Sound Waves. Answer (c). Every crest in air produces one crest in water immediately as it reaches the interface, so there must be 500 in every second.

Transcription

1 7 ound Wae CHAPTER OUTLNE 7. peed of ound Wae 7. Periodic ound Wae 7.3 ntenity of Periodic ound Wae 7.4 The Doppler Effect 7.5 Digital ound Recording 7.6 Motion Picture ound *Q7. ANWER TO QUETON Anwer (b). The typically higher denity would by itelf make the peed of ound lower in a olid compared to a ga. Q7. We aume that a perfect acuum urround the clock. The ound wae require a medium for them to trael to your ear. The hammer on the alarm will trike the bell, and the ibration will pread a ound wae through the body of the clock. f a bone of your kull were in contact with the clock, you would hear the bell. Howeer, in the abence of a urrounding medium like air or water, no ound can be radiated away. A larger-cale example of the ame effect: Coloal torm raging on the un are deathly till for u. What happen to the ound energy within the clock? Here i the anwer: A the ound wae trael through the teel and platic, traering joint and going around corner, it energy i conerted into additional internal energy, raiing the temperature of the material. After the ound ha died away, the clock will glow ery lightly brighter in the infrared portion of the electromagnetic pectrum. Q7.3 f an object i meter from the onic ranger, then the enor would hae to meaure how long it would take for a ound pule to trael one meter. ince ound of any frequency moe at about 343 m, then the onic ranger would hae to be able to meaure a time difference of under 3 econd. Thi mall time meaurement i poible with modern electronic. But it would be more expenie to outfit onic ranger with the more enitie equipment than it i to print do not ue to meaure ditance le than meter in the uer manual. Q7.4 The peed of ound to two ignificant figure i 34 m. Let aume that you can meaure time to econd by uing a topwatch. To get a peed to two ignificant figure, you need to meaure a time of at leat. econd. ince d t, the minimum ditance i 34 meter. *Q7.5 (i) Anwer (b). The frequency increae by a factor of becaue the wae peed, which i dependent only on the medium through which the wae trael, remain contant. (ii) *Q7.6 (i) (ii) Anwer (c). Anwer (c). Eery cret in air produce one cret in water immediately a it reache the interface, o there mut be 5 in eery econd. Anwer (a). The peed increae greatly o the waelength mut increae _7_ch7_p indd 449 /3/7 8:8: PM

2 45 Chapter 7 Q7.7 When litening, you are approximately the ame ditance from all of the member of the group. f different frequencie traeled at different peed, then you might hear the higher pitched frequencie before you heard the lower one produced at the ame time. Although it might be intereting to think that each litener heard hi or her own peronal performance depending on where they were eated, a time lag like thi could make a Beethoen onata ound a if it were written by Charle e. *Q7.8 Anwer (a). We uppoe that a point ource ha no tructure, and radiate ound equally in all direction (iotropically). The ound waefront are expanding phere, o the area oer which the ound energy pread increae according to A 4π r. Thu, if the ditance i tripled, the area increae by a factor of nine, and the new intenity will be one-ninth of the old intenity. Thi anwer according to the inere-quare law applie if the medium i uniform and unbounded. For contrat, uppoe that the ound i confined to moe in a horizontal layer. (Thermal tratification in an ocean can hae thi effect on onar ping. ) Then the area oer which the ound energy i dipered will only increae according to the circumference of an expanding circle: A π rh, and o three time the ditance will reult in one third the intenity. n the cae of an entirely encloed peaking tube (uch a a hip telephone), the area perpendicular to the energy flow tay the ame, and increaing the ditance will not change the intenity appreciably. *Q7.9 Anwer (d). The drop in intenity i what we hould expect according to the inere-quare law: 4π r P and 4π r P hould agree. (3 m) ( μw m ) and (95 m) ( μw m ) are 8 W and 8 W, agreeing with each other. *Q7. Anwer (c). Normal coneration ha an intenity leel of about 6 db. *Q7. Anwer (c). The intenity i about 3 W m. Q7. Our brae iberian aw the firt wae he encountered, light traeling at 3. 8 m. At the ame moment, infrared a well a iible light began warming hi kin, but ome time wa required to raie the temperature of the outer kin layer before he noticed it. The meteor produced compreional wae in the air and in the ground. The wae in the ground, which can be called either ound or a eimic wae, traeled much fater than the wae in air, ince the ground i much tiffer againt compreion. Our witne receied it next and noticed it a a little earthquake. He wa no doubt unable to ditinguih the P and wae from each other. The firt air-compreion wae he receied wa a hock wae with an amplitude on the order of meter. t tranported him off hi doortep. Then he could hear ome additional direct ound, reflected ound, and perhap the ound of the falling tree. Q7.3 A you moe toward the canyon wall, the echo of your car horn would be hifted up in frequency; a you moe away, the echo would be hifted down in frequency. *Q7.4 n f ( + o )f ( ) we can conider the ize of the fraction ( + o ) ( ) in each cae. The poitie direction i defined to run from the oberer toward the ource. n (a), n (b), 34 (34 5).8 n (c), 34 (34 + 5) 93 n (d), (34 + 5) 34.7 n (e), (34 5) n (f ), (34 + 5) (34 + 5) n (g), (34 5) (34 5). n order of decreaing ize we hae b > d > a f g > c > e. 3794_7_ch7_p indd 45 /5/7 6:33: PM

