Honors Classical Physics I

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1 Honors Classical Physics I PHY141 ecture 32 ound Waes Please set your clicker to channel 21 ecture 32 1

2 Bosch 36W column loudspeaker polar pattern Monsoon Flat Panel speaker: (5 db grid) 400 Hz: Real oudspeakers 1 khz: 3 khz: 7 khz: ecture 32 2

3 Example Your ears are sensitie to dierences in pitch, but they are not ery sensitie to dierences in intensity. You are not capable o detecting a dierence in sound intensity leel o less than 1 db. By what actor does the sound intensity increase i the sound intensity leel β increases rom 60 db to 61 db? 2 1 I (10 db)log I (10 db)log 1 db (10 db)log I I 10 I I I0 I0 2 (10 db)log I 10 I 1 I 1 db I1 ecture 32 3

4 Example A 3 db Increase in sound leel corresponds to: +3 db = 10 db log(i new /I old ) I new /I old = 10 3/10 = = 2.0 ecture 32 4

5 tanding ound Waes In open or closed air pipes sound waes will orm standing waes when the air is rhythmically excited at the appropriate requency This is the basis or all wind-instruments: organs, lutes, etc. Note that the elocity o sound in air is ixed ( = 344 m/s or normal conditions), thus the product o λ = is ixed, and and λ are NOT independent! As or standing waes on a string under tension, we expect the waelength λ o the standing wae, and the length o the pipe, to be similarly related. At an OPEN END o the pipe, the pressure is closely equal to the ambient atmospheric pressure: i.e. there must be a PREURE NODE NO such situation occurs or a COED END; in act there the PREURE has an ANTI-NODE (maximum amplitude) ecture 32 6

6 Wind Pipes Considering the aboe statements, we arrie at the ollowing picture (where we depict an OPEN-ENDED pipe, and graph the standing PREURE waes Δp(x,t) on top: λ 1 /2 λ 2 /2 λ 4 /2 A harmonic series appears (we show the 1 st, 2 nd, and 4 th harmonic), goerned by the statement that a whole number o hal-waelengths must it in length o the pipe : i.e.: = n(λ n /2), with: n=1,2,3, or: λ n = 2/n = λ 1 /n or: n = /λ n = n /(2) = n 1 the DIPACEMENT graph looks dierent: the place where pressure has a NODE, displacement has an ANTI-NODE and ice ersa, because there the motions o the molecules (displacing themseles to keep local pressure constant) is most iolent ecture 32 7

7 topped Wind Pipes Consider now a TOPPED pipe: a pipe with one OPEN end, and one COED END (or example the clarinet), and graph the standing PREURE waes Δp(x,t) on top: Careul inspection shows that a dierent harmonic series appears (we show the 1 st, 3 rd, and 7 th harmonic): λ 1 /4 λ 2 /4 i.e.: = (2n 1) λ n /4, with: n=1,2,3, (Note: (2n 1)is odd!) equialently: = n odd λ n /4, with: n=1,3,5, (ODD harmonics only) or: λ n = 4/n odd = λ 1 /n odd or: n = /λ n = n odd /(4) = n odd 1 λ 4 /4 ecture 32 8

8 Resonance arge displacement/pressure waes will occur when the exciting orce is acting in sync with the NATURA FREQUENCY o the pipe (or other type o instrument). Absent damping, the displacements can become uncomortably large! Resonance is used in many instruments to enhance particular (oer)tones (e.g. bass relex tube, iolin/guitar body case) Example: A 0.40 m long, one-side-closed organ pipe is in exact resonance with a 0.50 m guitar string; both ibrate at the undamental tone. Calculate string : pipe =λ pipe 1 /4 and string =λ string 1 /2 pipe = air /(λ pipe 1 =4 pipe ) = string = string /(λ string 1 =2 string ) string = air 2 string /(4 pipe ) = m/s = 215 m/s Calculate the λ and o the sound waes in the air: λ 1 = 4 pipe = 1.60 m, 1 = air /λ 1 =344/1.60 = 215 Hz Calculate the λ and o the standing wae on the guitar string: λ 1 =2 string =1.0 m; check: 1 = string /λ 1 =215/1.00=215 Hz ecture 32 10

