Result

Standing wave

y (x, t) = 2 A sin (k x) cos (ω t)

Statement

The superposition of two counter-propagating waves of equal amplitude and frequency.

Symbols and units

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

When it applies

Two waves of equal amplitude and frequency travelling in exactly opposite directions.

Don't use when

Requires two waves of equal amplitude and frequency traveling in exactly opposite directions — unequal amplitudes produce a partial standing wave instead.

Derivation

Add two identical waves travelling in opposite directions:

y = A sin (k x - ω t) + A sin (k x + ω t)

A trigonometric identity separates the sum into independent spatial and temporal factors:

y = 2 A sin (k x) cos (ω t)

Watch out for

A standing wave transports no net energy along the string — energy oscillates between nodes and antinodes but never moves along it, unlike a travelling wave.