3 ound Wae 45 *Q7.5 (i) (ii) Anwer (c). Both oberer and ource hae equal peed in oppoite direction relatie to the medium, o in f ( + o )f ( ) we would hae omething like (34 5)f (34 5) f. Anwer (a). The peed of the medium add to the peed of ound a far a the oberer i concerned, to caue an increae in λ f. (iii) Anwer (a). Q7.6 For the ound from a ource not to hift in frequency, the radial elocity of the ource relatie to the oberer mut be zero; that i, the ource mut not be moing toward or away from the oberer. The ource can be moing in a plane perpendicular to the line between it and the oberer. Other poibilitie: The ource and oberer might both hae zero elocity. They might hae equal elocitie relatie to the medium. The ource might be moing around the oberer on a phere of contant radiu. Een if the ource peed up on the phere, low down, or top, the frequency heard will be equal to the frequency emitted by the ource. OLUTON TO PROBLEM ection 7. peed of ound Wae *P7. ince >> we hae d ( 343 m)( 6. ) km light ound We do not need to know the alue of the peed of light. A long a it i ery large, the trael time for the light i negligible compared to that for the ound. B P7. ρ km *P7.3 The ound pule mut trael 5 m before reflection and 5 m after reflection. We hae d t d 3 m t 533 m 96 P7.4 (a) At 9 m, ΔT 9 (. C 5 ) 6. C o T 3 C Uing the chain rule: d d dt dx d dt ( ) 67 dt dt dx dt dt dx 5 47, o dt d ( 47 ) t dt ( 47 ) f i d f t ( 47 ) ln i ( 3 ) ( 47 ) ln ( 3 ) t 7. for ound to reach ground. (b) h t 9 [ ( 3 ) ] 5. 7 t take longer when the air cool off than if it were at a uniform temperature. 3794_7_ch7_p indd 45 /3/7 8:8:3 PM

4 45 Chapter 7 P7.5 ound take thi time to reach the man: ( m. 75 m) m o the warning hould be houted no later than before the pot trike. ince the whole time of fall i gien by y gt : m (. m )t t 93. the warning need to come into the fall, when the pot ha fallen ( 9. 8 m)(. 58 ). m to be aboe the ground by. m. m 78. m P7.6 t i eaiet to ole part (b) firt: (b) The ditance the ound trael to the plane i d h The ound trael thi ditance in., o + h h 5 h 5 d ( 343 m)(. ) 686 m giing the altitude of the plane a h ( 686 m ) 64 m 5 h (a) The ditance the plane ha traeled in. i (. ) 37 m 37 m Thu, the peed of the plane i: 53 m. P7.7 Let x repreent the cowboy ditance from the nearer canyon wall and x hi ditance from the farther cliff. The ound for the firt echo trael ditance x. For the econd, x. For the third, x + x. For the fourth echo, x + x + x. Then x x 9. and 34 m x + x x 34 m 47. Thu x 34 5 m.47 m and 34 (a) o x + x 86 m x m ; x 576 m (b) x + x + x x + x 34 m ( ) _7_ch7_p indd 45 /3/7 8:8:4 PM

