Physics of The Q Song, by Walter Fox Smith

Physics and history of "The Q Song"

There are many types of oscillators in the universe, from the simple mass hanging on a spring, to resonant electrical circuits used for cell phone transmissions, to the sinusoidal motions of a guitar string after it is plucked, to the swinging of a pendulum, to vibrations of the atoms in a molecule, to the swaying of a building during an earthquake. The "quality factor" Q describes the tendency of a system to oscillate for many periods before the oscillation dies away.

The term Q was coined in 1920 by Kenneth Simonds Johnson, a researcher at the engineering department of Westinghouse Electric (later Bell Labs), and author of a well-regarded book on telephone technology. He chose the letter because it was not already in common use for other physical quantities. It was only later that V. E. Legg began to refer to it as the "quality factor".

Johnson graduated from Harvard in 1907. In 1913, he wrote for his class notes, "I recently came to the conclusion that all fellows of the class of 1907 should get married before the sexennial, and have acted accordingly."

Kenneth Simonds Johnson (1885-1958) in 1939. Photo credit: Alfhild Roff.

The concept is important both for oscillators that are given some initial kick after which they oscillate with decaying amplitude, and for oscillators that are supplied with a drive force (e.g. periodic pushes to a child on a swing) and move with a steady-state amplitude.

Quantitatively, Q equals 2π times the number of oscillation cycles required for the energy to decay by a factor 1/e.

Verse 1: Q is also equal to ω₀/γ, where ω₀ is 2π times the resonant frequency and γ is the rate of energy decay. (Formally, γ is one over the time required for the energy to decay by a factor 1/e.)

Verse 2: If you draw a graph of power supplied to a driven oscillator vs. frequency (which is equal to the power dissipated), Q is the ratio of the peak frequency (aka the resonant frequency) to the full-width-at-half-maximum of the curve.

Verse 3: The response amplitude (e.g. the amplitude of motion of a mass on a spring) at the resonant frequency divided by the response amplitude at very low frequency also equals Q.

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Play recording:

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Vocals: Walter Smith & Marian McKenzie ... Instruments: Jared Cattoor

[Up and down arrows are pitch cues.]
Verse 1:
Put omega-naught ↓on the top
And gamma down below
For a mass and spring it’s a quality thing!
It’s a magical ratio-o-o
It’s a magical ratio.

Verse 2:
Draw a power curve peaked at f,
With a width delta-f just so,
Divide that f by that delta-f –
It’s the very same ratio-o-o
It’s the very same ratio.

[Slowing down at the start and middle of the verse, then a tempo at the end of the verse]
Verse 3:
Near zero frequency
The response is kind of slow
Divide that response into the peak response!
It’s again the same ratio-o-o
It’s again the same ratio.

Verse 4:
And who is this versatile fraction?
To whom is such ho↓nor due?
She’s the queen serene of the bouncing scene,
But we all just call her Q-oo-oo,
But we all just call her Q.

She’s the queen ser↑ene of the ↑bouncing scene,
But-- we all just call her Q.

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