Pressure and Displacement sound waves. Pressure always lead?

1. Jan 6, 2010

FireBones

A sound wave can be interpreted in any of four ways. Two of those involve the correspondence between (1)displacement and position and (2)pressure and position.

[By "position" here I refer to the distance from the source. Positive refers to going away from the source.]

These waves are 90 degrees out of phase. My questions are:
1) Is it always the case that the pressure wave "leads" [it attains a maximum a quarter period before the displacement attains a maximum.]

2) If it is not always the case, what determines whether the pressure leads or lags?

An important correlation to pressure leading displacement is that velocity is always negatively proportional to pressure. In other words, a particle is always moving fastest forward in rarefaction and fastest backward in compression. Is there a slick intuitive reason for this?

I'm wondering if any of this depends on the maximum displacement speed of a particle versus the speed of the wave itself.

A reference from page 477 of Physics for scientists and engineers: http://books.google.com/books?id=up...re wave time equilibrium displacement&f=false

2. Jan 6, 2010

FireBones

After doing some thought experiments with this, I think it is not hard to show that the key question is whether the sound wave began by something pulling back [causing an initial rarefaction] or pushed forward [causing an initial compression.]

Initial compression -> Pressure leads displacement
Initial Rarefaction -> Pressure lags displacement

3. Jan 7, 2010

FireBones

Alright, after more thought I realized the above is wrong.

I've figured out that pressure always leads. Here is why:

For ease of discussion, imagine a very long pipe with massless paper partitions. These partitions will vibrate with the same frequency as the surrounding air particles. The pressure at a given partition is, more or less, inversely proportional to the difference in positions between the partition in front of it and the one behind it. At rest, this distance is 2. If the distance is less than 2, the pressure is higher than normal; if the distance is less than 2, the pressure is less than normal.

The key point is that there is a lag in the vibrations of these papers. The air particles closer to the source began their vibrations slightly before the ones further away. This means that the paper to the left of a given paper is slightly ahead in its sinusoidal rhythm than the paper to the right.

Imagine the point in time when a given paper has moved as far to the right as it is going to move. It will begin moving back to the left. The paper to its left has ALREADY begun moving to the left while the paper to the right is still moving a bit to the right and will soon begin moving to the left. For the next quarter phase the general relationship will persist that the paper to the left is moving slightly faster to the left than the central paper, which is moving slightly faster to the left than the right-most paper.

Thus, the pressure will be least at the end of this quarterphase because the paper to the left has been moving away more and more and the paper to the right has not been moving as fast, so it is falling more and more behind.

At the end of this quarter phase, the paper to the left will begin slowing down, so we know that this point [which is the equilibrium point for the central paper) is the point with least pressure.

Conclusion: The rarefaction for a given point always occur when the point is on its reverse journey. [Similar argumentation shows that the highest pressure always occur at the equilibrium point where a paper is moving forward.]

This conclusion prescribes that the pressure wave always leads the displacement wave because that is equivalent to the _VELOCITY_ wave being 180 degrees out of phase with the pressure.