Chapter One: An Acoustics Primer
3. What is Sound? | page 3
Molecular displacement, velocity and pressure
While an overall sound wave propagates out from its origin at the speed of sound, consider the contribution of the single lowly air molecule constituent of this wave, moving first in one direction away from its origin at rest (equilibrium) and then moving in the other through its origin and out to another extreme.* What about its speed? Is it always traveling at the same speed of sound as the overall sound wave? Why no, it is not!
Just as you likely don’t wake up and suddenly move at full speed in the morning before your first cup of coffee or tea (unless you are late for my class), or instantaneously become comatose at day’s end, so the lowly air molecule participating in an idealized sound wave accelerates smoothly in one direction, slows down to a stop and then smoothly begins to accelerate in the opposite direction. Inertia, as minute as it may be, prevents the molecule from suddenly switching directions at a maximum velocity. A sound wave molecule’s changing velocity is demonstrated by the BLUE cosine wave on the graph below. Its oscillating position in space and the overall sound pressure at the point of the molecule is represented by the RED sine wave. The molecule reaches maximum speed as it passes through its origin, and it then gradually slows to a halt as it finds itself in a denser and denser clump of molecules at maximum compression (max pressure). From there, it slowly begins to accelerate in the opposite direction, reaching maximum velocity again at its origin and begins to slow down and stop as it finds itself very lonely in the state of rarefication (min pressure). It then begins to accelerate again towards it origin to continue the cycle.
It would seem, from the scenario above, that the air molecule is motionless (velocity = 0) at both the maximum local compression (highest pressure) of the overall sound wave, and also at the maximum local rarefaction (lowest pressure) of the overall sound wave. These are also the points of maximum displacement of the molecule from its origin. It may seem counter-intuitive that at the peaks and troughs of sound pressure, the molecules are not moving at all (or in the real world, moving minimally), yet they are moving the fastest at the point of minimal overall pressure change (equilibrium)!
Mathematically, as the chart demonstrates, if the motion of the molecule's displacement forms a sine wave, then the corresponding molecular velocity and rate of pressure change (or pressure delta) is in a cosine relationship (or an over-eager sine wave with a 90° head start).
Greater maximum molecular velocities mean greater amplitudes of the sound wave. In fact, amplitude in sound refers to the degree of change in atmospheric pressure above or below equilibrium. The major contributing factor to that is the maximum speed of individual molecules involved in the wave. To impart these individual particle velocities, a force as a transfer of energy (what physicists call work--ugh, what's that?!) is required, and this often occurs across mediums. For example, a bass drum head, when struck will impart its mechanical energy of physical oscillation to the adjacent air molecules, causing them to move to and fro as outlined above and transfering their kinetic energy to through the mechanism described earlier. As the sound wave propagates through space, perhaps a small portion of it will strike your ear drum causing it to oscillate in a proportional fashion as the original bass drum head. As described later in our psychoacoustics chapters, these tissue movements will be converted into fluid vibrations, and converted again to electrochemical nerve signals that will communicate these pressure changes to your brain. Your brain may decide the pattern of these oscillations seems familiar, and perhaps you will say to yourself, "I think I just heard a bass drum struck!"
A fantastic and concise description of the relationships between sound, energy, force, work, power, amplitude, pressure and velocity at a level easily understood without extensive scientific training can be found in Mark Ballora: Essentials of Music Technology, 2003.
*The following expanation is an idealized version of reality, assuming molecules of a sound wave move reciprocally back and forth along a single axis, where as in reality, they are bumped and jostled all over the place--but this model is useful for an overall generalization of the sound pressure wave model.
For further study, see Hyperphysics->Sound waves