5 ound Wae 453 ection 7. Periodic ound Wae *P7.8 (a) The peed gradually change from (33 m )( + 7 C 73 C) 347 m to (33 m ) ( + 73 C) 33 m, a 4.6% decreae. The cooler air at the ame preure i more dene. (b) (c) The frequency i unchanged, becaue eery wae cret in the hot air become one cret without delay in the cold air. The waelength decreae by 4.6%, from f (347 m ) (4 ) 86.7 mm to (33 m ) (4 ) 8.8 mm. The cret are more crowded together when they moe lower. *P7.9 (a) f f 4. MHz, 5 m λ 6 f mm (b) f f MHz, 5 m λ 6 f 5. mm f f MHz, 5 m λ μm P7. ΔP ρω ΔP ρω 4 3 Nm 3 (. kg m )( 343 m)( π ). ( ) ( ) m P7. (a) A. μm π λ 4 m 4 cm 5. 7 ω m k 5. 7 ( ). co (b) ( )( ) ( ) (c) (. )( Aω μm 858 ) 7. mm. 433 μm x t P7. (a) ΔP ( ) π 34π 7. Pa in ( unit) m The preure amplitude i: ΔP 7Pa. (b) ω π f 34π, o f 7 Hz π (c) k π m, giing λ. m λ (d) λ f (. m)( 7 Hz) 34 m 3794_7_ch7_p indd 453 /3/7 8:8:5 PM

6 454 Chapter 7 π π P7.3 k 6. 8 m λ ( m ) π π ( 343 m) ω 6. 4 λ ( m) Therefore, ΔP ( Pa) in 6. 8 x m. 6 4 t P7.4 (a) The ound preure i extra tenile tre for one-half of each cycle. When it become ( ) 8 ( 5% ) 3. Pa 6. 5 Pa, the rod will break. Then, ΔP ρω ΔP ρω 8 65 Nm 3 3 ( 89. kg m ) 5 m π 5. ( )( ) 463. mm (b) From ( kx ω t) co ω ( kx t t ) in ω ω π5 ( mm) 4. 5 ( ) m (c) ρ( ω ) ( 89 3 ρ. kg m 3 ) 5 m 4. 5 m W m ( )( ) P7.5 π ΔP ρω ρ λ ( ) πρ π (. )( 343) 5. 5 λ Δ P m ection 7.3 ntenity of Periodic ound Wae P7.6 The ound power incident on the eardrum i P A where i the intenity of the ound and A 5. 5 m i the area of the eardrum. (a) At the threhold of hearing,. W m, and 5 7 P. Wm 5. m 5. W ( )( ) (b) At the threhold of pain,. W m, and P 5 5 (. Wm / )(. 5 m ) 5. W 6 P7.7 β 4. log log. 66. db 3794_7_ch7_p indd 454 /3/7 8:8:6 PM

7 ound Wae 455 P7.8 The power necearily upplied to the peaker i the power carried away by the ound wae: ρa ω π ρaf P ( ) π (. kg m 3 ) ( )( ) π 8. m 343 m 6 m ( ). W P7.9 ρω (a) At f 5 Hz, the frequency i increaed by a factor of.5, o the intenity (at contant ) increae by ( 5. ) 65.. Therefore, 6. 5( 6) Wm (b) 6 W m P7. The original intenity i ρω π ρf (a) f the frequency i increaed to f while a contant diplacement amplitude i maintained, the new intenity i π ρ f ( ) o (b) f the frequency i reduced to f intenity i f π ρ or the intenity i unchanged. ( ) π ρ f π ρf f f or f f f while the diplacement amplitude i doubled, the new π ρ f ( ) 343 m P7. (a) For the low note the waelength i λ f m 343 m For the high note λ 39 m 88 We obere that the ratio of the frequencie of thee two note i 88 Hz Hz. nearly equal to a mall integer. Thi fact i aociated with the cononance of the note D and A. (b) β dblog 75 db gie W m Wm ΔP ρ 5 3 ΔP 36. W m. kg m 343 m 6 Pa ( ) for both low and high note. ( ). continued on next page 3794_7_ch7_p indd 455 /3/7 8:8:7 PM

8 456 Chapter 7 (c) ρ( ω ) ρ4π f π ρf Wm for the low note, 3 π. kg m 343 m for the high note, m 45. m 79. (d) With both frequencie lower (numerically maller) by the factor , the waelength and diplacement amplitude are made.93 time larger, and the preure amplitude are unchanged. 7 8 m m P7. We begin with β log Alo, P 4πr and and β log P 4πr giing o β β log r r r Then, β β r log log r r ( ) ( P7.3 (a). Wm ( β ). Wm ) 4 or. W m ( ) ( 8.. Wm ( ). Wm β ) 45 5 or.. Wm 3. 6 Wm When both ound are preent, the total intenity i Wm Wm. 3 4 Wm (b) The decibel leel for the combined ound i 3. β log. 4 Wm log Wm ( ) db 3794_7_ch7_p indd 456 /3/7 8:8:8 PM

