Suppose that a moving sound source emits a sound of frequency ƒs. Let v be the speed of sound, and let the source approach the listener or observer at speed vs, measured relative to the medium conducting the sound. Suppose further that the observer is moving toward the source at speed vo also measured relative to the medium. Then the observer will hear a sound of frequency ƒo given by:

ƒo = ƒs( v + vo ) / ( v - vs )

If either the source or the observer is moving away from the other, the sign on its speed must be changed.

When the source and the observer are approaching each other, more wave crests strike the ear each second than when both are at rest. This causes the ear to perceive a higher frequency thatn that emitted by the source. When the two are receding, the opposite effect occurs; the frequency appears to be lowered.

Because v + vo is the speed of a wave crest relative to the observer, and because v - vs is the speed of a wave crest relative to the source, an alternative form is

ƒo = ƒs · crest speed relative to the observer / crest speed relative to the source

While I was at University, I took part in a project to create some toys to demonstrate physical principles like the Doppler effect to children. I came up with this neat little toy called the Doppler Ball, which I built and demonstrated (this little sucker made the difference between graduating and having to repeat my final year).

When I was doing my PGCE training I actually used it in anger, and it worked quite well, so I present here a construction and user's guide for anyone who's interested.

Doppler Ball Construction Guide

This guide will enable you to construct your own Doppler Ball. In the interests of safety it should be noted that construction of the Doppler Ball involves the use of a soldering iron and sharp knives, and therefore school students should only construct the Doppler Ball under supervision.

Introduction

The Doppler Ball is a simple toy that can be used to demonstrate the Doppler Effect. It's a simple foam ball with a buzzer inside. When you throw the ball around, the motion distorts the sound produced by the buzzer, thereby demonstrating a Doppler shift.

Materials and Tools needed

• 1x sponge ball, between 15 and 20 centimetres in diameter
• 1x 9 volt buzzer (the ideal frequency for this is 60Hz, so get one at this frequency if you can)
• 1x 9 volt battery
• 1x Batter clip
• 1x Latching switch (the kind that you depress once to close the circuit, and again to open it)
• Connecting wires
• A sharp knife or scalpel
• Masking tape (or equivalent, eg gaffer tape)
• Soldering iron
• Glue (must be suitable for porous materials, e.g. PVA)
Estimated cost: £5.00

Instructions

1. The first thing to do is build the buzzer circuit. To do this, user the solder to connect the battery clip, switch and buzzer in series, making sure the connections are strong. The completed ball will need to endure a fair amount of stress. You may wish to mount these parts on a piece of plastic.

2. Place the battery in the clip and test the circuit.

3. Wrap the components together into a tight bundle and seal with the tape. You will need to make sure the switch is in a prominent position and can be accessed easily.

4. Next, prepare the sponge ball. To do this, cut a slit into the ball large enough for the buzzer circuit to be pushed through. The slit should go down to roughly the centre of the ball.

5. Excavate some of the foam at the centre to make a cavity for the circuit.

6. Place the circuit inside the ball and fill in any space with the foam you previously removed. The circuit shouldn't be allowed to move about too much.

7. Mark the position of the switch on the outside of the ball with a marker pen. You can then push on this spot to activate the buzzer.

8. Seal the slit using a suitable adhesive. When it's dry, the Dopper Ball is ready for use.

Using The Doppler Ball

You can use the Doppler Ball in a number of ways. One of the most effective is to tie a cord around the ball, and swing it around. As you increase the frequency of the swing, the Doppler shift observed will increase.

Alternatively, you can have two students (who are good throwers) stand a distance apart and throw the ball to each other. If all goes well, the thrower should notice a decrease in the pitch, and the receiver will notice an increase.

Project Contributors: Gavin Brown, Stuart Butler, Andy Franqueira, Pat Hargreaves, Gareth Scaddan, Adam Stow

The sound started almost right away.

Faint, very faint, just a whine in the air from a few miles away. Very familiar too, living in the city. You hear it all the time. I didn't give it a thought.

It was cold, at first. Deep cold, heavy, cold like damp, but not, a temperature that confuses the skin. Like clothes lined-dried in winter. You slip into your denim jeans regardless--figure the heat from your body will dry them the rest of the way, go out, walk around, squirm, tug, put your hands in your pockets to test the progress, rub your thighs to warm them up. Friction.

My jeans were not entirely dry when I put them on this morning; I don't think they are now. I cannot reach my pockets.

Two dozen pairs of feet in my field of vision. Boots, sneakers, loafers, a range of designers, styles, prices. Those complicated Nikes with the spring-heels, I've always wondered if they really felt any different than just shoes, but never had the nerve to try on a set; I'm too short, I don't exercise. My lifestyle must be consistent with my footwear, not the other way around. I like those Kenneth Coles, they look like a cross between wing-tips and bowling shoes. I've seen them in ads on the subway. That woman has nice legs.

They've all come out to watch. Rich and poor, shoulder-to-shoulder. What a way to break down the class barrier! But I was never really a Marxist.

Not everyone is stopping. Behind the treeline, there's intermittent movement. The pause, turn, turn back, move away. That would have been me. Background actor, an extra with a walk-through in the last scene. Man #4.

Now I'm the star.

It's getting colder.

"What happened? Did you see?" Someone said, one of the women, I think, her voice was in its forties.

"Nah--"

"I don't know."

"I missed it."

"His shit is fucked up."

Too many voices now. I want to hear an explanation; I hadn't seen it either, am curious to know what happened myself. But it becomes a noise, a cacophany. Cacophany was on my GRE test.

