lecture28 - beats- - What you will be able to do...

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Unformatted text preview: What you will be able to do a/er today … Explain … Calculate … •  the condi4ons for beats •  the origin of the ‘so/ ­loud ­so/ ­ loud’ pulsing of the modulated frequency •  the average frequency and why it ma?ers •  the average frequency and the modulated frequency Apply … knowledge of traveling wave superposi4on and interference to two waves of slightly different frequencies. CHALLENGE: extend to Doppler effect Noise Cancellation Headphones •  Based on destructive interference. •  Work best for constant lowfrequency noise. •  Noise is picked up by microphone, and played back over the headphones in real time but with a phase shift of π. Original noise and played back noise interfere destructively. Interference in Time: Beats •  So far: Superposition of waves with same frequency and similar amplitudes. •  What happens if waves have different frequency? •  We get a superposition of the two waves as usual but no continual constructive or destructive interference. •  Why? •  Two waves with different frequency cannot remain in phase over more than a few cycles. video   When two waves of slightly different frequencies combine at a point in space (e.g. at your ears) they periodically match up such that their crests (or troughs) coincide. At those instants the resultant amplitude is maximum possible and a maximum in the sound level is heard.   As the waves go out of phase they resultant amplitude decreases and the sound level decreases, passing through a minimum (when the crest of one matches up with the trough of the other).   The number of times per second the sound level becomes maximum is called the beat frequency   The beat frequency is equal to the difference of the two frequencies. •  Simulation D1(0, t ) = sin(10π t ) D2 (0, t ) = sin(12π t ) t D(0, t ) = sin(10π t ) + sin(12π t ) What does this sound like? Frequency that you hear = f avg f1 + f 2 = 2 Beat frequency = f beat = f1 − f 2   Beat frequency is the number of times one hears a maximum in loudness, PER second. Question •  On vista: •  A tuning fork and a piano emit f1 = 523 Hz and f2 = 527 Hz, respectively. What do you hear? 1) A sound at 1050 Hz disappearing every 4 seconds. 2) A sound at 1050 Hz disappearing every 2 seconds. 3) A sound at 2 Hz disappearing every millisecond. 4) A sound at 525 Hz disappearing every 4 seconds. 5) A sound at 525 Hz disappearing every 2 seconds. D = 2DM cos(ωmod t )sin (ωav t ) 1 favg = ( f1 + f2 ) 2 fbeat = 2 f mod = f1 − f 2 7 Interference in Time: Beats •  Superposition of two waves of equal amplitude and close frequencies at a fixed location. Choose x = 0 to eliminate k x part: D = D1 + D2 = DM sin(ω1t ) + DM sin(ω2t ) 1 1 •  Trigonometric identity: sin α + sin β = 2 sin (α + β ) cos (α − β ) 2 2 •  We obtain: ȹ 1 ȹ ȹ 1 ȹ D = 2 DM cosȹ (ω1 − ω2 )t ȹ sin ȹ (ω1 + ω2 )t ȹ ȹ 2 Ⱥ ȹ 2 Ⱥ Low frequency Slow modulation of the amplitude High frequency Average oscillation frequency D = 2DM cos(ωmod t )sin (ωav t ) 8 Sound Beats: DEMO •  Tuning an instrument: Strike a tuning fork and play the corresponding note at the same time. •  If the two frequencies are close, you hear beats as an oscillation in sound level. Demo •  Beat frequency  max humans can distinguish is fbeat = 10 Hz f beat = 2 f mod ȹ ωmod ȹ = 2ȹ ȹ = f1 − f 2 ȹ 2π Ⱥ favg usually > 100 Hz favg = ½ (f1 + f2) 9 Question You hear three beats per second when two sound tones are generated. The frequency of one tone is known to be 610 Hz. The frequency of the other is 1. 