s, what is the net displacement [in mm] of the string at [a] .r = 2. rs m and {b} .t = lid rn? Be sure to include the algebraic sign (+ or —,‘r with your answers. 5. m Two loudspeakers are vibrating in phase. They are set up as in Figure 11?, and point C is located as shown there. The speed of sound is 343 arts. The speakers play the same tone. What is the small- est frequency that will produce destructive interference at point E“? ti. @ Both drawings show the same square, each of which has a side of length L = i335 to. An observer CI is stationed at one corner of each square. 'l‘wo loudspeaka-s are located at corners of the square, as in either drawing 1 or drawing 2. The speakers produce the same single~frequency tone in either drawing and are in phase. The speed a This icon represents a biomedical application. of sound is 343 rats. Find the single smallest frequency that trill: duce both constructive interfereace in *drawing 1 an r =li interference in drawing 2. 'b. Drawing 1 Drawing 2 1'. corn m The drawing shows a loudspeaker A and um'f where a listener is positioned. A second loudspeaker B is .u-' somewhere to the right of A. Both speakers vibrate in phase and are playing a odd-Ha é? / tone. The speed of sound is '3" \Eflfl, ' 343 hits. What rs the closest to speaker A. that speaker B can _. be located, so that the listener 9* hears no sound'iI it. Suppose that the two speakers in Figure 13.3 are r: lit] at and are vibrating eaactly out of phase at a freq 42? Ha. The speed of sound is 343 this. Does the observer at; scrve consuuctive or destructive interference when his dis : u - speaker E is {a} 1.15 m and {b} Elli] m? *9. The two speakers in the drawing are vibrating in phase, tenet is standing at point P. Does constructive or destructive u=-,j_ ence occur at P when the speakers produce sound waves frequency is {a} i4tio Ha and [b] 9?? Ha? Justify your mg, with appropriate calculations. Take the speed of sound to be a listener is standing in front of two speakers that are proe ' us g sound of the same frequency and amplitude, except that they "yibrating out of phase. Initially, the distance between the listener speaker is the same {see the drawing]. its the listener moves the sound intensity gradually changes. wtten the distance fill drawing is [1.92 m, the change reaches the maximum amount 'Inine loud to soft, or soft to lend]. Using the data shown in the draw- '- 343 rats for tlte speed of sound. determine the frequency of _,-:-t I t coming from the speakers, 4.03 m Dut—of—phase speakers it - a . A and E are vibrating in phase. They are directly fac— E';-- bother, are 7.30 m apart, and are each playing a T3.fl-I-Ix tone. speed of sound is 343 tests. en the line between the speakers _;I} ate tltree points where constructive interference occurs. What distances of these three points from speaker tit? . its unit-seams issued exits a diffraction horn loudspeaker through a rectangular Tun: like a small doorway. Such a loudspeaker is mounted outside w . In winter, when the temperature is 273 K, the diffraction files a value of lift“. What is the diffraction angle fer the same on a summer day when the temperature is 3] l K? Consult Multiple-Concept Example 3 for background to this problem. A speaker has a diarneter of [130 m. ' Ilse-m g that the speed of sound is 343 mfs, find the diffraction __'Efor a aerate tene. tat What speaker attesetes it should be "it generate a fill-kHz tone whose diffraction angle is as wide as ate art—arts tone in part {a}? The following two lists give the diameters and sound fre- for three loudspeakers. Fair each diameter with a frequency, H the diffraction angle is the same for each of the speakers, and can use cotmnon difiraction angle. Take the speed of sound to rat‘s. Diameter, fl Frequencysf H.050 m til] kHz '3. 1 I3 m 41] kHz llifi m Ill} kHz _tip1e~Certeept Example 3 reviews the concepts that are im- -';-: in this problem. The entrance to a large lecture room consists i"; side-by-side doors, one hinged on the left and the other 1;: es the right. Each trees is once or wide. sense of frequency is coming through the entrance from within the room. The sound is 343 rats. What is the diffraction angle s of the Is it passes through the doorway when {a} one deer is {bl both doors ete open? FHDELEME 529 Iti. @ Fer one approach to problems such as this, see Multiple- Concept Example 3. Sound emerges through a doorway, as in Figure 11H]. The width of the doorway is T? cm, and the speed of sound is 343 mfs. Find the diffraction angle .9 when the frequency of the sound is (3)10 kHz and {h} ill at Ill2 Hz. * 1?. A. 3.00-kHz tone is being produced by a speaker with a diametm of [11?5 m. The air tefitpm‘uture changes from i} to 29 T. Assuming air to be an ideal gas, find the change in the diffraction angle 5'. * 13. A row of seats is parallel to a stage at a distance of 3.? m from it. At the center and front of the stage is a diffraction hem loud- speaker. This speaker sends out its sound through an opening that is like a small doorway with a width D of "3.5 cm. The speaker is play- ing a tone that has a frequency of 1.13 ht“ If!