Frequency
Frequency is the number of complete waves that pass a point each second, measured in hertz (Hz); it is the inverse of the period.
Formula
f = \frac{1}{T}
Definition
Frequency is how many complete waves go by each second, measured in hertz, where one hertz equals one cycle per second; a high frequency squeezes many waves into a second and a low frequency only a few, and for sound, frequency is pitch: a whistle is high frequency and a drum is low frequency. The period $T$ is the time for one cycle, so $f = 1/T$; frequency is set by the source and does not change when a wave enters a new medium, and humans hear frequencies from about $20\ \text{Hz}$ to $20{,}000\ \text{Hz}$. Formally, frequency is the temporal rate of oscillation, $f = 1/T = \omega/2\pi$, and for a wave $f = v/\lambda$; a photon's energy is proportional to frequency, $E = hf$, and the Doppler effect shifts observed frequency when source and observer move relative to each other, $f' = f\,(v \pm v_o)/(v \mp v_s)$ for sound, with resonance occurring when a driving frequency matches a system's natural frequency.
Example
A hummingbird beating its wings $50$ times per second makes a $50\ \text{Hz}$ hum with period $T = 1/50 = 0.02\ \text{s}$, and middle C on a piano at about $262\ \text{Hz}$ vibrates roughly every $0.0038\ \text{s}$. An ambulance siren at $700\ \text{Hz}$ approaching a stationary listener at $30\ \text{m/s}$ is heard at $f' = 700 \times 343/(343 - 30) \approx 767\ \text{Hz}$, dropping to $f' = 700 \times 343/(343 + 30) \approx 644\ \text{Hz}$ after it passes, a fall of about $123\ \text{Hz}$ that the ear registers as the familiar pitch drop.
Key Insight
Frequency is the one property a wave keeps when moving from air to water or glass; speed and wavelength both change, but the number of wiggles per second was fixed by whatever made the wave. Frequency is also the most precisely measurable quantity in physics: atomic clocks count the $9{,}192{,}631{,}770\ \text{Hz}$ transition of cesium, which is why the second itself is now defined by a frequency.