Every Halloween, a steady stream of trick-or-treaters visits my house. They’re looking for handouts, of course: Snickers, Pixy Stix, Reese’s, jawbreakers. A simple mathematical model of the children’s visits can help explain the Doppler effect.
Let’s say the children are dressed as zombies (zombies are all the rage). Let’s further assume that the children act like zombies—not from premature candy consumption and subsequent hyperglycemia—but from a desire for greater verisimilitude. Thus, we make the following assumptions:
- The children move at a uniform speed v.
- The children are separated by a uniform distance λ.
With what frequency f do the children visit my house?
It’s obvious that a greater speed v means that that children visit more often. So:
f ∝ v,
meaning that frequency is proportional to speed. Further, it seems clear that a larger distance λ between children means less visits, so that
f ∝ 1/ λ.
It is logical to take these proportionalities and combine them, giving
f = v/ λ.
In the study of waves, this is the fundamental relation between frequency and wavelength. We see that in this analogy, the zombie children are meant to represent successive peaks of a wave, and the distance between the children represents wavelength.
(I haven’t defined what a “wave” actually is. If you’re curious, a wave is something that is periodic in both space and time. The space periodicity of the zombie kids is codified by the number λ, since if you travel a distance λ in space, you get another zombie kid just like the first. The time periodicity is codified by the period T = 1/f, since if you’re at the house and you wait for time T, an identical zombie kid will show up at your door.)
So, what about the Doppler effect?
We now imagine that I live in a mobile home. ( I don’t, actually, but I was injured in a tornado in 2011, so I guess I am an “honorary” mobile home denizen.) What is the effect of me driving the mobile home either towards or away from the zombie kids?
Suppose I drive towards the kids at a gentle 1 m/s. If they are walking 2 m/s towards me, our relative speed is 3 m/s. The kids show up at my doorstep more frequently.
If instead I drive away, the kids aren’t visiting as often, since our relative speed is now just 1 m/s. In fact if I drive away at 2 m/s or greater, the kids never catch me and f drops to zero.
The same thing happens with sound. Regions of less dense/more dense air propagate from a source to a receiver (presumably, your ears). Each “pulse” is analogous to a zombie kid, and the frequency with which the pulses jostle your eardrums is interpreted by your brain as a pitch. Higher frequency, higher pitch. Now, in air the speed of sound is roughly constant, so the f that you hear depends upon one thing: the wavelength. The more separation between pulses, the lower the pitch you hear.
You can guess what comes next. If you run away from a sound source, the pulses can’t hit your eardrums as often; you hear a lower pitch. Conversely, running towards a sound source makes the pitch sound higher. The f has increased.
Of course its not just the house (the receiver) that can move; the source of the sound (i.e. the zombie kids) can move as well. I can implement this idea next October 31 by installing a moving walkway outside my house. If the kids walk at 2 m/s on my moving walkway, but the walkway itself is set at 1 m/s towards me, then the kids are actually moving at 3 m/s relative to me and will visit more often. (A moving sound source such as a siren sounds higher in pitch if moving towards you.) I could likewise set the walkway to move away from me, and have the kids visit less often or not at all. (A siren traveling away from you sounds lower in pitch.)
This analogy is not perfect by any means. For one thing, the zombie children (who are actual, flesh-and-blood material objects) represent wave maxima, which are mathematical abstractions. In the case of sound, if I shout and you hear it, that does not mean that any actual physical object traveled from me to you. A series of wave pulses traveled, sure; but no individual atoms or molecules went all the way across the room. In a typical wave, energy travels from A to B but matter does not.
If you have trouble seeing how this can be, recall the wave (also known as the Mexican wave) that appears in large sports stadiums. Wikipedia says it best: “The result is a wave of standing spectators that travels through the crowd, even though individual spectators never move away from their seats.” Similarly sound can travel from me to you, even though the individual oxygens and nitrogens don’t really move that far.
Which brings up another idea of mine, which I’d like to patent. Waves (in football stadiums) are always transverse: people raise their arms and then lower them, in a direction perpendicular to the motion of the wave itself. But if we really wanted to model sound with a stadium wave, we should instruct the audience to move their arms from side to side. This would set up a longitudinal wave. I think it would look a little different.
Next time you’re directing festivities for a crowd of 10,000+, get them to do a longitudinal wave. You’ll thank me for it. And if you have kids of an appropriate age, dress them as zombies this Halloween. At my house, zombie trick-or-treaters get the best candy.