There's something under it--a distinguishable sound, a pitch cutting through the human clutter, gaining intensity, getting closer. Ah. It's coming... Below black pants, black shoes appear, their heels facing me, smooth round heels against the field of laces and buckles, brass zippers and velcro flaps. The sound's getting louder and louder. I'd heard it before, though. One hears it all the time. It comes, then goes, makes a reflection of itself, if you're the mirror and can look both ways, out at what's coming and into yourself. Bad analogy. Simile or metaphor? Metaphor, remember, before the GREs, the SATs; before them, progress tests, then aptitude tests, then math quizzes, spelling bees, matching colors, putting the right shaped block in the right shaped hole. All those examinations suddenly becoming clearer; wait, they'll haze out again, dissipate into a universe of memory. The noise will go away. It always does.

I definitely think my pants are wet. They've moistened the bottom of my shirt--I don't usually tuck in my shirt--I can feel the fibers expanding with...dampness, clinging to me, crawling up my spine, shocking cold, you would not think it could be this cold, but warm air rises, doesn't it? Better to be standing. Definitely better to be standing, looking on. Not being looked upon. I can't see their eyes, it makes me nervous. I'm getting very nervous, and no one seems to want to touch me.

I would rather they would touch me.

It's getting hard to hear them. I can't really hear them, but I sense they must be speaking. The noise has gotten very close now. It's ringing in my ears, driving itself into my brain. I can't see the source yet either. It's approaching me from behind, I guess. I wish I could see it coming. I would feel better if I could see it coming.

It comes then goes. Always comes then goes, this familiar sound; the waves compress upon approach, build at a speed relative to the conducting medium. Air, I guess. Just air. Oxygen, nitrogen, carbon dioxide, some other other stuff too. Would it take longer to get here in Alaska? Or Los Angeles? Not just air. Time, too. Time is involved, it comes through time. Distance equals rate times time. It's more complicated than that, though. Than DiRT. I remembered it as dirt. Lots of dirt now, I think I must have swallowed some; I'm coughing.

All the feet are starting to back away. The black shoes are pushing them away. The noise is at a climax now. It's terrible. I've never heard it this loud for this long. I can't hear anything else, there's nothing else, I can't even see, the sound's so overwhelming. I can't get away from it, no matter how I try to twist away. So loud and clear.

And familiar. Familiar, on top and underneath. It's--I think it's compressing my chest. I recognize this. From before the city, I recognize--what is that I can hear? I've heard it before, faintly...faintly, a whine in the air from years and years away...I know I've heard it before, getting louder, getting closer.

The sound started right away.

It will go away from all those shoes. The source will move away from them. The sound will diminish, then come back, crescendo, decrescendo. Crescendo at the last. Break the set. Defy expectation. I had a pair of shoes. I think I was knocked out of them. I can see one. It's untied. No matter.

It's quiet now, and dry.

There is also a Doppler effect for light, that is partly responsible for the red shifts (and corresponding blue shifts) that astronomers observe in distant objects moving away from us (and moving toward us). The Doppler effect for sound varies when it is the source, observer, or both are moving, apparently violating the principle of relativity, which states that the only thing that should count is the relative motion between the source and the observer. However, because sound waves only occur in a material medium like air or water, this material medium serves as a frame of reference against which the absolute motion of the source and observer can be measured. For light however, once theories of a luminiferous ether were discarded, no such medium could be involved and so the Doppler effect for light must be very different from that for sound.

The equations governing the Doppler effect for light may be easily derived by special relativity. We can think of a light source with frequency f as a clock that ticks f times every second and emits a wave of light every time it ticks. There are three cases we need to consider: (1) when the observer is moving perpendicular to the light source, (2) when the observer is receding from the light source, and (3) when the observer is approaching the light source.

1. For the perpendicular case, the proper time is t0 = 1/f between one tick and the next. However, time dilation causes, from the point of view of the observer, the time between ticks to increase, and hence causes the frequency to decrease to: f' = f*sqrt(1- v2/c2), where v is the velocity of the observer, and c is the speed of light.

2. For the receeding case, the observer moves a distance vt away from the source between ticks, meaning that a light wave takes vt/c more time to reach the observer than the previous wave did. Hence, the actual time between the arrival of successive waves becomes:

T = t + vt/c = t0(1 + v/c)/sqrt(1 - v2/c2) = t0*sqrt((1+v/c)/(1-v/c))

and the frequency the observer sees thus becomes:

f' = 1/T = f*sqrt((1 - v/c)/(1 + v/c))

The observed frequency f' is thus smaller than the original frequency f, producing a red shift.

3. When the observer is approaching the source, each light wave takes vt/c less time than the previous wave, so following a similar derivation as for the receding case, we have the following formula:

f' = f*sqrt((1 + v/c)/(1 - v/c))

This results in a higher frequency than that emitted, or a blue shift.

Note that, unlike the Doppler effect for sound, it makes no difference in any case as to whether it is the observer that is moving and the light source is stationary or the light source moving and the observer stationary. All that matters is the relative motion between source and observer. Verification of this fact is left as an exercise for the reader.

This phenomenon occurs for all forms of electromagnetic radiation, including radar and radio waves, and is used in Doppler radar sets such as those used by the police to catch speeders, and Doppler shifts in radio waves transmitted from a constellation of satellites forms the basis for some satellite navigation systems like the old Transit marine navigation system. As previously mentioned, it is also partially responsible for the red/blue shifts observed in astronomy, but at cosmological distances, more accurate formulae based on general relativity should really be used (thanks to tdent for pointing this last one out). See here for the proper derivation of the cosmological redshift.