Wave Formula Audits

Wave Speed Closure Audit

solve v, f, or wavelength

The relation v = f x wavelength is a speed budget: any two values force the third.

Example

The relation v = f times wavelength is solved three ways so the speed budget is visible from every pair of known values. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The same formula can close from any two values

The wave relation is one budget: speed equals frequency times wavelength. If any two values are fixed, the third is forced. This audit solves for each variable once.

v=fλ,f=vλ,λ=vfv=f\lambda,\quad f=\frac{v}{\lambda},\quad \lambda=\frac{v}{f}
A repeated spatial patternA wave snapshot where wavelength is read across space.space ->

Given speed and wavelength, frequency is forced

Ten metres per second spread over five metres per cycle means two cycles pass each second. The diagram's wavelength and the table row are the same value.

f=vλ=10 m/s5 m=2 Hzf=\frac{v}{\lambda}=\frac{10\ \text{m}/\text{s}}{5\ \text{m}}=2\ \text{Hz}
Five metre wavelengthA wave snapshot with a long repeated spacing.space ->

Given speed and frequency, wavelength is forced

Twelve metres each second divided among three cycles each second leaves four metres for each cycle. This is the same relation, read in the wavelength direction.

λ=vf=12 m/s3 Hz=4 m\lambda=\frac{v}{f}=\frac{12\ \text{m}/\text{s}}{3\ \text{Hz}}=4\ \text{m}
Four metre wavelengthA wave snapshot with a medium repeated spacing.space ->

Given frequency and wavelength, speed is forced

Two cycles each second, each six metres long, moves twelve metres of wave each second. This closes back to the usual speed form.

v=fλ=2 Hz⋅6 m=12 m/sv=f\lambda=2\ \text{Hz}\cdot6\ \text{m}=12\ \text{m}/\text{s}
Six metre wavelengthA wave snapshot with the wavelength used in the speed row.space ->

Every row satisfies the same speed budget

Read the last column as the audit column. In each row, frequency times wavelength exactly reproduces the speed column.

vfλfλ10 m/s2 Hz5 m10 m/s12 m/s3 Hz4 m12 m/s12 m/s2 Hz6 m12 m/s\begin{array}{c|c|c|c}v & f & \lambda & f\lambda \\ \hline 10\ \text{m}/\text{s} & 2\ \text{Hz} & 5\ \text{m} & 10\ \text{m}/\text{s} \\ 12\ \text{m}/\text{s} & 3\ \text{Hz} & 4\ \text{m} & 12\ \text{m}/\text{s} \\ 12\ \text{m}/\text{s} & 2\ \text{Hz} & 6\ \text{m} & 12\ \text{m}/\text{s}\end{array}
waves Solve for frequency, wavelength, and speed in separate exact rows, then check the same product.