9 ound Wae 457 P P7.4 n 4π r, intenity i proportional to, o between location and : r n ρ( ω ), intenity i proportional to, o Then, r r or r r But, r ( 5 m) + d yield d m P7.5 (a) db dblog Wm P. Wm 4πr P 6. W r 4π 4π. W m ( ) 6 9 m We hae aumed the peaker i an iotropic point ource. r r giing r r ( 5 m) m (b) db dblog Wm. P r 4π Wm 6. W 4π. W m ( ) 69 We hae aumed a uniform medium that aborb no energy. P7.6 We preume the peaker broadcat equally in all direction. km (a) r AC m 5. m 3 P. W 38. 4πr 4π ( 5. m) Wm β dblog Wm β db db 6 Wm (b) r BC 447. m 3 5. W π ( m) β dblog β db 6 Wm (c) 38. μw m μw m 95. β dblog db 3794_7_ch7_p indd 457 /3/7 8:8:9 PM

10 458 Chapter 7 P7.7 ince intenity i inerely proportional to the quare of the ditance,. and The difference in ound intenity leel i At 4 km, At 4. km, ΔP ( ) ρ (. )( 343) Wm Δβ 4 km log (. ) db 4 km Wm Wm β 4. log 8 db β4 β4. + Δβ ( 8 ) db 9 8 db Allowing for aborption of the wae oer the ditance traeled, β4 β4 ( 7. db km)( 3. 6 km) db Thi i equialent to the ound intenity leel of heay traffic. ( ) P7.8 (a) E P t 4πr t 4π( m) 7. W m ( ) 76. kj 7. (b) β log. 8 db P7.9 β log ( β ) ( ) Wm ( db). Wm; db. Wm; db. ( ) (a) P 4πr o that r r ( ) Wm r r 3 (. m).. 3 m (b) r r. 3 9 (. m) m. 3794_7_ch7_p indd 458 /3/7 8:8: PM

11 ound Wae 459 P7.3 Aume you are m away from your lawnmower and receiing db ound from it. The intenity of thi ound i gien by db dblog ; Wm. f the lawnmower Wm P radiate a a point ource, it ound power i gien by 4πr P 4π ( m) W m 6 W Now let your neighbor hae an identical lawnmower m away. You receie from it ound with 6 W 5 intenity 5. Wm. The total ound intenity impinging on you i 4π ( m) 5 5 Wm +. Wm. 5 Wm. o it leel i 5 dblog. db f the mallet noticeable difference i between db and db, thi cannot be heard a a change from db. P7.3 (a) The ound intenity inide the church i gien by β ln db ( db) ln Wm. ( ). 9 Wm Wm 6 Wm We uppoe that ound come perpendicularly out through the window and door. Then, the radiated power i P ( )( ) A 6. Wm. m 77. W Are you urpried by how mall thi i? The energy radiated in minute i E t ( )( ) 6 P 77 J min. min 33 J (b) f the ground reflect all ound energy headed downward, the ound power, P 77. W, coer the area of a hemiphere. One kilometer away, thi area i 6 A πr π( m) π m The intenity at thi ditance i P 77. W 6 A π m Wm and the ound intenity leel i Wm β ( db) ln. Wm db 3794_7_ch7_p indd 459 /3/7 8:8: PM

12 46 Chapter 7 ection 7.4 The Doppler Effect P7.3 (a) ω π 5 min f π 6. rad. min ( ) 3 ωa (. rad ). 8 m 7 m (b) The heart wall i a moing oberer. + O ( ) f f Hz Hz (c) Now the heart wall i a moing ource. f f ( ) 5 9 Hz Hz f ( + o ) *P7.33 (a) f ( ) f ( + ) ( ) 34. khz ( 5. ) (b) f ( 4 ).8 khz (c) ( 5. ) f khz while police car oertake f ( 4 ).4 khz after police car pae P7.34 (a) The imum peed of the peaker i decribed by m ka k m A Nm ( 5 m). m 5. kg The frequencie heard by the tationary oberer range from fmin f + where i the peed of ound. 343 m 44 Hz 343 m+. f min to f f m 343 m 44 Hz 343 m. m f 439 Hz 44 Hz continued on next page 3794_7_ch7_p indd 46 /3/7 8:8: PM