9 Intererence Intererence is the destructie or constructie addition o displacements by waes arriing rom two or more sources at a set o spatial locations. Intererence is a consequence o the UPERPOITION PRINCIPE, which says that disturbances caused by indiidual waes at any gien point simple ADD this is a consequence o the linear character o the wae equation Thus, a trough rom one wae may coincide with an equally high peak o another, with the result there is no dis-placement at all: destructie intererence. uch a dead spot will persist i the two waes hae exactly the same requency. I the waes hae a slightly dierent requency, beat waes may occur: ( kx1t) ( kx2t) ( kx1t) ( kx2t) y(,) xt Acos( kx1t) Acos( kx 2t) 2A cos cos Acoskx t cos t 2Acos( kxt)cos 2 2 t 2Acos 2 t 2 cos( kxt) This is a traeling wae, with a requency equal to the aerage requency o the initial waes, and an amplitude which is modulated (i.e. aries in time) with a much smaller requency equal to hal the dierence o the original requencies. The INTENITY is proportional to the amplitude squared, and thus the beat requency we hear is simply equal to the (absolute alue o) the dierence in original requencies: I cos t 1 cos( t ) 2 double-angle ormula 2 ecture 32 11

10 A B λ maxima (twice the amplitude) minima (dead spots) y x Intererence by ame- Frequency Waes Intererence o two synchronous equalrequency, equalamplitude sound sources, ignoring relections rom walls, loor, ceiling, etc P Moie Maximum in P: AP BP = nλ, n=0,1,2, ; Minimum in P: AP BP = (n+ ½)λ ecture 32 12

11 Doppler Eect: = (+ )/(+ ) Doppler eect: requency o receied sound depends on the relatie motion o the source or receier with respect to the medium (e.g. air): λ λ λ λ <λ > Doppler Formula: +e or approaching λ >λ < +e or moing away rom ecture 32 13

12 Doppler Eect - Deriation Doppler eect: requency o receied sound depends on the relatie motion o the source or receier with respect to the medium (e.g. air): E.g. moing towards the source o the sound wae, my ears will catch more pressure ariations per second than i I stay still or moe away. Thus, the requency I perceie depends on my elocity with respect to the air. imilarly, the motion o the sound source with respect to the air aects the waelength λ o the air waes. For the simple case where all motions are along the x-direction: Assume a sound wae in e x-direction; the speed o sound is = 343 m/s ITENER who has elocity in +e x-direction: λ relatie speed ( ) waelength x OURCE with elocity in +e x-direction: the source traels a distance T per period, so that the eectie waelength is increased by that amount: λ T Combining: Doppler Formula: x +e or approaching +e or moing away ecture 32 rom 14

13 Example Note, that i = then = (e.g. when the wind blows rom source to receier, nothing changes a bat emits a high-pitched chirp at 80 khz (ultra-sound) when it approached a ixed wall with elocity Bat =10 m/s calculate the requency o the relected chirps the bat receies incident on wall Bat relected rom wall Bat Bat Bat receied by Bat relected rom wall Bat Bat Bat 354 Bat 80 khz 84.8kHz 334 Bat +e or approaching +e or moing away rom Note that the IGN o the elocities is CRUCIA! the problem is een more complex when the wall is a MOVING INECT! This is an example o ONAR ecture 32 15

Physics 107 TUTORIAL ASSIGNMENT #7

Physics 107 TUTORIAL ASSIGNMENT #7 Physics 07 TUTORIL SSIGNMENT #7 Cutnell & Johnson, 7 th edition Chapter 6: Problems 5, 65, 79, 93 Chapter 7: Problems 7,, 9, 37, 48 Chapter 6 5 Suppose that sound is emitted uniormly in all directions