dB, wattage and SPL

People often get confused when it comes to decibels especially since it is a dimensionless unit. We all know how long 1 metre is (or 1 foot) but a decibel is like a percentage. It means different things in different situations (because it is relative). 50% of 10 is not the same as 50% of 100. The same goes for decibels.

To cause further confusion, a decibel is also a logarithmic unit which means something like 2dB does not always mean half of 4dB (it depends on the context). This will be explained later on.

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Read only if you need clarification on other uses of the decibel unit:
We must remember that the decibel is not a standard index. This means that is it can be used to express other measurements such as distance, power, voltage, current, etc..... In this case, we are talking about decibels of Sound Pressure Level, or dBspl (i.e. dB).
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First, a little background in how the human ear perceives loudness. Sound Pressure Levels are actually measured in Pascals (pa). Let's say that the human ear registers 20 micropascals as a certain loudness, it requires 200 micropascals to sound TWICE as loud and 2000 micropascals (or 2 milipascals) to sound THRICE as loud! This is logarithm.

Now let's convert that to decibels.

200 micropascals is a 10dB increase from 20 micropascals.
2000 micropascals (or 2 milipascals) is a 20dB increase from 20 micropascals.

Now it's beginning to make sense. For the original sound to sound twice as loud, it requires a 10dB increase and for it to sound three times as loud, it requires a 20dB increase.

As you can see, a 10dB increase in loudness will always provide an order of magnitude increase in loudness (to the human ear).

Now we see why we measure Sound Pressure Levels (loudness) in dBs and not pascals.

The confusing but simplifying decibel
For "everyday usage", the decibel actually makes things a lot easier. From the previous section we see that a 10dB increase from a certain loudness increase the loudness (to the human ear) by an order of magnitude. It makes it so much easier.

In the next section we will learn that a 3dB increase from a certain loudness will double the power requirements BUT a 6dB increase will QUADRUPLE it!

As we can now clearly see, the confusion with the decibel may come because the decibel can mean many things. In terms of loudness, 20dB is only TWICE as much as 10dB but in terms of power, 6dB is FOUR TIMES as much as 3dB!

Power
In the audio world, decibels are closely linked to wattage (power). The wattage that we are talking about here can be described as the amount of power needed to drive speakers (or other circuitries) to the required loudness.

Unlike in an earlier section where finding out how loud a sound appeared to the human ear was as simple as dividing the decibels by 10, here we see an exponential growth pattern in terms of power usage for an increase of loudness as little as 3dB and this requires just a little bit of math.

In terms of power, a 3dB increase in loudness will require twice as much power but a 6dB increase in loudness will require four times as much power. Don't ask me why it works that way because I find goatse more appetising but you can check it out here if you are interested.

Basically, for every 3dB increase in loudness, you need to double your power requirements, compounded.

As an example, let's say we are using 100 Watts of power and want to increase our loudness by 9dB.

WRONG EXAMPLE:
100Watts × (9dB/3dB) × 2 = 300Watts
[initial_wattage × number_of_times × double_it]

CORRECT EXAMPLE:
100Watts × 2^(9dB/3dB) = 100Watts × 2 × 2 × 2 = 800Watts
[initial_wattage × double_it^number_of_times]
(number of times = db_increase divided by 3)

This is known as binary logarithm. You may be wondering where I found the 3dB? As mentioned, everytime you hit 3dB you need to double the wattage so dividing the dB increase by 3 will tell you how many times you need to double the wattage.

Somewhere in the previous lines you can find the wiki link to the formula describing how much power you will use should you want to increase your loudness from an existing power.

But I simplified it to something much simpler:
total wattage = existing_wattage × 2 ^ (dB Increase / 3)

Conclusion
We now understand that while the decibel is used to hide non linear math, it's usability made it so useful to so many applications that it began to confuse some of us.

Summary
Increase 10dB(spl) = twice as loud (human ear)
Increase 20dB(spl) = three times as loud (human ear)
∴ loudness increase(human ear) = dB_increase / 10

Increase 3dB(spl) = twice as much power
Increase 6dB(spl) = four times as much power
∴ total wattage = increase_wattage × 2 ^ (dB_increase / 3)