Wave Formula Audits

Period-Frequency Inverse Audit

fT stays one

Frequency counts cycles per second. Period counts seconds per cycle, so their product stays one.

Example

Frequency and period are reciprocal views of one repeating motion; three rows make the inverse pattern visible. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Frequency and period are reciprocal

Frequency counts cycles each second. Period counts seconds per cycle. Because those are reciprocal views of the same repeat, their product must stay one. This audit scans three exact rows instead of trusting a single worked example.

T=1f,fT=1T=\frac{1}{f},\quad fT=1
One point watched over timeA history graph of one point moving up and down as time passes.time ->

One cycle each second

At one hertz, each full cycle takes one second. Multiplying the frequency row by the period row gives one, so the pair closes.

T=11 Hz=1 s,fT=1 Hz⋅1 s=1T=\frac{1}{1\ \text{Hz}}=1\ \text{s},\quad fT=1\ \text{Hz}\cdot1\ \text{s}=\hl{1}
One repeat in the same time windowA time trace with one complete up-and-down cycle.time ->

Twice the frequency, half the period

When two cycles fit in the same second, each cycle uses half as much time. The table shows the fractional period directly, with no decimal rounding.

T=12 Hz=12 s,fT=2 Hz⋅12 s=1T=\frac{1}{2\ \text{Hz}}=\tfrac{1}{2}\ \text{s},\quad fT=2\ \text{Hz}\cdot\tfrac{1}{2}\ \text{s}=\hl{1}
Two repeats in the same time windowA time trace with two complete cycles.time ->

Four times the frequency, one quarter the period

With four cycles per second, the period is one quarter second. The scan is the useful habit: as frequency rises, period falls, but the product remains one.

T=14 Hz=14 s,fT=4 Hz⋅14 s=1T=\frac{1}{4\ \text{Hz}}=\tfrac{1}{4}\ \text{s},\quad fT=4\ \text{Hz}\cdot\tfrac{1}{4}\ \text{s}=\hl{1}
Four repeats in the same time windowA time trace with four complete cycles.time ->

The inverse check is visible in every row

Read across: frequency times period is one in all three cases. Read down: doubling frequency halves period; doubling again halves period again.

fTfT1 Hz1 s12 Hz12 s14 Hz14 s1\begin{array}{c|c|c}f & T & fT \\ \hline 1\ \text{Hz} & 1\ \text{s} & 1 \\ 2\ \text{Hz} & \tfrac{1}{2}\ \text{s} & 1 \\ 4\ \text{Hz} & \tfrac{1}{4}\ \text{s} & 1\end{array}
waves Three exact rows make the inverse pattern visible: when frequency rises, period falls.