Frequency and Period - Waves, Oscillations, and Resonance Step by Step

Frequency counts cycles per second at one point and the period is its reciprocal; these are time measurements, not read off the snapshot.

waves At 2 Hz the period is a clean 1/2 second; this is the time view, not the space snapshot.

Example

Frequency counts cycles per second at one point and the period is its reciprocal — these are time measurements, not read off the snapshot. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Now watch one point over time

Wavelength was a SPACE measurement off the snapshot. Frequency and period are TIME measurements: fix your eye on one point of the string and watch it bob up and down. One complete up-and-down is one cycle.

\text{watch one point: its up-and-down is one cycle}

Frequency: cycles per second

Frequency is how many full cycles happen each second, measured in hertz. This point completes 2 cycles every second, so the frequency is 2 hertz.

f = 2\ \text{Hz}

Period: seconds per cycle

The period is the time for one cycle — just the frequency flipped over. At 2 cycles per second, one cycle takes one over 2 of a second.

T = \frac{1}{f} = \frac{1}{2\ \text{Hz}} = \hlmath{\tfrac{1}{2}}\ \text{s}

More cycles each second means less time per cycle

Frequency and period are reciprocals. A faster repeating point has a shorter time for each cycle.

\begin{array}{c|c|c}f & \text{one cycle} & T \\ \hline 1\ \text{Hz} & 1 & 1\ \text{s} \\ 2\ \text{Hz} & 1 & \tfrac{1}{2}\ \text{s} \\ 4\ \text{Hz} & 1 & \tfrac{1}{4}\ \text{s}\end{array}