13 ound Wae 46 (b) β dblog dblog P 4πr The imum intenity leel (of 6 db) occur at r rmin m.. The minimum intenity leel occur when the peaker i farthet from the litener (i.e., when r r r + A m). min. Thu, or β β Thi gie: P βmin dblog 4π r P βmin dblog 4π r min min log db 4π r P P 4π r r dblog r 6 db β dblog ( 4. ) 6. db and β min 54. db min f P7.35 Approaching ambulance: f ( ) f Departing ambulance: f ( ( )) ince f 56 Hz and f 48 Hz ( 343) m 4 m 33 m 6 ( C) 35 m C P7.36 (a) ( ) + min 6. 4 m (b) Approaching the bell, the athlete hear a frequency of + After paing the bell, he hear a lower frequency of + ( ) f f O The ratio i f O 5 f + O 6 which gie m o o O P7.37 f f d ( 9. 8t ) fall ( ) + ( )( ) ( )( ) t f t f gt f 8. 3 m: t return The fork continue to fall while the ound return. ttotal fall tf + treturn dtotal gttotal fall 9. 3 m 9. 5 m 3794_7_ch7_p indd 46 /3/7 8:8: PM

14 46 Chapter 7 P7.38 (a) ound moe upwind with peed ( 343 5) m. Cret pa a tationary upwind point at frequency 9 Hz. Then 38 m λ f m (b) By imilar logic, ( ) m λ f m (c) The ource i moing through the air at 5 m toward the oberer. The oberer i tationary relatie to the air. f + f o Hz Hz (d) The ource i moing through the air at 5 m away from the downwind firefighter. Her peed relatie to the air i 3 m toward the ource. + o f f 9 Hz 343 ( 5) 9 Hz Hz P7.39 (b) inθ 3. ; θ 9. 5 tanθ h x ; x h tanθ (a) m 4 x 566. m 566. km tan x 566. m t take the plane t m ( ) to trael thi ditance. x t h h a. Oberer b. FG. P7.39(a) Oberer hear the boom P7.4 θ in in P7.4 The half angle of the hock wae cone i gien by inθ light 8 light 5. m 8. 8 m inθ in ( 53. ) 3794_7_ch7_p indd 46 /3/7 8:8:3 PM

15 ound Wae 463 ection 7.5 ection 7.6 Digital ound Recording Motion Picture ound P7.4 For a 4-dB ound, 4 db dblog 8 Wm ΔP ρ Wm ΔP ρ ( kg m )( 343 m ). 8 Wm Nm Nm (a) code 8. 7 Nm (b) (c) For ound of 4 db or ofter, too few digital word are aailable to repreent the wae form with good fidelity. n a ound wae Δ P i negatie half of the time but thi coding cheme ha no word aailable for negatie preure ariation. Additional Problem *P7.43 The glider tick together and moe with final peed gien by momentum coneration for the two-glider ytem: 5 kg.3 m + (5 + ) kg 986 m The miing mechanical energy i (/ )(5 kg)(.3 m /) (/ )(35 kg)(986 m /) 397 J 7 J 7 J We imagine one-half of 7 mj going into internal energy and half into ound radiated iotropically in 7 m. t intenity 8 m away i E At 5(7 J) [4π(8 m) 7 ]. W m t intenity leel i β log(. ) 3 db The ound of air track glider latching together i many order of magnitude le intene. The idea i unreaonable. Nearly all of the miing mechanical energy become internal energy in the latch. 3794_7_ch7_p indd 463 /3/7 8:8:4 PM

