To get the nasty part out of the way, I'm not fond of the word "amplitude" being used the way you are using it. No doubt this came about as a result of the fact that amplitude is most often associated with voltage, which varies as the square root of power, thus leading to the notion or practice of classifying any quantity that varies as the square root of power as "amplitude". The word that comes to mind is "yuch."

The first excerpt you took from what I wrote was purely a mathematical statement correlating a doubling scale to a base 10 logarithmic scale. Neither power nor root-power quantities were implied. Perhaps there is another way to show why a difference of 3.01 dB implies a doubling (of power), but my way of doing it has the advantage of mathematical purity.

Here is one of the basic properties of logarithms:

log( X ^ n) = n x log( X)

Since power is proportional to the square of voltage, power ratios are obtained by squaring voltage ratios. If we are given two voltage values and want to express the implied power ratio in decibel, we would first square the voltage ratio in order to obtain the power ratio, then take the base 10 log and multiply by 10. It is appropriate to think of the multiplication by 10 as a conversion from Bel to decibel. But of course the above logarithm property reveals that this calculation is the same as taking the log base 10 of the voltage ratio and then multiplying by 20 instead of by 10.

Nearly a century ago, some engineers at Bell Labs proposed a way of expressing power losses in transmission lines using a logarithmic scale. The essential advantage of using a logarithmic scale was that accumulation of power losses could be computed via direct addition and subtraction in lieu of multiplication. They defined the "Transmission Unit" or TU as 10 x the base 10 log of the power ratio. Subsequently they changed the name from TU to "deciBel", which implicitly defined the Bel as the base 10 log of the power ratio. Even though there is no inherent mathematical reason that the deciBel should only be used in a way whereby 3 dB implies a 2:1 ratio in power, this became a de Facto rule for use of the decibel. For root-power quantities in particular, e.g., voltage and sound pressure, this means that an increment or decrement of 3 dB does not correspond to a doubling or halving of the root-power quantity itself. Ratios of root-power quantities are generally squared before taking the logarithm, thereby translating the ratio of the root-power quantity to the corresponding ratio of power. And of course this is equivalent to multiplying the base 10 log of the ratio (of the root-power quantity) by 20.

Thus, in the formula we use to convert sound pressure in Pascal to decibel (SPL), we divide the sound pressure value in Pa by the reference level (.00002 Pa), then take the common log, then multiply by 20. (The definition of Pascal is 1 Newton of force per square meter.) It is apparent that a 3 dB difference in SPL values implies that the acoustic

*power* values (in Watts or milliWatts, etc.) are in the ratio 2:1, and that the sound

*pressure* values for a 3 dB difference in SPL are in the ratio 1.4:1. (Even though SPL is pressure, not power.) When two SPL values (in decibels) differ by 6 dB, the two sound pressures (in Pa or in any alternative units for force per unit area) are in the ratio 2:1.

Since +/- 3 dB is a spread of 6 dB, it follows that for a speaker with sensitivity remaining within a +/- 3 dB window over some range of frequency, the ratio for sound pressure (force per unit area) at the upper and lower boundaries of the window is 2:1. The strongest sound pressure encountered in this range of frequencies will be not greater than twice the weakest sound pressure encountered in this range of frequencies. I regret that at the tail end of my prior post I muddled this so badly as to have spoiled a post that otherwise was probably half decent