Decibels
The logarithmic ratio between the sound-pressure of a given sound-wave and a reference value \(p_ref\).
Sound pressure is commonly measured using [see page 4, Pascals].
The range of human hearing ranges from around \(20/1*10^6\) pascals upto
ear-splitting at \(20\) Pascals. This huge range is really inconvenient
to work with so we commonly use decibels instead.
\(0 dB\) is commonly [see page 6, scaled] to be the threshold of human hearing, with \(140 dB\) being the threshold of pain.
Technically it's wrong to say x is 50 dB loud because dB is a ratio and a ratio
always needs a reference value.The formula for converting the power/amplitude of a wave to Decibels is: \(x \log_10 (p1 / p_ref)\) where \(x\) is some multiplier we choose to scale the range of decibel values. Observe that this means at:
- $p1 = \(p_ref\) the power is exactly \(0\)
- $p1 = 10 \(p_ref\) the power is \(x\)
- $p1 = \(\frac{p_ref}{2}\) the power becomes negative
- $p1 = \(p_ref * 2\) the power becomes positive
Power vs. Amplitude
For power we normally set \(x = 10\) and for amplitude we set it to \(x = 20\) (because we square \(\frac{p1}{p_ref}\) by expanding it out of the log).