16 464 Chapter 7 *P7.44 The wae moe outward equally in all direction. (We can tell it i outward becaue of the negatie ign in.36 r 3 t.) t amplitude i inerely proportional to it ditance from the center. t intenity i proportional to the quare of the amplitude, o the intenity follow the inere-quare law, with no aborption of energy by the medium. t peed i contant at f λ ω k (3 ) (.36 m).49 km. By comparion to the table in the chapter, it can be moing through water at 5 C, and we aume that it i. t frequency i contant at (3 ) π 33 Hz. t waelength i contant at π k π (.36 m) 4.6 m. t preure amplitude i 5. Pa at radiu m. t intenity at thi ditance i ΔP ρ ( 5 N/m ) 9 μw/m 3 ( kg/m )( 49 m/ ) o the power of the ource and the net power of the wae at all ditance i P 4π r 9 4. W/m 4π( m) 63. mw. ( ) *P7.45 Model your loud, harp ound impule a a ingle narrow peak in a graph of air preure eru time. t i a noie with no pitch, no frequency, waelength, or period. t radiate away from you in all direction and ome of it i incident on each one of the olid ertical rier of the bleacher. uppoe that, at the ambient temperature, ound moe at 34 m ; and uppoe that the horizontal width of each row of eat i 6 cm. Then there i a time delay of 6. m 34 m ( ) between your ound impule reaching each rier and the next. Whateer it material, each will reflect much of the ound that reache it. The reflected wae ound ery different from the harp pop you made. f there are twenty row of eat, you hear from the bleacher a tone with twenty cret, each eparated from the next in time by 6 (. m) 4 34 m ( ) Thi i the extra time for it to cro the width of one eat twice, once a an incident pule and once again after it reflection. Thu, you hear a ound of definite pitch, with period about 4, frequency 3 Hz ~ a few hundred Hz 3 5 waelength and duration λ ( 34 m ) f ( 3 ) ( 4 ) ~. m ~ m (b) Ye. With the tep narrower, the frequency can be cloe to Hz. f the peron clapping hi hand i at the bae of the pyramid, the echo can drop omewhat in frequency and in loudne a ound return, with the later cycle coming from the maller and more ditant upper rier. The ound could imitate ome particular bird, and could in fact contitute a recording of the call. 3794_7_ch7_p indd 464 /3/7 8:8:4 PM

17 ound Wae 465 *P7.46 (a) (b) (c) The ditance i larger by time. The intenity i 6 time maller at the larger ditance, becaue the ound power i pread oer a 4 time larger area. The amplitude i 4 time maller at the larger ditance, becaue intenity i proportional to the quare of amplitude. The extra ditance i (4 6) 45 4 waelength. The phae i the ame at both point, becaue they are eparated by an integer number of waelength. P7.47 ince co in θ + θ, inθ ± co θ (each ign applying half the time) ΔP ΔP in( kx ωt) ± ρω co ( kx ωt) Therefore ΔP± ρω co ( kxωt) ± ρω 343 m P7.48 (a) λ 3 m f 48 (b) β 8. db dblog Wm ( Wm).. Wm. 6 Wm ρω 4 (. 6 Wm) ρω (. kg m )( 343 m ) 4 ( 48 ) 8 m π 343 m (c) λ 46 m Δλ λ λ 3. 8 mm f 397 P7.49 The truck form a train analogou to a wae train of cret with peed 9. 7 m and unhifted frequency f 667 min 3. min (a) The cyclit a oberer meaure a lower Doppler-hifted frequency: + o ( ) ( 4. 47) f f 667 min min + (b) f f o ( 667 min ) (. 56) min The cyclit peed ha decreaed ery ignificantly, but there i only a modet increae in the frequency of truck paing him. P7.5 (a) The peed of a compreion wae in a bar i Y Nm m 3 ρ 7 86 kg m (b) The ignal to top pae between layer of atom a a ound wae, reaching the back end of the bar in time L 8 m t m continued on next page 3794_7_ch7_p indd 465 /3/7 8:8:5 PM

18 466 Chapter 7 (c) (d) A decribed by Newton firt law, the rearmot layer of teel ha continued to moe forward with it original peed i for thi time, compreing the bar by The train in the rod i: 4 3 ΔL t (. m) m.9 mm i ( ) 3 ΔL L 9. m m 3 (e) The tre in the rod i: σ ΔL ( )( Y. Nm ) L 476 MPa ince σ > 4 MPa, the rod will be permanently ditorted. (f ) We go through the ame tep a in part (a) through (e), but ue algebraic expreion rather than number: The peed of ound in the rod i Y ρ L The back end of the rod continue to moe forward at peed i for a time of t L traeling ditance Δ L after the front end hit the wall. The train in the rod i: The tre i then: t i ΔL L it i L ρ Y ρ σ Y Δ L L Y Y i i ρy For thi to be le than the yield tre, σ y, it i neceary that ρ, Y i ρy y < σ y or i < σ ρy With the gien number, thi peed i m. The fact that the length of the rod diide out mean that the teel will tart to bend right away at the front end of the rod. There it will yield enough o that eentually the remainder of the rod will experience only tre within the elatic range. You can ee thi effect when ledgehammer blow gie a muhroom top to a rod ued a a tent take. P7.5 (a) f f ( ) dier dier f o f f dier f with 343 m, f 8 Hz and f 5 Hz (b) we find dier 343 m f the wae are reflected, and the kydier i moing into them, we hae ( + ) dier ( + dier ) f f f f ( dier ) o f ( + ) ( ) 5 Hz. 3794_7_ch7_p indd 466 /3/7 8:8:6 PM

