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,  = 90 dB  = (10 dB) log (I/I 0 ) I = I 0 10 (  /10) I = I = 1X10 -3 W/m 2 We need to relate I to s m : I = ½  v  2 s m 2 s m =  Air density =  = 1.21 kg/m 3  Velocity of sound = v = 343 m/s

PAL #10 Music (cont.) s m = (I/(½  v(2  f) 2 )) ½ s m = (1X10 -3 /(½)(1.21)(343)(2  440) 2 ) ½ s m =  Even the loudest sounds only produce very small motions  What if the distance is doubled?  Since I = P s /4  r 2, then  but s m => √I, so  The displacement is ½ as great

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 

Stationary Source

Moving Source

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 

Doppler Effect

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

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

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

General Doppler Effect  We can combine the last two equations and produce the general Doppler effect formula: f’ = f ( v±v D / v±v S )  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

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 

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 = (  ) c  c, the speed of light in vacuum, is constant (3 X 10 8 m/s)

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 

Red Shifted Spectrum

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 

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/  ) ½

Summary: Wave Equations  The equations for the amplitude and pressure of a sound wave are: s = s m cos (kx-  t)  p =  p m sin (kx-  t)  p m = (v  ) s m  Waves from two sources will interfere based on the path length difference between the sources and detector  L = m (fully constructive)  L = (m+½) (fully destructive)

Summary: Intensity and Music  The intensity of sound falls off with a inverse square law: I = P s /4  r 2 I =½  v  2 s m 2  The sound level is:  = (10 dB) log (I 0 /I)  Harmonic frequencies of a pipe f = nv/2L (open at 2 ends) f = nv/4L (open at 1 end)  Beat frequency = f beat = f 1 - f 2

Summary: Doppler Effect  Relative motion together produces an increase in frequency  Relative motion apart produces a decrease in frequency f’ = f ( v±v D / v±v S )  For light: u = (  ) c