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### Breaking the Sound Barrier

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Date: 10/21/97 at 23:40:51
From: Gabrielle Mills
Subject: Breaking Sound Barrier

This is not really a math question, but I am a teacher and I am trying
to find a way to incorporate breaking the sound barrier into a math
lesson for my third grade class. Do you have any suggestions?
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Date: 10/22/97 at 14:24:49
From: Doctor Tom
Subject: Re: Breaking Sound Barrier

Well, it might be more of a physics lesson, but that's okay - there's
certainly some math involved.

To break the sound barrier, you basically have to go faster than sound
does in the fluid you're moving through (usually, of course, that
"fluid" is air). The speed of sound in air is about 1100 feet per
second, which is tough to demonstrate in a class since it's so fast.

But any time you have waves moving in a medium, they move at a certain
rate. If you take a big pan of water and touch the surface, the water
waves move away forming a set of circular waves, and they move at some
fixed speed that depends on the depth of the water, among other
things.

Next, drag your finger slowly across the surface of the water in a
straight line and look at the shape of the rings. They are no longer
uniformly spaced - those in front of your finger are more closely
packed and those behind are spread out.

If you move your finger a little faster, the waves in front are packed
more and more closely, and when you move your finger at exactly the
rate that the wave moves, the front edge of the waves is no longer
curved - it becomes a cone with straight edges. If you move even
faster, the cone just gets narrower.

The edge of the cone is called a "shock wave" and since it's
together, it has a relatively large amount of energy, and when it hits
things, they are more strongly affected than they would be by the
ripples from just touching the water surface or from dragging your
finger slowly.

Sound is a 3D wave in air, and for a fixed noise, it forms a set of
spherical waves going in all directions. For a slowly moving noise,
the sphere is distorted, just as in the water, and when the noise is
going as fast or faster than sound, it turns to a conical, high-energy
shock-wave that can jolt your ear-drums, or even occasionally break
windows.

It's been a while since I was in third grade, so you'll have to use
your judgement about how much math to put in. I'm pretty sure the kids
can at least understand the explanation above.

High-speed power boats make shock-waves all the time, and if you've
ever gotten hit by the wake (shock-wave) of a big boat when you're in
a little boat, it's obvious that it's different from just having some
ripples pass you by.

It's not hard (given the speed of the waves and the speed of your
finger) to figure out the shape of the v-shaped shock-wave before you
do it.

The same thing can happen with light, but it's a little weird. Of
course nothing can travel faster than the speed of light in a vacuum,
but when light is travelling through certain substances, it does go
significantly slower. Certain particle accelerators can get them going
almost the speed of light in a vacuum, and if you then slam these
particles into glass or something where light goes slower, you can
actually make light shock-waves.

Of course you need a few million dollars worth of accelerator to do
this.

-Doctor Tom,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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