“ He. The speed of sound is 343 mr's. What is the distance between two seats, located near the center of the row, at which the tone cannot be heard? Section 1?.4 Beats 19. earn Two pure tones are sounded together. The drawing shows the pressure variations of the two sound waves, measured with re- spect to atrnospheric pressure. What is the beat frequency? {3.1320 a Pressure Pressure [1.024 a 2i]. Two pianos each sound the same note simultaneously, but they are both out of tune. Do a day when the speed of sound is 343 refs, piano A produces a wavelength of 3359 to, while piano I3 produces a wavelength of {1136 m. How much time separates successive heats? 21. Two outsof—tune flutes play the same note. Due produces a tone that has a frequency of 262 He, while the other produces can He. When a tuning fork is sounded together with the 262~Hz tone, a beat frequency of 1 He is produced. When the same tuning fork is sounded together with the flee-Ha tone. a beat frequency of 3 He is produced. What is the frequency of the tuning fork? 22. team When a guitar string is sounded along with a Mil-Hz tun- ing fork, a beat frequency of 5 He is heard. When the same string is sounded along with a 433R: tuning fork, the beat frequency is 9 Hz. What is the frequency of the string? 13. In Concept Simulation 112 at www.wlley.eomfeollegefcutnell you can explore the-concepts that are imphrtattt in this problem. A 44llfl-Ha tuning fork is sounded together with an out-of-tune guitar string, and a beat frequency of 3 Hz is heard. When the string is tight- ened, the frequency at which it vibrates increases, and the beat fre- quency is heard to decrease. What was the original frequency of the guitar suing? “‘24s @ Two cars have identical horns. each emitting a frequency of f, = 395 He. flue of the cars is moving with a speed of 1233.I mils toward a bystander waiting at a corner, and the other car is parked. The speed of sound is 343 nu's. What is the beat frequency heard by the hystander? *25. ® at. sound wave is traveling in seawater, where the adiabatic bulk modulus and density are 2.3] at 1ft“ Pa and 1:325 kgfmj, respec- tively. The wavelength of use sound is 3.35 m. A tuning fork is struck under water and vibrates at 4401} Hz. What would be the beat fre- quency heard by an underwater swinuuer? H“ 215. Two carpenters are hammering at the same time, each at a differ- ent. hammering frequency. The hammering frequency is the number 53ft] fiHAFTEH 11' THE PRINCIPLE DF LINEAR SUPERFDSITIDN AND INTERFERENCE PHENGMENA ef hammer blews per secend. Every 4.15 s. beth carpenters strike at the same instant. producing an effect very similar te a beat frequency. The first carpenter strikes a blew every 11.25 s. Hew many secends elapse between the secend carpenter’s blews if the secend carpenter hammers ta} mere rapidly than the first carpenter. and {b} less rapidly than the first carpenter? Seetinn 12.5 Transverse Standing Waves 2?. 1f the string in Figure 12.15e is vibrating at a frequency ef 4.11 1-1: and the distance between twe successive nedes is (1.313- m. what is the specd cf the waves en the string“?I .23. A string is fixed at betit ends and is vibrating at 131] He. which is its third harmenic frequency. The linear density ef the string is 5.15 it Ill"3 ltgfm. and it is under a tensien ef 3.3 N. Determine the length cf the string. 29. fill! The appreach te selving this preblcrn is similar te that taken in Multiple-Cenccpt Example 4. [in a celle. the string with the largest linear density (Lift its: 113'“2 kgt'm} is the C string. This string preduees a fundamental frequency ef fi5.4 He and has a length ef ilhitfl m between the twe fixed ends. Find the tensien in the string. 31]. @ Twn wires. each ef length 1.2 m. are stretched between twe fixed supperts. fin wire A there-is a secenduharmenic standing wave whese frequency is net] Ha. Hewever. the same frequency ef titifl He is the third harrnenic en wire E. Find the speed at which the individ- ual waves travel en each wire. 31. sent The A string en a string bass vibrates at a fundamental frequency ef 551i 11:. 1f the string's tensien were increased by a faeter ef feur. what weuld be the new fundamental frequency? 