The physics of sound in air, from a statistical point of view (part 1)

This is the first part of a study on the statistical nature of sound.

Considering the phenomena of sound propagation at the very basic level, I think, is the best way for me to understand the very nature of sound. I'm planning on investigating the nature of sound in gases at standard temperature and pressure, yet resorting to as little math as possible (to keep things as clear as I can). I think surprisingly many attributes of sound, which otherwise might be difficult to grasp, can be explained from a statistical point of view (this area of physics is called statistical mechanics). And then there's always the saying "If you can't explain it simply, you don't understand it well enough". So I shall give it a try!┬áSimplicity can also be relative, something to keep in mind ­čÖé

An ideal gas

I'll start from the assumption that sound propagates in an ideal gas. I've found that the concept of an ideal gas is best understood through the kinetic theory of gases.

The movement of molecules in an ideal gas, according to the kinetic theory of gases (source: Wikipedia)

By viewing the great picture provided by Wikipedia, one can see that an ideal gas consists of particles (molecules) which are constantly in motion. By isolating the system (like in the picture) and assuming that no energy is lost in any of the collisions, the particles will continue colliding forever. This seems like a starting point which is relatively easy to understand.

A small portion of air, like in the box, contains a certain amount of kinetic energy, distributed throughout the particles in the form of different velocities. The velocities of the particles in an isolated system will follow a statistical distribution called the Maxwell speed distribution. I will not investigate the derivation of this distribution any further, and just assume it holds true.

Probability distribution of the velocity of a single particle

Probability distribution of the velocity of a single particle

We can illustrate the statistical distribution further by doing a plot of the most probable velocity a random particle will have in the system. We place the particle in the middle, and consider the color surrounding it as describing the most probable place to draw a velocity vector, with red corresponding to the most probable choice and purple the least probable choice.

There are a few familiar ways to measure the amount of kinetic energy stored in the isolated particle system:

  • pressure, which describes the average force per unit area, caused by the particles colliding with the walls of the container.
  • temperature, which describes the average kinetic energy of the particles.
Probability distribution of the velocity of a single particle with an average particle velocity to the left

Probability distribution of the velocity of a single particle with an average particle velocity to the left

There's also another really important concept  here which relates to acoustics; the average velocity of the particles. In the case of the isolated system above, the average particle velocity will be 0, as the velocities are equally distributed in all directions. But if the box was moving relative to some fixed point, the average particle velocity would differ from 0, and instead be equal to the velocity of the box. This is illustrated in the figure to the right, where the added average velocity simply shifts the probability distribution to some direction by the amount of the average particle velocity. The average velocity is relative; it's fully dependent on the point of reference.

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