Beats and Doppler Chapter 17

1
Beats and Doppler Chapter 17 18

• PHYS 2326-28

2
Concepts to Know

• Beats
• Doppler Effect
• Sign Convention

3
Beats

• Listening to a band or orchestra of relative
  beginners, youll notice the problem that the
  music just doesnt sound good. If you even
  listen to two people playing the same thing it
  could sound bad as well.
• Quite often this is due to tuning of the musical
  instruments. If one is at a different pitch from
  the other there is a beat that may be fairly low
  to very high speed.

4
Beats

• This beating ruins the effect of the music.
• It is caused by the instruments playing at
  slightly different pitches and the intensity
  changes with time rather than with space.
• Beating is the periodic variation in amplitude
  caused by the superposition of two waves of
  slightly different frequencies
• The frequency of the beat equals the difference
  in frequency of the two sources

5
Beats

• Section 18.7 shows that given two wave functions
  of different frequencies can be superimposed
  using a trigonometric identity cos a cos b 2
  cos((a-b)/2)cos((ab)/2)
• The superposition gives y 2a cos (2pi
  ((f1-f2)/2)t) cos(2pi((f1f2)/2)t) eqn 18.10
• This shows there is an average frequency that is
  beating at the difference frequency.

6
Beats

• The envelope that beats becomes
• 2Acos(2 pi ((f1-f2)/2)t)
• the beat frequency (f1-f2) since cosine goes to
  zero twice every 2pi radians

7
An Interesting Effect

• Since this qualifies regardless of the actual
  frequencies and differences, one can achieve a
  fascinating musical effect under some conditions.
  Given 2 flutists playing a duet, music can (and
  has) been written that makes use of these sum and
  difference of frequencies so that rather than a
  irritating beat due to poor tuning, one can
  achieve what sounds like 3 or 4 instruments
  playing different notes played at one time with
  even a different melody line

8
Doppler Effect

• So far, weve dealt with observing waves emitted
  from and observed from positions at rest
• We know that the waves travel at a speed relative
  to the medium
• What happens if the transmitter and / or the
  receiver are traveling relative to the medium?
• Weve all heard train and car horns shift
  frequency as they travel past us

9
Moving Observer

• Eqns 17.9 and 17.10 show what happens to the
  frequency if an observer moves relative to the
  medium. Since vf ?
• f v/? and for a moving observer fv/ ?
  (vvo)/ ? where v is the velocity of the wave and
  vo is the velocity of the observer
• f ((vvo)/v) f moving towards the source and
  f ((v-vo)/v) f moving away from the source

10
Moving Source

• When a source moves, it moves a distance vsT
  during each period vs/f and the wavelength is
  shortened to ? ?-?? ?- vs/f
• The observer hears f v/ ? v/(v/f - vs/f)
• f (v/(v vs)) f source moving towards
  observer and
• f (v/(v vs)) f for the source moving away

11
Sound Wave Doppler

• The general equation becomes eqn 17.13
• f ((vvo)/(v-vs))f
• SIGNS these signs are for the observer or
  source moving towards the other and for the case
  where one is moving away from the other the
  appropriate sign is reversed.

12
For Electromagnetic Waves

• fr sqrt((c-v)/(cv)) fs
• where c is the speed of light and v the velocity
  difference between transmitter and receiver.
  This is the relativistic equation and cannot
  distinguish between which is in motion wrt the
  medium or even if there is a medium.

13
Example 1

• A car traveling at 20m/s blows its horn, f300Hz
  as it passes a second car at rest. If the speed
  of sound is 345m/s, find a) the frequency heard
  in the second car before it passes, b) the
  frequency heard after it passes, c) the
  wavelength ahead of the car, d) the wavelength
  behind the first car, e) If the second car blows
  its horn after the first passes, what is the beat
  frequency heard by those in the second car, f) by
  those in the first car?

14
Example 1

• a) observer at rest vo0,f ((vvo)/(v-vs))f
• f ((3450)/(345-20))300 318.5Hz
• b) f((3450)/(345-(-20)) 345/365 284.4Hz
• c) wavelength v/f 345/317.5 1.083m
• d) wavelength v/f 345/284.4 1.213m
• e) fbeat fa-fb 300Hz-284.4Hz 15.6Hz
• f) f ((345(-20))/(3450) 282.6Hz
• f-f 300 282.6 17.4Hz

15
Example 2

• Student skating at 3m/s towards a wall, blows a
  500Hz whistle at 100m from the wall and times the
  echo. a) How far does he travel before hearing
  the echo, b) what is the echo frequency? c) what
  is the beat frequency with the whistle? assume
  v345m/s

16
Example 2

• in time t, sound travels vt and skater travels
  v2t. The distance traveled by the sound is D D-
  v2t since the skater is that much closer to the
  wall when the sound returns. hence 2D v2t vt
  2(100) 345t 3t 348t, t 0.5745 s
• x vt 3 0.5745 1.72m
• b) f ((vvo)/(v-vs))f ((3450)/(345-3))500
  504.4 Hz heard by the wall and reflected
• For the skater f((3453)/(3450))504.4 508.8
  Hz
• c) fbeat fa-fb 508.8-500 8.8 Hz

17
Example 3

• Hydrogen gas emits spectral lines in the visible
  in what is called the Balmer series. The first
  two are the H-alpha 656.1nm (nice and red) and
  H-beta 486.1nm (bluish green). a) how fast is a
  galaxy moving away from us if its beta line is
  shifted all the way to the alpha wavelength b)
  what is the fractional change what is the
  fractional change in wavelength if the star is
  moving away from us at 0.5c, c) what is the
  observed shift in the alpha line for a star
  moving towards the earth at a speed equal to that
  of the earth around the sun orbital radius
  1.5E11m

18
Example 3

• source wavelength 4.861E-7, cfs?s
• received wavelength 6.563E-7, cfr ?r
• fr sqrt((c-v)/(cv))fs, fr/fs
  sqrt((c-v)/(cv)) ?s/ ?r, square both sides
• (?s/ ?r)2 (c-v)/(cv), solving for v
• v (1-(?s/ ?r)2)c/(1 (?s/ ?r)2) 0.291c

19

• b) v 0.5c, fr sqrt((c-v)/(cv))fs
• fs/fr sqrt((cv)/(c-v)) ?r/?s
  sqrt(1.5c/0.5c) 1.732
• fractional change in wavelength ??/? ?r/?s-1
  0.732
• c) t 1 year 3.156E7 sec
• ?s 6.563E-7 s, 2pi r vt,
• v2pi r/t 29863 m/s
• ??/?s sqrt ((c-v)/(cv)) 1, multiply sqrt by
  c-v and
• get sqrt((c-v)2/(c2-v2)) and v2 ltltc2 so
  result
• is (c-v)/(c) 1-v/c 1v/c 29863/3.0E8
  9.96E-5
• ?? (v/c) ?s
• ?? 9.96 E-5 6.563E-7 6.531E-11 m

20
(No Transcript)