QUICK AND DIRTY AUDIO MATH

We are faced will kinds of thing to compare or calculate everyday in audio. We could of course pull out a calculator to find the answer (assuming we know the proper formula to use), or we can get “close enough” for practical usage, by simply understanding some basic concepts and a very few numbers or relationships. Hopefully this will help some of those who are “math challenged”, by making it simple.
For the math inclined: dB (watts)=10log W1/W2 dB(distance or volts)= 20log D1/D2
Most of the time we are not interested in exact numbers, but rather the dB relationship between the numbers. Once you start to think in terms of the dB differences, rather than the actual numbers, things begin to become clearer and make more sense.

WATTAGE: Yes, more watts is louder (assuming all other metrics stay the same), but does it matter, and how much of a difference does it make or how loud will it get? If you can remember that twice or half the power is a 3dB change, and can count the number of zeros in the difference (each 0 accounts for a 10dB increase), you can easily figure out, close enough) in your head things like the dB gain of a loudspeaker, differences is amplifier sizes etc. Let’s say that a loudspeaker has a power handling of 2000 watts. How much louder is that than the rated sensitivity. You should be able to answer that in about 2 seconds. First of all, 1000 has 3 zeros in the number, so that is 30dB greater than 1 watt. (1 second). Since 2000 watts is twice that of 1000 watts, that is 3dB. So 30+3=33dB. (2 seconds). So if a loudspeaker has a sensitivity of 103dB with 1 watt, then it should produce 136dB with 2000 watts applied. YES, there are a number of other factors, but we are keeping it simple here, for the basic simple math.

Let’s go the other way. A loudspeaker has a power handling of 250 watts and a sensitivity of 98dB with 1 watt applied. How loud can it get? You can approach it two different ways. Since 250 is a little more than twice 100, and since 100 has 2 zeros, 100 watts is a 20dB gain over 1 watt, and 250 watts is a little over twice 100, so it is a little more than 3dB extra, the answer would be a little over 23dB gain. Or we could go backwards and see that 250 is ¼ of 1000. Since we know 1000 watts is 30dB gain (again over 1 watt), and half that (500 watts) is 3dB less, and half of 500 is 250, then we subtract 6dB (3+3) from 30 and end up with 24dB gain. 24dB is a little bit more than 23dB (as done the first way) So when looking at the wattage of amplifiers, generally a useful change is considered 3dB, so twice the wattage (assuming the loudspeaker can safely handle the extra power). So while the difference between a 3000 watt amplifier and a 4000 watt amplifier may at first seem like a lot, it is no different than the difference between a 3 watt and 4 watt amplifier, in terms of dB change. A 3dB change to a 3000 watt amplifier would be to go to 6000 watts. So don’t think in terms of ONE THOUSAND WATTS, think of it as just a little tad bit more than the current 3000 watts. It is generally not worth discussing. Technically it is 1.25dB louder, but turn up a signal 1.25dB on a digital console or DSP and see if that increase of level is worth spending money on.
DISTANCE from the source.

The same things happen when thinking of distances, but the math is a little different. Double or half the distance = 6dB change in level. Now that assumes that you are not too close to the source. When you are in the “nearfield” (this changes with different models) of a loudspeaker, these rules do not apply
So let’s say the FOH mix position is 50’ from the loudspeakers, and you want to know approx. how loud it will be at the rear seats, which are 200’ away If the SPL is 100dB @ FOH. There are number of other variables (humidity, temp etc) involved in sound traveling through the air, so we have to say “approximately”. The thought process would be: Going from 50’ to 100’ would be a loss of 6dB, and going from 100’ to 200 would be another 6dB, so a total loss of 12dB from 50’ to 200’, or 88dB at the seats 200’ away.

HOWEVER-this is “assuming” that the sound is pointed along the same axis as the ground, and started out at head height, and often it is not. So the actual numbers could vary a bit. The actual numbers could be more or less loss, depending on things such as height of the loudspeakers, where they are aimed, pattern of the loudspeakers and so forth. But it is a good starting point.

And it works for longer distances also. The difference between 500’ and 1000’ is 6dB. That could be good or bad, depending on “whose side” you are on. If you trying to cover a long way out, that is a good thing, but if the neighbors are there, it is a bad thing, you would want more loss over distance.
Ten times or 1/10th the distance would be equal to a 20dB change in SPL.

So with a little understanding, and just a couple of numbers to remember, you can get “close enough” for audio, without having to grab a calculator. Understanding the relationships of “all things audio” is as important as the detailed specifics. Often some quick “math in the head” will do the job.