32. Multiple-Cencept Example 4 deais with the same cencepts as this preblem. A 41-cm length ef wire has a mass ef Iii] g. it is stretched between twe fixed supperts and is under a tensien cf let} hi. 1|What is the fundamental frequency ef this wirc'.JI 33. A string has a linear density ef 3.5 it ltji'iL ltgtm and is under a ten. sien ef 2311 H. The string is 1.3 in lung. is fixed at beth ends. and is vibrating in the standing wave pat- tern shewn in the drawing. Determine the {a} speed. {1}} wave- length. and (cl frequency ef the traveling waves that make up the standing wave. 34. Te review the cencepts that play rules in this preblem. ccnsuit Multiple-Cencept Example 4. Semetimes. when the wind blews acress a leng wire. a law-frequency “meaning” seund is preduced. This seund arises because a standing wave is set up en the wire. like a standing wave en a guitar string. Assume that a wire {linear density = fl.l314fl ltgt‘m) sustains a tensien [if 323 N because the wire is stretched between twe peles that are 7".6I} tn apart. The iewcst frequency that an average. healthy human ear can detect is 2t}.tl He. 1|tiit'hat is the lewest harmcnic num- ber n that ceuld be respensible fer the “meaning” settnd’? Ies. @ a. cepper black is sus— pended frern a wire. as in part 1 cf the drawing. A centaincr cf nter— cury is then raised up arcund the Part 2 Frttbfcm 35 Part. 1 blecl-t. as in part 2. se that sees: ef the bleclt's velume is It;- in the mercury. The density ef cepper is 32th] kgt'mi“. and that :_.'-;' cury is 13 fiflll kgi‘mf. Find the ratie cf the fundamental f H -I'1'_. the wire in part 2 in the fundamental frequency in part 1. *3d. Twe strings have dif- ferent lengths and linear densities. as the drawing shews. They are jeined te- gethcr and stretched se that the tensien in each string is 19111] bl. The free ends ef the jeined string are fixed in place. Find the lewest frequency that permits '.'j.'; waves in beth strings with a nede at the junctien. The stan' .;.-. pattern in each string may have a different number cf leaps. I 1‘32. we The E string en an electric bass guitar has a length at and. When producing the mic E. vibrates at a fundamental freq 4| .2 Ha. Players semetimes add in their instruments a device is." “D~tuner.” This device allews the E string te be used tn produce D. which has a fundamental frequency ef 3b.? He. The 1;; - by extending the length ef the string. keeping all ether factetsthe Ey hew mach dees a D—tuner extend the length ef the E string? ' *3tl. ® Standing waves are set up en twe strings fixed at mi as shewu in the drawing. The twe strings have the same te mass per unit length. but they differ in length by [1.52 cm. en the sherter string prepagate with a speed ef41.t3 rats. and Hi" damental frequency ef the sherter string is 225 He. Dete uni-.1 beat frequency predeced by the twe standing waves. 4-. 3.?5 rn 5.01:! x ttt‘E tigtrn Lana ti “39. item we! The arrangement in the drawing shews 3.- tniass = lit} kg) that is held in ' pesitien en a frictienicss incline by a cerd [length = [1.511] m]. The mass per unit length ef the cerd is |.2fl ht lil—2 l-tgt'm. se the mass ef the cerd is negligible ccntpared tn the mass ef the bleck. The cerd is being vibrated at a frequency cf [63 Ha {vibratien seurce net . F t: the drawing}. What are the values ef the angle if between i 1:5 913.11“ at which a standing wave exists en the cerd? inf41]. Review Cehceptual Example 5 befere attempting this n fits the drawing shews. the length ef a guitar string is [1.62% L; ' frets are numbered fer cenvenience. .4. perferrner can play a "1'; scale en a singie string because the spacing frenveen the i signed accerding te the fellewing rule: When the string is against any fret j. the fundamental frequency pat; the shertenetl-.__ is larger by a facter ef the twelfth reet ef twe t T2] than itis t__:;. string is pushed against the fret j - 1. Assuming that the! the string is the same fer any nete. find the spacing {it} been I and fret I} and tb} between fret 2' and fret ti. {3.623 M 213543211] its Longitudinal Standing Waves, In] on campus Sound Waves T_ a tube of air is open at only one end and has a length of _'-’Ihis tube sustains a standing wave at its third harmonic. What distance between one node and the adjacent antinode? Sound enters the ear, travels through the auditory canal, and I reaches the eardrum. The auditory canal is approximately a '5': at only one end. “the other end is closed by the eardrum. A length for the auditory canal in an adult is about 2.9 cm. The lfluf sound is 343 n'tt's. What is the fundamental frequency of the (Interestingly, the fundamental frequency is in the frequency human hearing is most sensitive.) _' organ pipe is open at both ends. It is producing sound at its _Q__tiarmonic, the frequency of which is 262 He. The speed of -"I 343 rt'tl's. 1What is the length of the pipe? One method for measuring the speed of sound uses standing a cylindrical tube is open at both ends, and one end admits Thorn a tuning fork. A movable plunger is inserted into the :3, at a distance l. from the end of the tube where the tuning For a fitted frequency, the plunger is moved until the smallest l. is measured that allows a standing wave to he formed. :.'[;.