19 ound Wae 467 P7.5 Let P(x) repreent abolute preure a a function of x. The net force to the right on the chunk of air i + P( x) A P( x+ Δ x) A. Atmopheric preure ubtract out, leaing ΔP( x+ Δx) + ΔP( x ) ΔP A Δ xa. The ma of the air i Δm ρδv ρaδ x and it x acceleration i. o Newton econd law become t ΔP Δ xa ρa Δ x x t B P(x)A P(x + Δx)A ρ x x t B ρ x t nto thi wae equation a a trial olution we ubtitute the wae function x, t co( kx ω t) We find ( ) k ( kx x t ) in ω k co( kxωt) x + ω t ( kx ωt ) in ω co( kx ωt) t B ρ x t become B k ( kx t ) ω ω ( kx ω t ρ ) co co Thi i true proided B 4π 4π f ρ λ B The ound wae can propagate proided it ha λ f ; that i, proided it propagate with ρ peed B ρ O P7.53 When oberer i moing in front of and in the ame direction a the ource, f f where O and are meaured relatie to the medium in which the ound i propagated. n thi cae the ocean current i oppoite the direction of trael of the hip and Therefore, ( ) ( ) O 45. km h km h 55. km h 5. 3 m, and 64. km h km h 74. km h 55 m (. Hz) f 5 m5. 3 m 5 m 55 m 4. P7.54 Ue the Doppler formula, and remember that the bat i a moing ource. f the elocity of the inect i x, ( ) ( 34 x ) ( ) ( 34 + x ) oling, x 33. m Therefore, the bat i gaining on it prey at.69 m. Hz FG. P _7_ch7_p indd 467 /5/7 6:4:3 PM

20 468 Chapter 7 P db dblog (a) P7.56 (a) Wm. Wm 4πr 4π (. 6 m) P 64 W P P ound output power (b) efficiency total input power 64 W W. (b) 343 m λ f 343 m (c) λ ( ) m f f 33 m (d) λ + ( ) m f f 383 m (e) O ( ) ( ) m f f Hz ( ) m 3. khz FG. P7.56(a) *P7.57 (a) (b) (c) The ound through the metal arrie firt, becaue it moe fater than ound in air. Each trael time i indiidually gien by Δ t L. Then the delay between the pule arrial i Δt L L cu and the length of the bar i L air cu Δt L (365 m )(7 ) 46.3 m cu air air cu air cu air ( m ). ( ) ( 3 m ) m Δt ( 365 m/) Δt (d) Δt The anwer become L where r i the peed of ound in the rod. A r 33 m r goe to infinity, the trael time in the rod become negligible. The anwer approache (33 m )Δ t, which i jut the ditance that the ound trael in air during the delay time. P7.58 P P P β β log P 8 β log + 3. β 67. db 3794_7_ch7_p indd 468 /3/7 8:8:8 PM

21 ound Wae 469 P7.59 (a) θ in 33 in ound obj (b) θ in P7.6 Let T repreent the period of the ource ibration, and E be the energy put into each waefront. Then P a E. When the oberer i at ditance r in front of the ource, he i receiing a pherical T waefront of radiu t, where t i the time ince thi energy wa radiated, gien by t t r. Then, r t 4π r The area of the phere i 4π ( t) i uniform with the alue E P T a A 4π r ( ) ( ) Doppler hifted frequency f f intenity T. The energy per unit area oer the pherical waefront. The oberer receie parcel of energy with the, o the oberer receie a wae with ( ) ( ) E A f T P a 4π r T( ) P a 4πr Y P7.6 For the longitudinal wae L ρ T For the tranere wae T μ f we require L T 8., we hae T μ Y 64. ρ where ma m ρ olume πrl μ m L and Thi gie 3 πry π (. m) 6. 8 N m T ( ) N 3794_7_ch7_p indd 469 /3/7 8:8:9 PM

