One of the concepts that comes up in the Introduction to Phonetics course that I'm teaching this semester — first meeting yesterday — is SNR ("Signal to Noise Ratio"). This is the ratio between the power of the "signal" (defined as the stuff you care about, essentially) and the power of the "noise" (the stuff that you aren't interested in).
And at this point, there are a few things that students need to learn. Since SNR is a ratio of power to power, it's a dimensionless quantity. Similar ratios of physical quantities come up elsewhere in acoustics, like "sound pressure level" (SPL), defined as the ratio of sound pressure to the some reference level, usually taken to be the nominal threshold of human hearing. Because additive scales are more intuitive, we generally take the log of such ratios. And because powers of ten are inconveniently far apart, we generally multiple log10(whatever ratio) by 10 to get "decibels".
Now comes the part that I'm interested in this morning: the power of a sound wave is proportional to the square of its amplitude. And I'm looking for a simple and correct way to justify this statement, and to explain why we generally quantify "levels" of physical signals as ratios of powers rather than as ratios of amplitudes.
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