Alas, there's little actual mathematics in this episode, so we'll explore the next best thing: some physics. In particular, we'll look at the Doppler effect, what it is, when it occurs, and how we can use it.

A light wave (or electromagnetic
wave, as it's
properly called) differs from other types of waves in that it
does not require a medium in which to propagate: electromagnetic
waves travel through empty space, while for example sounds waves
can only travel through a gas, liquid, or a solid. Another,
related peculiarity of electromagnetic waves is that they act
both like waves and like particles! This puzzling phenomenon
has motivated much of the development of modern physics and the
discovery that *all* particles have
wave-like properties.

A traveling wave, whether occurring in water (e.g. sea wave), air (sound
wave), or vacuum (light wave), looks something like this
in action (courtesy of wikipedia). You can see a snapshot
representation of a wave to the left below. The distance
between two consecutive peaks or troughs is called the
*wavelength* is usually represented by the Greek letter lambda
(λ). The time it takes for a wave to travel one wavelength, the
distance λ, is called a *period*, and we'll denote it by
T. In the picture to the right below, a source is
producing a continuous wave of a given wavelength. The
circles represent the peaks of the wave, the distance between them
equaling the wavelength.

- What is the relation between a wave's speed, wavelength, and period? (Hint: it's not hard, try. If you're stuck, look here.)
- The quantity 1/T is called the
*frequency*of the wave. It corresponds to the pitch we hear when a sound wave hits our eardrums and the color we see when a light wave hits the retina. Rewrite the above relation in terms of frequency instead of period.

Suppose now that the source of the wave is moving toward you, as in the picture in activity 2 below. While the source is still emitting a fixed frequency wave, the perceived wavelength now depends on where the observer is located with respect to the source.

- Assume the source is moving toward you, the observer, with
speed v
_{0}, and the wave emitted is a sound wave of frequency f. Then the corresponding wavelength λ, by activity 1 above, is equal to v_{s}/f, where v_{s}is the speed of sound in air (around 345 meters/second). What does the perceived wavelength λ' equal to in terms of v_{0}, T, and λ? - What does λ' equal to in terms of the speed of sound
v
_{s}and the*perceived*frequency, which we'll denote f’? (Hint: use activity 1.) - Write the equation from part 1 above in terms of f,
f’, v
_{s}and v_{0}.

If you did the algebra correctly, you should have gotten something
equivalent to

for #3 above. What this tells us is that perceived sound frequency, the
pitch, increases relative to the original frequency as v_{0}
increases.

- Suppose that instead of moving toward you, the source of the
sound moves away from you with speed v
_{0}. How does the equation for the perceived frequency f’ change? - What happens if the sound source is stationary, but you, the
observer, are moving toward it with speed v
_{0}? Draw a picture, work it out! - What if the sound source approaches you with a speed greater
than v
_{s}? I.e. what if v_{0}≥ v_{s}?

- Suppose you're standing by a sound source which emits a a single frequency tone for a split second. How fast will you need to run away from the sound source to hear the tone for as long as you keep running? (Ignore the issue of volume, which corresponds, by the way, to a wave's amplitude or height.)
- The orchestra has just finished playing Beethoven's 5th symphony, and you, a member of the audience, suddenly feel a strong urge to hear it again, but backwards! How fast would you have to run away from the concert hall to hear it replayed from finish to start at the pitch it was originally performed?

As you may have noticed, we have so far ducked the original question! All of the above discussion does not explain why we hear the pitch of a siren change as it passes us. The reason is that we, the observer, are not in the path of the sound source as that happens: the ambulance passes by us, not through us (one may hope). Thus while the pitch did not change for the observers in the diagram in activity 2, it would if the observers were not on the line of motion, as in the illustration to the right. The key to describing how the perceived frequency changes is to note that we only care about the speed of the source relative to the observer. Thus the only modification needed is to figure out what this speed is in the case where the observer is not on the line of motion (see illustration below).

- Recall a bit of trigonometry: in a right triangle to the right, what is length of side a in terms of angle θ and the hypotenuse c?
- As in the picture above, the sound source is moving West
with speed v
_{0}. What is the speed of the source relative to the observer at an angle θ North of it? (Hint: use part one and see illustration above.)