604 Hz. 2. 607 Hz. 3. 613 Hz. 4. 616 Hz. 5. Either 2 or 3. Question You hear three beats per second when two sound tones are generated. The frequency of one tone is known to be 610 Hz. The frequency of the other is 1. 604 Hz. 2. 607 Hz. 3. 613 Hz. 4. 616 Hz. 5. Either 2 or 3. 11 Worksheet •  •  A.  B.  C.  D.  E.  A piano tuner strikes his tuning fork (f = 523.3 Hz) and strikes a C-note at the same time. The two have nearly the same frequency and he hears 3.0 beats per second. As he tightens the piano string, the beat frequency decreases to 2.0 beats per second. What is the frequency of the piano string before tightening the string? 520.3 Hz 521.3 Hz 523.3 Hz 525.3 Hz 526.3 Hz Worksheet •  •  A.  B.  C.  D.  E.  A piano tuner strikes his tuning fork (f = 523.3 Hz) and strikes a C-note at the same time. The two have nearly the same frequency and he hears 3.0 beats per second. As he tightens the piano string, the beat frequency decreases to 2.0 beats per second. What is the frequency of the piano string after tightening the string? 520.3 Hz 521.3 Hz 523.3 Hz 525.3 Hz 526.3 Hz Lecture Activities beats & standing waves Relevant textbook sections covered: 21.8 Worksheet 1(a) & 1(b) 1. A piano tuner strikes his tuning fork (f = 523.3 Hz) and strikes a C-note (2"d halmonic) at the same time. The two have nearly the same frequency and he hears 3.0 beats per second. As he tighlens the piano string, the beat frequency decreases to 2.0 beats per second. The length of the piano string is 1.8 m and a linear mass density (¡r) of 8.3 x 10-a kg/m. (a) What are the TWO OPTIONS for the frequency of the piano string beþre tightening? al^^r-r"'-__r {o*r, = Z.o Hz= la,(L"o f, | à {,= szz.e¡z't Z It+ =laz",z l,= l¿.";^¡€-.r. = ça8." rì¿) lt (b) What is the frequency of the piano string beþre and after tightening? Hint: Use the information about the change in tension and the change in beat frequency. -Ll¡ç.n"', t*+\ =L6'z vtâl + q þlx1o!.r* t".t.= Z.ôüz= l€,-f,,**i :> f,,^"*= Szz,zwzLzHz = 571,1 Ylz oR 5 z5 '7\\e kn"w, Ç._0."* > f , (o.: "";.n"ll5) ì+ 6.u'5¡ ho,^ ! o uu., Gzo ZU*l onot ìntvc:'J'r'J +" ^+ TqãA'i*^f is rhe rension in the srring belure tightening, Y tin.<- ,nre (c.) whar V=}'{ [-:-r- ,'./=Å-i t+ ,'¡2 c L n -.s- ,\. r - ,rlI -.:i. ----/-'1--r -7 .# -: t, -r' -l a 14 = 571,1 Ylz oR 5 z5 '7\\e Worksheet Y tin.<- ,nre k1(d)._0."* > f , (o.: "";.n"ll5) 1(c) & n"w, Ç (c.) whar ì+ 6.u'5¡ ho,^ ! o uu., Gzo ZU*l onot ìntvc:'J'r'J +" ^+ TqãA'i*^f is rhe rension in the srring belure tightening, V=}'{ 2ü !r^.*,,. ^; e. t+ [-:-r- ,'¡2 c L n -.s- ,\. r - ,rlI -.:i. ----/-'1--r -7 .# -: t, -r' -l a ,'./=Å-i t f--/ .\ I _ //\ _ I.¿)fi\' /-." <> t-- -t -__, _'/ L + T,= (r,2"\õ+#)(r.B =f+zu q )f (szo'27$ =à % .\.-^3. = ( tu'?\---tT n \.,oo, = E\ .ì \ 428rt I J I 15 Worksheet A street-performing violinist is tuning the A-string (440 Hz - fundamental) on their violin on Granville St. They listen for beats with a tuning fork played simultaneously with a frequency 440 Hz. They hear 4 beats per second. When slightly increasing the tension in the A-string, the beat frequency increases to 5 beats per second. What is the frequency of the string a"er 2ghtening? A.  435 Hz B.  437 Hz C.  400 Hz D.  443 Hz E.  445 Hz Worksheet A street-performing violinist is tuning the A-string (440 Hz - fundamental) on their violin on Granville St. They listen for beats