Breaking the Sound Barrier

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?

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 
effectively made of a whole bunch of circular wave edges added 
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/