Could "0 1 0 0 0 0 0" actually represent a wave? That looks like a compression without a rarefaction. I thought that the 'cycle', a compression and rarefaction, was a necessary characteristic of a sound wave. Can you have just a compression or just a rarefaction?
Well, I would certainly call it a wave, but let's avoid any potential terminological debates and say the following:
- The above disturbance could be created in air.
- The disturbance would be propagated and described by the wave equation.
I used that simple example because I wanted to demonstrate the behavior of the wave equation. In practice, both a compression and rarefaction will usually occur because total matter has to be conserved. If I want to compress the gas in one place, then I must take particles from elsewhere.
honestrosewater said:
Which wave equation did you mean? The only 'wave equation' I understand is v = \lambda f. I understand some of its concepts, but I don't speak calculus, so I can't really read the others.
Unfortunately, the wave equation I'm referring to does involve calculus, but let's see if we can step through it. For simplicity, we'll continue to work in one dimension:
\frac{\partial^2f}{\partial^2x}=\frac{1}{v^2}\frac{\partial^2f}{\partial^2t}
What this equation basically says is that the shape of the function in space is equal to the shape of the function in time multiplied by a normalization constant (it's a little more complicated than that, but that's the basic idea). To visualize this, imagine you're in a boat on a flat lake and there's a wave coming at you. If the wave were governed by the above equation, then the following two things would be equal:
1) Sitting motionless and letting the wave pass under you (it's moving at speed v).
2) Freezing the wave and passing over it at speed v.
Notice that I didn't say anything about the character of the wave. It could be a single hump, a wiggle, or a sine function. The point is that the shape is maintained as it moves forward. In practice, water waves are not well described by the above equation, but sound waves are, so let's move on to the issue at hand...
What are the minimum requirements that a disturbance must meet in order to count as a sound wave?
I'm sure it depends on who you ask. I would say that any disturbance that satisfies the wave equation could be called a wave, but these things aren't written in stone. Some people would probably require, as you say, at least a single cycle.
Sheesh, sorry, I have a lot of questions, but I'll try to make the rest easy to answer.
1) In
0 1 -2 4 0 0 0 0
the numbers represent amplitude?
The numbers are meant to be representations of the density relative to the ambient ("normal") value. However, they could just as easily represent pressure or temperature.
2a) Is amplitude measured from ambient density (or pressure*) to the peak of the compression (or rarefaction)?
Amplitude is traditionally used to refer to a property of simple waves, like sine waves, where the symmetry makes the definition is unambiguous. In the case of sound, they usually talk about the "root mean square" amplitude, which is the square root of the average of the square of the deviation from the ambient value. A mouthful, I know. In your case:
0 -1 3 0 0 0
this would be given by:
A_{rms}=\sqrt{\bar{A^2}}=\sqrt{\frac{(-1)^2+3^2}{2}}=\sqrt{5}
*2b) Can I use density and pressure interchangeably?
Well, they're not numerically equal, but they both are governed by the wave equation.
3) Could
0 -1 3 0 0 0
be a wave? That is, assuming amplitude is measured from ambient pressure, must the amplitudes of the compression and rarefaction be equal?
I would give the same answer as above -- it depends on who you ask. I would say it could be a wave (and so would scienceworld), but I've seen a variety of definitions.
4) Could
0 -2 2 0 0 0
be traveling in either direction, i.e., must rarefaction follow compression, or can rarefaction precede compression?
It can go either way. Both are solutions to the wave equation, but are distinguished by the initial conditions.
5) How fine-grained are the changes in pressure? Could I safely expand
0 -2 2 0[/color] 0 0
into, for instance,
0 0 0 0 -.5 -1 -1.5 -2 -1.5 -1 -.5 0 .5 1 1.5 2 1.5 1 .5 0 0 0 0[/color]
In theory, it could show any amount of small scale variation (down to intermolecular distances). This goes back to the graph with which you started the thread, because it's the spectrum that tells you how fine-grained it is. If the spectrum shows a lot of power at high frequencies, the expansion will likely be more like:
0 0 0 -0.5 -1.5 -2.5 -1.5 -2.5 -1.5 -1 0.5 -0.5 1 1.5 2.5 1.5 2.5 1.5 2.5 0 0 0 0[/color]
6) Does the pressure change, increasing or decreasing, in a predictable way? Perhaps at a certain rate depending on frequency and wavelength?
It depends on the circumstance. Musical notes are very predictable, while a talking crowd would likely produce a very erratic sound.
7) I may be able to figure this out, but I guess I may as well ask. If you know the speed of the wave and the ambient pressure, do you need to hear an entire cycle in order to determine the frequency, wavelength, and amplitude, or can you determine them from only a segment of a cycle? What is the smallest segment that you need?
That's an excellent question. I don't know how sophisticated the software in our ear is (you should ask in the biology forum), but in astronomical systems, we regularly infer periodic motions from only a small portion of a cycle. Making this inference does, of course, require assumptions, so we could easily interpret something as having a certain frequency when it really doesn't!
Anywho...
Yes, that makes sense.
) thanks for taking the time to answer. I think I'm comfortable enough now that I can move on and just learn more as needed.
