Wave Equation — v = fλ

The wave equation v = fλ explained: how speed, frequency and wavelength lock together for every wave, with a worked sound example.

v = f\lambda

Wave speed equals frequency times wavelength.

What it means

This is the master key to all wave phenomena: the speed of a wave equals how many cycles pass per second times the length of each cycle. Frequency is decided by the source — a guitar string's tension and length — while speed is decided by the medium. Wavelength is whatever it must be to satisfy the equation, which is why sound waves stretch out in warm air (faster medium, same frequency, longer wavelength). One equation governs ripples, sound, light and earthquakes alike.

Variables

Symbol	Name	Unit	Meaning
$v$	Wave speed	m/s	Speed at which the wave disturbance travels through the medium
$f$	Frequency	Hz	Number of wave cycles passing a point per second
$λ$	Wavelength	m	Distance between two consecutive identical points of the wave

Assumptions

Wave travels through a uniform medium (constant speed)
Frequency is set by the source and does not change between media
Non-dispersive medium for the simple form (all frequencies same speed)

Derivation

In one period $T$ , a wave advances exactly one wavelength $\lambda$ . Speed is distance over time, so $v = \lambda/T$ . Since frequency $f = 1/T$ , substitution gives $v = f\lambda$ .

Worked example

Problem

A tuning fork produces a 440 Hz tone with wavelength 0.78 m in air. What is the wave speed?

Given:

f = 440 Hz (concert A)
λ = 0.78 m

Write the wave equation: $v = f\lambda$
Substitute: $v = 440 \times 0.78$
Multiply: $v = 343.2\ \mathrm{m/s}$

≈ 343 m/s — the speed of sound in air

Multiplying 440 cycles per second by 0.78 meters per cycle gives 343 meters per second — matching the known speed of sound. The equation is self-consistent: measure any two of the three quantities and the third is fixed.