Pressure and Displacement sound waves. Pressure always lead?

In summary, a sound wave can be interpreted in four ways, with two involving the correspondence between displacement and position, and pressure and position. These waves are 90 degrees out of phase and the pressure always leads the displacement. The velocity is negatively proportional to pressure, and this correlation is dependent on whether the sound wave began with something pulling back or pushing forward. The explanation for this is that there is a lag in the vibrations of the particles, causing the pressure to be least at the equilibrium point and highest at the points of maximum displacement. This means that the pressure wave always leads the displacement wave, as the velocity wave is 180 degrees out of phase with the pressure.
  • #1
FireBones
103
0
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
 
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  • #2
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
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.
 

1. What is the difference between pressure and displacement sound waves?

Pressure sound waves are characterized by a fluctuation in air pressure, while displacement sound waves are characterized by a movement of air particles. In other words, pressure waves involve a change in the density of air, while displacement waves involve a change in the position of air particles.

2. How do pressure and displacement sound waves travel through a medium?

Both pressure and displacement sound waves travel through a medium as longitudinal waves, meaning they propagate in the same direction as the wave motion. As the wave travels, air particles are compressed and expanded, creating areas of high and low pressure that correspond to the wave peaks and troughs.

3. Why does pressure always lead in a sound wave?

This is because of the nature of sound propagation. As a sound wave moves through a medium, it causes air particles to compress and expand. This compression leads to a buildup of pressure in front of the wave, while the expansion causes a decrease in pressure behind the wave. Therefore, the pressure always leads the displacement in a sound wave.

4. Can pressure sound waves and displacement sound waves occur simultaneously?

Yes, it is possible for pressure and displacement sound waves to occur simultaneously. In fact, this is how most sound waves behave in reality. While the pressure wave is leading, the displacement wave is following closely behind, creating the characteristic oscillating pattern of a sound wave.

5. How does the amplitude of a sound wave affect its pressure and displacement?

The amplitude of a sound wave, which refers to the maximum displacement or pressure of the wave, directly affects the intensity or loudness of the sound. A larger amplitude means a higher intensity and thus a louder sound. This also means that the pressure and displacement of the wave will be larger for a higher amplitude sound wave.

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