with a tuning fork played simultaneously with a frequency 440 Hz. They hear 4 beats per second. When slightly increasing the tension in the A-string, the beat frequency increases to 5 beats per second. What is the average frequency heard a"er 2ghtening? A.  435 Hz B.  437 Hz C.  400 Hz D.  443 Hz E.  445 Hz beats when thís note is played simultaneously with a tuning fork of frequency 440 Hz. She hears 4 beats per second. She notices that, when she increases the tgnsion in the string Worksheet 2(a) & 2(b) slightly, the beat frequency increases to 5 beats per second. The length of the violin string is 32.5 cm and has a linear mass density (¡r) of 6.1x 104 kg/m. ' (a) What is the frequency of the string ajler fightening? lv"otr= 4,0 H¿= [c,-ç, 1 , G+ .ç,= l¿.."r^e+"r = 44o t|+ à Ç.,o"*r: {{oH+ I5 x 445 oe '+65 ïlg ^o"u b*r "7u Frru'", Srrr.or > f*,"r¿ =;z f-.*.* = LTjg\ t\"^ {rt so * 4u*-'t, 5.o t+z ( W What is the average frequency heard string to obtatn the (b)b) hat percent change in tension is needed for the after tightening?440 Hz sound? Should the tension be increased or decrcased? * 1 tla , ,¡,¿ ryg$' {- 'ttlo Hr + Þng'o^ rr.eJ s *o 1 favg = ( f1 + f2 ) = ( 440 + 445) = 442.5Hz 2 2 Te -'¡u\t Ç' (r"e + 1-.) fz¡o- - 4¿tE \_ de cre,.trrì- \, % t\o^oe t 1.. ,- l.-:.'r^r '- -*'Ii--j' l;.;à;o :. -r: r ' r4¡-ñ . I ¿:'¡, = r,7 'l ^¿^L y'.ñ Ír,,'-^, --^t/n-C*;,,, (..*.., I ¿V¿\t liJ +7 +,^.": * 'toc'' {-,'^ ',L1^ t' 18 " , Worksheet EXTRA practice x 445 oe '+65 ïlg 4u*-'t, * 5.o t+z à Ç.,o"*r: {{oH+ I5 t\"^ {rt so ^o"u b*r "7u Frru'", S The > f*,"r¿ still f-.*.* EXTRA PRACTICE:rrr.or performer=;zneeds to=tune their violin. What percent LTjg\ change in tension is needed for the string to obtain the 440 Hz sound? Should the tension be increased or decreased? The length of the violin string is 32.5 cm (b) What percent change in tension is needed for the string to obtatn the 440 Hz sound? and the a linear ncreased or decrcased? Shouldhas tension be imass density (µ) of 6.1x10-4 kg/m. fz¡o- - 4¿tE tla , ,¡,¿ ryg$' {- * 'ttlo Hr + Te -'¡u\t Ç' (r"e \_ + cre,.trrì- 1-.) \, % t\o^oe 1.. ,- l.-:.'r^r '- -*'Ii--j' l;.;à;o . I ¿:'¡, = r,7 'l ^¿^L y'.ñ Ír,,'-^, --^t/n-C*;,,, (..*.., +7 I - t ¿V¿\t +.' ,! +,^.": * 'toc'' {-,'^ ',L1^ " t' ¿¿qo P'{ :. -r:ârr ' r4¡-ñ liJ (+¿1o + Þng'o^ rr.eJ s *o de Z ç.iq^"r"¡"- ì^ T-e r¡r sì o r'r = H zJz- ( ys e,rt 1..)Õf = I _ Z'.Z %l -Å ({.rçu+-)= _ ,l -;___:__.:_, (u ¡.. t""sì.^ sl-.-, 4 19 Ì-5 Ve J-e..¿as eé-. e,V 7 'lt.rce,"!¡ Worksheet CHALLENGE fvvùvli,{r s.ßsæffiä* 'tprqu,,r4,Æ#s .tüLlptå q"/s /,h CHALLENGE: The halfmarathon passes by Granville St. The winner in 2011 averaged a speed of 20km/hr. (c) What frequency would they hear when running towards the street-performing violinist?   Think about: What is the source frequency (or frequencies) in this case? -( (, one source t#r {a % g Y?¿tÉ source one cr= f(r (+) l(t+ +) = yToìf¿-( r+ ä) {' l'= Vq7 $¿ {r l*^* 2 ltSz-'lt+ r\ kt*u t uv-) (d) Will the runner hear more beats (higher fbeat), less beats (lower fbeat) or the same fbeat as the violinist? Lt ( z-+ '1"1+ L4'1 1. S )/- 14 {Þ fr,iç ìs wu*uå -å.bru- Würugî-ø- 'n^É'qd'å ^'tL s aw-Ê- twwfu¿" fu4e N ld*" .bnu}i'"**r n"f' y+ '-'/ ç, b&&,Åu* 20 \ ...
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This note was uploaded on 01/15/2012 for the course PHYS 101 taught by Professor Bates during the Winter '08 term at The University of British Columbia.

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