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Doppler Effect

Doppler Effect. Physics 202 Professor Lee Carkner Lecture 11. PAL #10 Music. How much would your eardrum move from a tuning fork sound? Example: f = 440 Hz, b = 90 dB b = (10 dB) log (I/I 0 ) I = I 0 10 ( b /10) I = I = 1X10 -3 W/m 2 We need to relate I to s m :

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Doppler Effect

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  1. Doppler Effect Physics 202 Professor Lee Carkner Lecture 11

  2. PAL #10 Music • How much would your eardrum move from a tuning fork sound? • Example: f = 440 Hz, b = 90 dB b = (10 dB) log (I/I0) I = I0 10(b/10) I = I = 1X10-3 W/m2 We need to relate I to sm: I = ½ rvw2sm2 sm = • Air density = r = 1.21 kg/m3 • Velocity of sound = v = 343 m/s

  3. PAL #10 Music (cont.) sm = (I/(½ rv(2pf)2))½ sm = (1X10-3/(½)(1.21)(343)(2p440)2)½ sm = • Even the loudest sounds only produce very small motions • What if the distance is doubled? • Since I = Ps/4pr2, then • but sm => √I, so • The displacement is ½ as great

  4. The Doppler Effect • If there is any relative motion between the two, the frequency of sound detected will differ from the frequency of sound emitted

  5. Stationary Source

  6. Moving Source

  7. How Does the Frequency Change? • If the source and the detector are moving closer together the frequency increases • If the source and the detector are moving further apart the frequency decreases

  8. Doppler Effect

  9. Doppler Effect and Velocity • The greater the change the larger the velocity • Let us consider separately the situations where either the source or the detector is moving and the other is not

  10. Stationary Source, Moving Detector • In general f = v/l but if the detector is moving then the effective velocity is v+vD and the new frequency is: • but l=v/f so, • If the detector is moving away from the source than the sign is negative

  11. Moving Source, Stationary Detector • In general l = v/f but if the source is moving the wavelengths are smaller by vS/f l’ = v/f - vS /f f’ = v / (v/f - vS/f) • The the source is moving away from the detector then the sign is positive

  12. General Doppler Effect • We can combine the last two equations and produce the general Doppler effect formula: f’ = f ( v±vD / v±vS ) • What sign should be used? • For motion toward the sign should be chosen to increase f • Remember that the speed of sound (v) will often be 343 m/s

  13. The Sound Barrier • A moving source of sound will produce wavefronts that are closer together than normal • At the speed of sound the wavefronts are all pushed together and form a shockwave called the Mach cone • This is dangerous because passing through the shockwave makes the plane hard to control

  14. Doppler Effect for Light • However, at low speeds (u<<c, where u is the relative velocity between source and detector) the equations reduce to the classical form: f’ = f (1 ± u/c) u = (Dl/l) c • c, the speed of light in vacuum, is constant (3 X 108 m/s)

  15. Spectral Line Shifts • When we observe a spectrum of a object, we compare the observed wavelengths to standard ones • For objects moving away from us the spectral lines move to larger wavelengths • For objects moving towards us the spectral lines move to shorter wavelengths

  16. Red Shifted Spectrum

  17. Expansion of the Universe • All galaxies are moving away from all others • In the past, everything in the universe must have been much closer together

  18. Summary: Sound Waves • Sound waves are longitudinal or pressure waves • The medium oscillates in the direction of travel • The speed of sound depends on the density and the bulk modulus (compressibility ) of the medium: v = (B/r)½

  19. Summary: Wave Equations • The equations for the amplitude and pressure of a sound wave are: s = sm cos (kx-wt) Dp = Dpm sin (kx-wt) Dpm = (vrw) sm • Waves from two sources will interfere based on the path length difference between the sources and detector DL = ml (fully constructive) DL = (m+½)l (fully destructive)

  20. Summary: Intensity and Music • The intensity of sound falls off with a inverse square law: I = Ps/4pr2 I =½rvw2sm2 • The sound level is: b = (10 dB) log (I0/I) • Harmonic frequencies of a pipe f = nv/2L (open at 2 ends) f = nv/4L (open at 1 end) • Beat frequency = fbeat = f1 - f2

  21. Summary: Doppler Effect • Relative motion together produces an increase in frequency • Relative motion apart produces a decrease in frequency f’ = f ( v±vD / v±vS ) • For light: u = (Dl/l) c

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