# GAIN11

## Inverse Square Law

The inverse square law is probably one of the most widely used fundamental of physics used in the field of audio. As you advance through your career you will continue to use this particular formulation on an increasingly frequent basis. You will use it at the very beginning of your system designs when trying to determine what speakers to select and what amplifiers will be required to power them. It can also be used to assist in the setting of delays speakers and much more. It will be a very valuable tool to have in your audio knowledge toolbox. Simply stated the definition of the inverse square law is as such.

For each doubling of the distance from the source of sound, the measured dBspl will drop by 6dB.

So to clarify this if you measure 90dBspl 50’ from a speaker the sound intensity will measure 84dBspl at 100’. As a result of doubling the distance from the sound source the sound intensity has inversely diminished by 6dBspl (90dBspl @ 50’ – 6db = 84dBspl @ 100’).

Okay so let’s lay some groundwork to explain just why this is. Imagine a sphere with any given radius. Every time we double the radius the surface area of the sphere quadruples. We know this because it’s an established law of physics which reads in an equation as follows or . Let’s say with 1watt of power we achieve 94dBspl across the entire surface of our sphere. When we double the radius we now know that the surface area will quadruple meaning our 1watt of power now has to cover four times the area. Inversely this translates into a level of intensity ¼ of our original value.

So how does all of this translate into a 6dB reduction in sound pressure level? For this we need to refer back to another known principal of sound. Every time you double your power the result is a gain of 3dB. The exact opposite applies as well, if you cut your power in half you lose 3dB. For example if we started out with a 100watt amplifier and replaced it with a 200watt amplifier we would achieve a 3db gain in output. Adversely if we start out with a 100watt amplifier and replaced it with a 50watt amplifier we would lose 3dB. What this tells us is after doubling our radius our same output will only achieve ¼ its’ original value at any given point on the surface of our sphere. Effectively this is the same as if we were operating with ¼ the power we started with or otherwise replacing our 100watt amp with a 25watt amp. Let’s apply the power ratio we learned above and the result would look something like this.

((100W / 2 = 50W)+(50W / 2 = 25W)) or (-3dB)+(-3dB) = -6dB.

Of course as with all things in life there are exceptions to the inverse square law. The law only holds true in free field. Or otherwise a sound source in a space which has no obstacles to impede or alter the path of the soundwaves. Imagine suspending a speaker from a crane in mid air and that would represent a free field until which time the sound reached the ground plane. So as you can imagine in reality we will seldom (virtually never) be working in a free field environment. The majority of the time we are forced to work with a ½ space, ¼ space or ⅛ space. A ½ space is when the speaker is near one boundary such as a wall. A ¼ space would be the speaker placed where two boundaries meet such as a corner with two walls. Lastly the ⅛ space is equivalent to a speaker being placed in a corner where three boundaries meet like two walls and the floor or two walls and the ceiling. Typically speaking the only type of speaker that will benefit from being placed in such a space is a subwoofer due to the fact that the more boundaries you are near it will increasingly reinforce the low frequency engergy. However and this is a big however; it will not do so evenly across the entire low frequency spectrum so it may do more harm than good. Always approach this method with caution and either a good ear or using a test instrument such as TEF with a good waterfall display to analyze the effects of the placement on the frequency response of the sub.

A fun fact worthy of noting is that the inverse square law also applies to other forms of energy as well. For instance light will also diminish at a known rated stated as where E = illuminance and I = pointance. This would otherwise read as Illuminance equals pointance divided by the radius squared.