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.
The Wikipedia article on Amplitude tells us,
The amplitude of a periodic variable is a measure of its change over a single period (such as time or spatial period). There are various definitions of amplitude (see below), which are all functions of the magnitude of the difference between the variable's extreme values. [...]
Root mean square (RMS) amplitude is used especially in electrical engineering: the RMS is defined as the square root of the mean over time of the square of the vertical distance of the graph from the rest state.
For complex waveforms, especially non-repeating signals like noise, the RMS amplitude is usually used because it is both unambiguous and has physical significance. For example, the average power transmitted by an acoustic or electromagnetic wave or by an electrical signal is proportional to the square of the RMS amplitude (and not, in general, to the square of the peak amplitude)
At this point, it's natural to ask why "power" is proportional to "amplitude" squared. And why do we take the ratio of powers rather then the ratio of amplitudes?
Of course, after we've taken the log of the ratio, this distinction is just a scaling factor, since log(x^2) = 2*log(x). But still, it's natural to ask why.
If you look this issue up on the internet, you mostly find answers like those that I got when I first asked this question at the age of 16 or so, back in neolithic times. For instance, a student asks in the Physics Forums asks
My book states: "Power is proportional to amplitude squared"
How can this be derived?
He gets a bunch of answers that mostly just offer more elaborated versions of the statement he is trying to understand, e.g.
… the power of the electricity in a resistor is the current-squared times R (watts), or the voltage-squared divided by R (watts). We also talk about the product of voltage and current as power. So current or voltage are amplitudes, and the power is watts.
This is also true in electromagnetic waves. We talk about the transverse E vector or the H vector (amplitudes) in an electromagnetic wave. [Note that E is volts per meter and H is amp-turns per meter]. E and H are amplitudes, while the Poynting vector, which is E x H , is power (per square meter).
The power of a signal or waveform is the signal or waveform multiplied by its conjugate.
Power = U x U*, where U is the function describing the signal.
For example; U = Ae(t/b) Where U is the signal function, A is the amplitude, t is time (or what ever you want it to be), and b is just a coefficient.
Since P = U x U*
P = Ae(t/b) x Ae(-t/b) = A2
None of this really explains why the power of a signal is proportional to the square of its amplitude — it just persuades the student to stop asking and respect the authority of those responding.
Poking around in Wikipedia is not much more helpful. Thus the link on power tells us that
In physics, power (symbol: P) is defined as the amount of energy consumed per unit time. [...] The integral of power over time defines the work done.
And later in the same article we learn that
This tells us again that power is related to amplitude squared, adding references to energy and work. But why isn't power related to amplitude cubed? or the square root of amplitude? or the square root of amplitude cubed? or just plain amplitude?
Additional levels of authoritative mystification can be derived from ISO-80000-1, where Wikipedia tells us that:
According to Annex A, "[t]he logarithm of the ratio of a quantity, Q, and a reference value of that quantity, Q0, is called a level". For example LP = ln(P/P0) is the level of a power quantity P.
Annex C introduces the concepts of power quantities and root-power quantities.
So to justify the statement that power is proportional to amplitude squared, we could explain that amplitude (like pressure) is a "root-power quantity". As an explanation, this would rival the insight of the Sçavantissimi doctores in Moliere's Le Malade Imaginaire, who answer the question "quare Opium facit dormire ?" ("Why does opium cause sleep?") by explaining that "Quia est in eo Virtus dormitiva" ("Because it contains a dormitive property").
There are no doubt many simple explanations for the basic distinction between "power quantities" and "root-power quantities" (previously "field quantities"), and for the fact that the amplitude of a sound waveform is a "root-power quantity", and for the practice of defining levels in terms of the ratio of "power quantities". One story might depend on the fact that this all makes the conservation of energy work out. Another might point to sound as pressure variation considered over a surface rather that at a point.
But I'm having trouble coming up with a simple, memorable, and correct story to put into my lecture notes.
In the end, of course, people just have to remember the factor of 20 (rather than 10) in the expression for SNR in dB:
SNR = 20*log10(RMS(signal)/RMS(noise))
Still, it would be nice to explain the factor of 2 in terms of something other than its virtus dormitiva. So I'm offering a lifetime subscription to Language Log as a prize to the commenter telling the best story about this.
One of the interesting-but-not-helpful things that I learned, in the course of poking around on the internet for the right way to explain why amplitude is a root-power quantity, was the story of where the decibel came from. I previously knew the vague outlines, but not the details.
From the Wikipedia article on the decibel:
The decibel originates from methods used to quantify reductions in audio levels in telephone circuits. These losses were originally measured in units of Miles of Standard Cable (MSC), where 1 MSC corresponded to the loss of power over a 1 mile (approximately 1.6 km) length of standard telephone cable at a frequency of 5000 radians per second (795.8 Hz), and roughly matched the smallest attenuation detectable to the average listener. Standard telephone cable was defined as "a cable having uniformly distributed resistance of 88 ohms per loop mile and uniformly distributed shunt capacitance of .054 microfarad per mile" (approximately 19 gauge).
The transmission unit (TU) was devised by engineers of the Bell Telephone Laboratories in the 1920s to replace the MSC. 1 TU was defined as ten times the base-10 logarithm of the ratio of measured power to a reference power level. The definitions were conveniently chosen such that 1 TU approximately equaled 1 MSC (specifically, 1.056 TU = 1 MSC). In 1928, the Bell system renamed the TU the decibel. Along with the decibel, the Bell System defined the bel, the base-10 logarithm of the power ratio, in honor of their founder and telecommunications pioneer Alexander Graham Bell. The bel is seldom used, as the decibel was the proposed working unit.
"Miles of Standard Cable" has a certain steampunk charm that almost makes me sorry that we lost it. But from the same article, I learned that the decibel itself has been challenged as a confusing residue of slide-rule thinking:
According to several articles published in Electrical Engineering and the Journal of the Acoustical Society of America, the decibel suffers from the following disadvantages:
* The decibel creates confusion.
* The logarithmic form obscures reasoning.
* Decibels are more related to the era of slide rules than that of modern digital processing.
* They are cumbersome and difficult to interpret.
Hickling concludes Decibels are a useless affectation, which is impeding the development of noise control as an engineering discipline.