What is the easiest way to convert watts to decibels?

An Audio Masterclass website visitor wants to convert watts to dB. Is this even possible?

Here is a question, which I quote exactly, from an Audio Masterclass website visitor...

Whats the easiest way of converting watts to dB, without using a table.i mean...using maths calculations solely

On the surface it seems like the kind of thing that could easily be looked up on Google. You might find an answer similar to this (once again an exact quote)...

db=10*log(w)

Oh dear. This is far from correct. It gives the impression that it is possible to convert from watts to decibels, but it isn't.

You can't measure the absolute quantity of anything in decibels - watts, volts or digital signal levels. A measurement in decibels is always a ratio between one level and another. So you can say that one signal or sound is 10 dB higher in level than another, but you can't say that a signal or sound has a level of 10 dB, or any other number of decibels.

What you can do however is give a measurement in decibels compared to a standard reference level. We commonly use a standard reference level of 1 picowatt (a millionth of a millionth of a watt) per square meter and call this 0 decibels sound pressure level, or 0 dB SPL.

(Note for the pernickety... sound pressure is measured in pascals, where one pascal = one newton per square meter. 0 dB SPL is 20 µPa, which clearly is a measurement of sound pressure. This corresponds to 1 pW/m2, which is a measure of sound intensity. It is possible to speak of SIL or sound intensity level, but it isn't common.)

If your PA system achieves a level of 100 dB SPL, then the level is 100 dB louder than 0 dB SPL. (0 dB SPL, by the way, is commonly taken as the lowest sound level that is audible to the human ear).

The real question

Although all of the above is interesting, I suspect that it doesn't answer the real question - the one we would all like to know the answer to. And that is...

How can I convert my amplifier's power rating to dB SPL?

So for instance, is there a method of working out how loud 500 watts of amplifier power would be when converted into sound energy? Would it be, say, 100 dB SPL?

Well if it were possible to be able to calculate this easily, a lot of PA operators would be very happy indeed, and a lot of acoustic consultants would be out of a job.

The thing is that although in theory it might be possible to make such a calculation, there are too many variables for it to be either easy to work out, or accurate to any useful degree.

The first variable is the method the amplifier manufacturer uses to specify power output. Is the amplifier capable of putting out a 500 watt sine wave on a continuous basis, or can it only manage 500 watts on momentary peaks?

The second variable is the efficiency of the loudspeakers. What proportion of the input electrical power will they convert to sound?

Next, the directivity of the loudspeakers. How much sound will they send to the audience, and how much will they waste on the walls and ceiling of the venue? (Or the sky, if it's an outdoor gig.)

Next... How much of the sound energy will be absorbed in the room, and what proportion will be reflected back? It takes more power to achieve any given sound intensity in a room that has more absorbent surfaces.

One last point (there are additional factors, but I don't want to try your patience too much)... How far is the listener from the loudspeakers? I have seen people put their head inside a bass bin, so that's the closest approach (but not to be recommended). Sound intensity will fall off with increasing distance.

Even if you could calculate all of the above, the total absorption in the room will change according to the number of people in the audience.

As you can see, there are a lot of variables here. PA operators, through experience, get to know what amplifier power they will need for different venues, with practical hands-on experience of their loudspeakers' efficiency and directional characteristics. But even if a precise calculation may be impractical, it doesn't hurt to get a feel for what kind of amplifier power will be necessary. Here's a calculator...