The superposition principle at work

Below is a series of animations illustrating the superposition principle. The length of the string selects the allowed modes: they are numbered according to the number of antinodes in the oscillation (mode n has n antinodes and n-1 nodes, excluding the ends). Every possible motion of the string is always given by the superposition of the allowed modes, "weighted" by their amplitudes.

component modes					sum mode
${\displaystyle \frac{1}{2}}$		${\displaystyle +\frac{1}{2}}$		${\displaystyle =}$
${\displaystyle \frac{1}{2}}$	as above	${\displaystyle +\frac{1}{2}}$	as above with phase ${\displaystyle \frac{\pi }{2}}$	${\displaystyle =}$
${\displaystyle \frac{1}{2}}$	as above	${\displaystyle +\frac{1}{2}}$	as above with phase ${\displaystyle \pi }$	${\displaystyle =}$
${\displaystyle \frac{1}{2}}$	as above	${\displaystyle +\frac{1}{2}}$		${\displaystyle =}$
${\displaystyle \frac{1}{2}}$		${\displaystyle +\frac{1}{2}}$	as above	${\displaystyle =}$
${\displaystyle \frac{1}{3}}$	as above with phase ${\displaystyle \frac{\pi }{2}}$	${\displaystyle +\frac{2}{3}}$	as above with phase ${\displaystyle \pi }$	${\displaystyle =}$

Mode 1 is

${\displaystyle A_1\sin (\frac{\pi }{L}x)\sin (\omega t+{\varphi }_1)}$

Mode 2 is

${\displaystyle A_2\sin (\frac{2\pi }{L}x)\sin (2\omega t+{\varphi }_2)}$

Mode 3 is

${\displaystyle A_3\sin (\frac{3\pi }{L}x)\sin (3\omega t+{\varphi }_3)}$

Further reading and links

• To learn about the range of applicability of the superposition principle, visit the page superposition principle.
• To fully understand how the string selects the allowed modes, visit the sections interference between reflected waves, normal modes, normal modes of a string, and also natural frequencies of the string).
• On this page you will find a series of ready-made experiments to carry out in our virtual laboratory to synthesize waves of any shape. The synthesis you will obtain is static (in the sense that it is not animated as on this page): being able to vary the amplitude and phase of the various allowed modes at will, you will become more aware of the contribution of the various modes to the final result.