Science and Math for Audio Humans – Inverse Squares

by Danny Maland

It's time again for the standard disclaimer: Everything that I set before you should be read with the idea that “this is how I've come to understand it.” If somebody catches something that's just flat-out wrong, or if you just think that an idea is debatable, please take the time to start a discussion via the comments.

In the previous part of this series, the topic was the characteristics of a sound pressure wave. Now that we've been through amplitude, frequency, and wavelength, it's time to talk about how we experience sound (and its interactions with objects) in real life.

A good place to start with this is the experience of sound's intensity being lessened as we put more distance between ourselves and the source of the sound pressure wave. This is a fairly intuitive thing – if you've ever been right up against the stacks at a concert and found that life was a little too loud, you probably realized that an easy way to remedy the problem was to walk away from the mountain of loudspeakers. The basic mechanics of why that works go like this:

Sound pressure waves radiating from a point source into unobstructed air propagate in a spherical pattern. (This bit was discussed in more detail within the “Sound Itself” article.) Effectively, the energy of the sound pressure wave is spread over the surface of that spherical pattern. An observer in the path of the pressure wave is exposed to some section of that sphere, and thus, some portion of the energy spread across that imaginary surface. An observer that is close to the source of the pressure wave gets a far more concentrated dose of energy than someone standing at a distance. Figure 1 offers a visual example of this, using a blue “sound balloon” and a gray, cubic “observer.” The left side of the picture is an observer that is relatively close to a sound source, and the right side is an observer that is farther away.

  • Yes, it's entirely true that we are very unlikely to experience this “idealized” model of sound propagation. Real life just presents too many boundaries and obstacles for a perfect sphere to be our observation of propagating sound pressure waves. However, that natural tendency, shall we say, for sound to “want” to expand in a sphere yields a tidy mathematical model of sonic intensity versus distance. That model is the inverse-square law, which states that, under “ideal” conditions, the intensity of a sound pressure wave experienced by an observer is QUARTERED when the distance to the source is DOUBLED. Figure 2 shows this written out as an equation.

In action, it looks like this:

Intensity 1 = 1/(1*1) (The observer is distance “1” away from a source.)
Intensity 1/4 = 1/(2*2) (The observer is distance “2” away from a source.)

That's all fine and good – but why is it like that? Well, it all comes down to the sphere.

The bit to take notice of is that “radius squared” at the end. For our purposes, “radius” and “distance” are interchangeable – the observer is, by necessity, sitting at a point on the end of some arbitrary radius. If we write out some equations, we get the following situation:

Intensity 1/4pi = 1 Power/4pi*(1*1) (The observer is radius “1” away from a source.)
Intensity 1/16pi = 1 Power/4pi*(2*2) (The observer is radius “2” away from a source.)

Since 1/16pi is a quarter of 1/4pi, the inverse-square law holds up. The only thing we've changed is the actual number of the answer – the proportionality is the same. The inverse-square law is an audio-centric shorthand for the more generalized physics equation. (It's not just for audio. Everything I've been taught indicates that inverse-square works in the lighting world as well.)

At this point, you may be asking if this is actually useful in real life. After all, a truly spherical radiation pattern is something that most pro-audio folks will never run across. My answer to the “usefulness” question is “you're darn tootin' it's useful!” Although we'll need to go over decibels and how they work in order to get “real numbers,” the inverse-square law is an approximation that I've used many times. It's a great “worst case scenario” rule-of-thumb for predicting how much output a PA system might have at a certain distance, and I'm a big proponent of using conservative estimates when predicting how much coverage you'll get from an audio setup. Further, this idea of power radiating over an area can awaken the question of whether or not we can focus power into a smaller area, for purposes of greater intensity. All of us have probably done it to light, by way of a magnifying glass – and yes, you can do it with sound pressure waves as well. Horn-loaded loudspeakers aren't only for purposes of pure directivity...and have you ever noticed that some manufacturers refer to a horn as an acoustical lens?

I've said it before, and I'll probably say it again. Even the most theoretical elements of how sound works are the foundation of everything that we run across in pro-audio.