Definition

Travelling wave

y (x, t) = A sin (k x - ω t)

Statement

A sinusoidal disturbance moving in the +x direction, with wave number k = 2π/λ and angular frequency ω = 2π/T.

Symbols and units

$y (x, t)$	displacement of the medium	m
$k$	wave number (2π/λ)	rad/m
$ω$	angular frequency	rad/s
$A$	amplitude	m

When it applies

A non-dispersive medium, where wave speed does not depend on frequency.

Don't use when

Assumes a non-dispersive medium where wave speed is independent of frequency — in a dispersive medium different frequency components travel at different speeds and this single-wavelength description no longer suffices.

Derivation

A shape that repeats with wavelength λ and period T, moving without changing form, is described by:

y (x, t) = A sin (k x - ω t)

The speed at which the pattern itself travels follows from the ratio of these two rates:

v = \frac{ω}{k} = λ f

Watch out for

The −ωt sign indicates propagation in the +x direction; flipping to +ωt reverses the direction of travel.