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Simple Doppler effect

  • To: mathgroup at
  • Subject: [mg120662] Simple Doppler effect
  • From: David Harrison <david.harrison at>
  • Date: Wed, 3 Aug 2011 07:05:00 -0400 (EDT)
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I am trying to generate a Doppler effect for a sound wave. The source is moving by the observer.  The observer is stationary relative to the air.  The results I'm getting are bizarre, and 2 physicist colleagues are as stumped as I about what simple stupid mistake I'm making.  Here is the code that initialises the variables:

vSound = 343; (* speed of sound *)
vSource = 50; (* speed of source *)
xObserver = 300; (* x coord of observer *)
yObserver = 50;  (* y coord of observer *)
fSource = 440; (* frequency of source *)
 The source is initially at x = 0.
 The source is a y = 0 always. 

Then I define a function of the position of the source as a function of time:

x[t_] := vSource * t

Now define the Doppler frequency.  It depends on the cosine of the angle between the source and the observer. I'm careful to use the cosine in a way to avoid possible singularities.

fDoppler[t_] := Module[ 
  {xPosn, r },
  xPosn = x[t];
  r = Sqrt[ (xObserver - xPosn)^2 + yObserver^2]; 
  fSource * vSound / ( vSound - vSource * (xObserver - xPosn) / r)

Finally, play what the observer hears, ignoring the fact that the intensity of the wave increases as the source is approaching and decreases as it is receding.

Play[ Sin[2 Pi fDoppler[t] t], {t, 0, 12}]

The sound starts at the higher Doppler frequency, then dips to a very low frequency as the source is at closest approach to the observer at 6 s, then rises to the lower Doppler frequency!  Weird.  I expect that the frequency around t = 6 s to smoothly decrease from the higher Doppler shifted frequency to the lower Doppler shifted one. 

In order to "see" what is going on, make a plot for the frequency of the source = 1.

fSource = 1;
Plot[ Sin[2 Pi fDoppler[t] t], {t, 0, 12}, PlotPoints -> 100000]

At t = 6 you can see the same bizarre behaviour.

If you can point out what is going on here, you can show how stupid I (and two colleagues) are.

-- David Harrison

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