22 47 Chapter 7 P7.6 (a) f the ource and the oberer are moing away from each other, we hae: θ θ 8, and ince co8, we get Equation (7.3) with negatie alue for both O and. (b) f O m / then f f coθ Alo, when the train i 4 m from the interection, and the car i 3 m from the interection, o f 343 m 343 m 8 5. m or f 53 Hz coθ 4 5 ( ) ( 5 Hz ) Note that a the train approache, pae, and depart from the interection, θ arie from to 8 and the frequency heard by the oberer arie between the limit 343 m f f co 343 m5. ( 5 Hz) 539 Hz m 343 m fmin f ( 5 Hz) 466 Hz co8 343 m+ 5. m P7.63 (a) The time required for a ound pule to trael L L ditance L at peed i gien by t Y ρ Uing thi expreion we find L L L 3 FG. P7.63 t t L L Y ρ 7. N m 7 ( ) ( kg m ) 3 5. m L 5. m L Y ρ 3 3 ( 6. N m ) (. 3 kg m ) 3 4 or t L t t 3 3 ( ) 5. m (. N m 3 ) ( 8 8 kg m 3 ) We require t + t t3, or L L Thi gie L 3. m and L m The ratio of length i then L L ( 96. L ) (b) The ratio of length L L i adjuted in part (a) o that t + t t3. ound trael the two path in equal time and the phae difference Δφ. 3794_7_ch7_p indd 47 /5/7 6:43:3 PM

23 ound Wae 47 ANWER TO EVEN PROBLEM P km P7.4 (a) 7. (b) longer than 5.7, becaue the air i cooler P7.6 (a) 53 m (b) 64 m P4.8 (a) The peed decreae by 4.6%, from 347 m to 33 m. (b) The frequency i unchanged, becaue eery wae cret in the hot air become one cret without delay in the cold air. (c) The waelength decreae by 4.6%, from 86.7 mm to 8.8 mm. The cret are more crowded together when they moe lower. P m P7. (a). 7Pa (b) 7 Hz (c). m (d) 34 m P7.4 (a) 4.63 mm (b) 4. 5 m (c) Wm P7.6 (a) 5. 7 W (b) 5. 5 W P7.8. W P7. (a) f f (b) P7. ee the olution P m P7.6 (a) 65. db (b) 67.8 db (c) 69.6 db P7.8 (a). 76kJ (b) 8 db P7.3 no P7.3 (a). 7 cm (b) 8. 9 Hz (c) Hz P7.34 (a) 44 Hz; 439 Hz (b) 54. db P7.36 (a) 35 m (b) 9. 5 m P7.38 (a) 364 m (b) 398 m (c) 94 Hz (d) 938 Hz P P7.4 (a) 7 (b) For ound of 4 db or ofter, too few digital word are aailable to repreent the wae form with good fidelity. (c) n a ound wae ΔP i negatie half of the time but thi coding cheme ha no word aailable for negatie preure ariation. P7.44 The wae moe outward equally in all direction. t amplitude i inerely proportional to it ditance from the center o that it intenity follow the inere-quare law, with no aborption of energy by the medium. t peed i contant at.49 km, o it can be moing through water at 5 C, and we aume that it i. t frequency i contant at 33 Hz. t waelength i contant at 4.6 m. t preure amplitude i 5. Pa at radiu m. t intenity at thi ditance i 9 μw m, o the power of the ource and the net power of the wae at all ditance i.63 mw. 3794_7_ch7_p indd 47 /3/7 8:8:3 PM

24 47 Chapter 7 P7.46 (a) The intenity i 6 time maller at the larger ditance, becaue the ound power i pread oer a 4 time larger area. (b) The amplitude i 4 time maller at the larger ditance, becaue intenity i proportional to the quare of amplitude. (c) The phae i the ame at both point, becaue they are eparated by an integer number of waelength. P7.48 (a) 3 m (b) 84. nm (c) 3.8 mm P7.5 (a) 5. 4 km (b) 59 μ (c).9 mm (d) 38 (e) 476 MPa (f) ee the olution P7.5 ee the olution P7.54 The gap between bat and inect i cloing at.69 m. P7.56 (a) ee the olution (b) 343 m (c) 33 m (d) 383 m (e). 3kHz P db P7.6 ee the olution P7.6 (a) ee the olution (b) 53 Hz 3794_7_ch7_p indd 47 /3/7 8:8:3 PM