Period-Frequency Relation

Period and frequency are one fact counted in opposite directions. The period $T$ is seconds per cycle; the frequency $f$ is cycles per second; and $T = 1/f$ is not a discovery about nature but unit algebra. If something completes four cycles in a second, each cycle takes a quarter of a second, and no experiment was required to establish that. What earns this relation a page of its own is that almost every oscillation formula you will meet returns one member of the pair while the question in front of you wants the other, so this conversion sits quietly in the middle of nearly every wave calculation.

North American mains alternates at 60 Hz, so one full cycle takes $1/60 = 16.67$ ms — the interval behind the familiar hum in audio gear. European mains at 50 Hz gives 20 ms. Concert A at 440 Hz gives 2.27 ms per cycle. Run it the other way and a resting heart beating once every 0.8 s is oscillating at 1.25 Hz, which is the same statement as 75 beats per minute.

The unit is younger than the idea. Frequency was written "cycles per second" well into the twentieth century; the International Electrotechnical Commission proposed hertz in 1930 and SI adopted it in 1960, honouring Heinrich Hertz, who between 1886 and 1888 generated and detected radio waves in his Karlsruhe laboratory and so turned Maxwell's equations from mathematics into an observed fact. The hertz is dimensionally just $\mathrm{s}^{-1}$, and it is reserved by convention for periodic phenomena — the becquerel is also $\mathrm{s}^{-1}$ and counts random decays, which is precisely why the two units are kept apart despite being numerically identical. Combined with $v = f\lambda$, this page also gives the equally useful $v = \lambda/T$.

Three traps. The first is angular frequency: $\omega = 2\pi f$ in radians per second, and the pendulum and spring formulas carry their $2\pi$ for exactly this reason. Substituting an $\omega$ where an $f$ belongs makes the answer wrong by a factor of 6.283, which is large enough to notice and small enough to rationalise. The second is revolutions per minute: 3600 rpm is 60 Hz, not 3600, and the conversion is a division by 60. The third catches people with pendulums — a "seconds pendulum" ticks once per second but has a period of two seconds, because a full cycle is out and back. Count a cycle as a return to the starting state moving in the starting direction, and the reciprocal will behave.