Increasing amplifier power to increase sound level

Question

I am reading a book by Everest and I dont understand one of the paragraphs:

Increasing amplifier to achieve higher acoustic level can be frustrating. Doubling amplifier power from 1 W to 2 W is a 3dB increase in power level, but this yields a very small increase in loudness. Similarly an incerase in power level from 100W to 200W or 1000W to 2000W yields the same increase in level.

What I dont understand is that how can an increase in power level from 1000 W to 2000 W have the same increase in loudness effect as from 1 W to 2 W ?

Answer 1

3dB is a doubling of the power as expressed in decibels.

Decibels are a logarithmic representation. As such, it "compresses" the numeric difference. The log of 10 to the base of 10 is 1. The log of 100 to the base of 10 is 2. So, you have a ratio of 10, but a difference in the logarithm of only 1. From 100 to 1000 is again a factor of 10, but the difference of the logarithms (2 to 3) is still just one.

So, yes, going from 1W to 2W is the same ratio as going from 1000W to 2000W. In both cases, you are doubling the power.

Your ears also have a (sort of) logarithmic response. It allows you to detect very quiet things, and very loud things.

Your nerves have a limited range of intensity they can convey. If the sound intensity were transmitted linearly to the real loudness (simple proportion,) then you would have a much smaller range of volume that you could hear. Say, maybe just quiet sounds, and loud sounds are just a mash. Or, maybe just really loud sounds but you couldn't hear an animal sneaking up on you.

With the logarithmic response your ears have, you can hear very quiet things and very loud things.

The extended range comes at the cost of not being able to detect small changes very well.

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Some numbers:

Human ears have a dynamic range of something like 130dB. That's a factor of over 3 million times.

Using a linear representation, your nerves would have to be capable of conveying a range of intensities from 1 to over 3 million.

As it is, you can't really detect differences smaller than about 3dB. 130dB/3dB gives you a range of intensities from 1 to about 43.

So, your nerves only have to deal with a range of around 40 intensities.

Much simpler to convey than over 3 million.

Answer 2

The ear's perception of sound pressure (volume) is logarithmic.

From your comments to Sunny's answer:

So, the loudness effect what the author is talking about in case of 1000 W to 2000 W increase is at a considerable larger distance compared to the 1 W to 2 W increase?

Only because you may suffer irreversible hearing damage if you stand too close to the 1000 W source.

... at a distance say 10 m away from the loudspeaker, increasing the power from 1000 to 2000 W will just have a minor 3 dB increase in loudness effect, although the power has been doubled.

It's not a 'minor' effect but rather the way the ear is designed. Having logarithmic sensitivity allows us to hear anything from a pin-drop to a truck engine in close proximity (without suffering pain) despite a million to one change in sound pressure level. It also means that while one pneumatic road hammer may be bearable at a certain distance, two operating in tandem is not perceived twice as loud. Similarly the solo vocalist can be heard against the background of a large choir.

Answer 3

1W to 2W can have the same effect in a Headset as 1kW to 2kW on an open stage far away.

Power is usually measured in dB or relative levels like dBm, or A weighted bandwidth, dBA for A weighted to match our hearing BW, because path loss is logarithmic and so is our sensation of hearing and sight as well.