=-:-- that the tuning fort-t produces a 4Efi—Hs tone. and that the value observed for L is b.2t34 m. What is the speed of sound - in the tube? l3? - to Interactive Solution 1145 at wttltrvttwiley,contl'cclings.tr :— to review a method by which this problem can be solved. The 'iiu a : frequencies of two air columns are the same. Column a 'i: at both ends. while column B is open at only one end. The column A is 0.?0 to. What is the length of column B? Divers working in underwater chambers at great depths . must deal with the danger of nitrogen narcosis [the in which nitrogen dissolves into the blood at toxic levels. - to avoid this danger is for divers to breathe a mixture con- ; : only helium and oxygen. Helium, however, has the effect of 33;; voice a high-pitched quality, like that of Donald Buck's 'flh see why this occurs, assume for simplicity that the voice is ';_:-n=+- by the vocal cords vibrating above a gas—filled cylindrical ill.- is open only at one end. The quality of the voice depends on '.. n. in frequencies generated by me tube; larger frequencies "higher-pitched voices. Consider two such tubes at 2D 1UPl3. Doe I Ifllltll. PHDBLEMS - fundamental frequency of a string fixed at both ends is 2515 Hz. flu; does it take for a wave to travel tlte length of this string? range of human hearing is roughly from twenty beds to .hilchertz. Based on these limits and a value of 343 mfs for the El: sound, what are the lengths of the longest and shortest pipes both ends and producing sound at their fundamental fre- _'-}'i:- that you expect to find in a pipe organ? l'teyiew Example I in the text. Speaker at. is moved further ' while ABC remains a right triangle. What is the separation 3-1»: the speakers when constructive interference occurs again at ADDITIDHAL PRDELEME 531 is filled with air, in which the speed ofsound is 343 mix. The other is filled with helium, in which the speed of sound is Hill a: HT" nus. To see the effect of helium on voice quality, calculate the ratio of the nth natural fretiuency of the helium-filled tube to the nth natural fre- quency of the air-filled tube. 4?. com The fundamental frequency of a vibrating system is 4llfl Ha. For each of the following systems, give the three lowest frequencies {excluding the fundamental} at which standing waves can occur: {a} a string fixed at both ends, [h] a cylindrical pipe with both ends open, and [c] a cylindrical pipe with only one end open. * 4d. a thin 1.3m aluminum rod sustains a longitudinal standing wave with vibration antinodes at each end of the rod. There are no other antinodes. The density and Young‘s modulus of alu minum are, respec- tively, 2TH] kgfmi" and ti? it“ It}m Hfmi. What is the frequency of the rod‘s vibration? * 49. Review Multiple—Concept Example “I for background that is rele- vant to the kind of approach needed to solve this pmblem. Two ideal gases have the same temperature and the same value for y {the ratio of the specific heat capacities at constant pressure and constant volume]. A molecule of gas A. has a mass of 131 K Ill”? kg. and a molecule of gas 13 has a mass of Lon it to”? ltg. When gas A (speed of sound = 259 mils} fills a tuhe that is open at both ends, the first overtone fre— quency of the'tube is ass Ha. Gas E fills another tube open at both ends, and this tube also has a first overtone frbquency of 3315 He. What is the length of the tube filled with gas E? ‘50. @ com m A vertical tube is closed at one end and open to air at the other end. The air pressure is tall] it ill-3 Pa. The tube has a length of {L'lfi m. Mercury [mass density = [3 tillo kgfmi'] is poured into it to shorten the effective length for standing waves. What is the absolute pressure at the bottom of the mercury column, when the fun- damental frequency of the shortened, air-filled tube is equal to the third harmonic of the original tube? *51. A person hums into the top of a well and finds that standing waves are established at frequencies of 42. Till}. and '33 He. The fre- quency of 42 He is not necessarily the fundamental frequency. The speed of sound is 343 mfs. How deep is the well? “'52. a tube, open at only one end, is cut into two shorter [nonequall lengths. The piece that is open at both ends has a fundamental fre- quency of 425 He, while the piece open only at one end has a funda- mental frequency nf fi'i'fi Ha. 1|tilt'hat is the fundamental frequency of the original tube? as. A pipe open only at one end has a fundamental frequency of use He. Fl. second pipe, initially identical to the first pipe, is short- ened by cutting offa portion of the open end. Now, when botlt pipes vibrate at their fundamental frequencies, a beat frequency of 12 He is heard. How many centimeters were cut off the end of the second pipe? The speed of sound is 343 this. 5?. at. tube is open only at one end. A certain harmonic produced by the tube has a frequency of am He. The next higher harmonic has a frequency of ran He. The speed of sound in air is 343 mts. {a} What is the integer n that describes the harmonic whose frequency is 45b Ha? {bl What is the length of the tube? ...
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17 - E i 525 fiHAFTEH 1'." THE PRINCIPLE ClF...

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