Friday, May 11, 2007

Doppler

The Doppler effect is an effect that we encounter in our lives rather often: When an ambulance or any vehicle with a siren approaches us, we can hear the pitch of the sound (i.e. the frequency of the sound) increase as the vehicle approaches, reaching a peak frequency when it is directly beside us, and then taking a swift plunge to a lower frequency when it recedes from us.

Let's have a look at what's going on. So far in class we have only been looking at stationary sources, like in a ripple tank. The source of the wave, i.e. the dropper or the damper in the ripple tank does not move with respect to the wave. It just taps the water surface, that's all.

The Doppler effect comes into play when the source begins to move. Here's a diagram to give a quick illustration:


You can see very clearly the bright spot is the where the source of the wave is, and the source is moving towards the left. As the source moves, you can see how each wave produced is centred slightly to the left, and so the wavelength of the waves on the left hand side is comparatively smaller to the wavelength of the waves on the right.

If you happen to be standing somewhere on the left, you would hear (if the waves were sound waves) a shorter wavelength sound, i.e. higher frequency sound, since the speed of sound is rather fixed in air. As the source passes you, and you suddenly end up to the right of the source, the frequency takes a plunge. This explains the ambulance approaching and receding you perfectly.

Now what is so splendid about this? A quick flick of the mathematics wand will give you some rather involved equations that in principle are quite simple, but in reality some pretty tedious stuff. Anyway, the cool thing about this effect is that it leads to some wonderful applications, such as in the radar.

As you probably can remember (or should remember), radar works by shining radio waves in all directions, and waiting for the reflected radio waves to come back to produce a blip on the screen. As the radio waves arrive at the airplane, they reflect off the surface of the moving airplane, which gives the effect of a moving source. The Doppler effect in this case will actually increase the frequency of the approaching aircraft, and a quick calculation (done preferably by the computer) gives us the speed of the airplane. This procedure is also used by the traffic police to determine the speeds of vehicles.

This is also how missiles decide when you to explode when they are in midair. Contrary to popular belief, missiles do not go off when they come into contact with the plane. Instead, they usually have a radar of their own built into the missile. As the missile approaches the plane, the frequency increases, and then suddenly drops when the plane starts to recede. This is one of the signals for the missile to detonate itself.

A common occurrence when jet planes fly past is the sonic boom. This is the Doppler effect at its extreme: the source of the wave is moving at pretty much the same speed or faster than the wave itself:

You can see that the waves are unable to move away from the source, because the source is travelling at the same speed as the waves being produced, and by the time the waves reach the observer, all the waves will simultaneously arrive and be heard, which would probably make for a rather loud sound.

Finally, I've been talking so much about sound, sound and sound, but the Doppler effect actually applies across the board to all forms of waves, including light waves.

One of the most important discoveries made in the 20th century was that galaxies and other distant objects in our universe were actually moving away from the Earth. You have to realise that before the 20th century at the advent of advanced astronomy techniques, many people held a firm belief in the static nature of the universe. Scientists thought that the universe was eternal and never changing, until a few brilliant ideas (see Olber's Paradox) and the Doppler effect observed in light helped us realise that the universe was expanding.

Using spectroscopic techniques, scientists back then were able to deduce what elements were present in gas clouds in space. When the light arriving from these gas clouds were dispersed by a prism (split up into the rainbow colours) astronomers were able to see certain missing colours, which corresponded to certain elements.

They soon realised that many of the lines did not appear where they were supposed to, but rather they all seemed to be squashed towards the red side of the spectrum:

This meant that when the light arrived on Earth, the wavelengths were longer than expected. If you observe the Doppler effect carefully, this would mean that the object that was being studied was moving away from the Earth. Astronomers called this phenomenon the redshift, because the spectral lines were all shifted towards the red side of the spectrum.

What was even more incredible was the fact that this redshift became more pronounced the further away astronomers looked, which suggested that the further away an object was, the faster it was moving away from us! An excellent astronomer named Edwin Hubble (of Hubble Telescope fame) was the discoverer of the law in 1912 (known as Hubble's Law) that the speed v at which something in our universe was receding from the Earth, is directly proportional to the distance d from the Earth:


Where H0 is the Hubble's constant. Let's consider a galaxy that is a distance D from us right now. Here's something amazing: since we believe that everything began with the Big Bang which basically says that everything in our universe started from a single point, the distance D is very roughly the distance travelled by the galaxy since the time when the galaxy and the Earth were still together at the start of the Big Bang. The velocity of the galaxy, v, let's assume to be constant. If we take D divided by v, we should get the time taken for the galaxy to go from the Big Bang to where it is now, i.e. we would roughly get the age of the universe!

So from the equation, t = D/v = 1/(H0). From astronomical observations, we do have a rather good idea of the value of H0, and it turns out that our estimate of the age of the universe from this method is about 15 billion years old.

Incredible? I hope you think so!

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