= Standing wave
{wiki}
A <standing wave> results from the <superposition principle> applied to equal-<frequency>, equal-<amplitude> <waves> travelling in opposite directions. Its spatial <nodes of a standing wave> remain fixed while its <displacement> oscillates in time.
For a pure <standing wave>, the two <travelling waves> also have equal <amplitudes>: their <interference> leaves a fixed <mode shape>, rather than a profile moving along the string. Choose the position origin and time origin so that their transverse <displacements> are
$$
y_+(x,t)=a\sin(kx-\omega t),\qquad y_-(x,t)=a\sin(kx+\omega t).
$$
Here $a>0$ is each constituent's <amplitude>, $k=2\pi/\lambda$ is the <wavenumber>, $\lambda$ is the <wavelength>, and $\omega=2\pi f$ is the <angular frequency> corresponding to <frequency> $f$. Their <wave speed> is $c=\omega/k$. The <superposition principle> and the <sine addition formula> give
$$
y=y_++y_-=2a\sin(kx)\cos(\omega t).
$$
The spatial factor stays put; the temporal factor changes its size and sign. Other constant <wave phases> shift the position or time origin. Unequal constituent <amplitudes> generally leave moving <displacement> at the would-be <nodes of a standing wave>, rather than this exact factorization.<40.75 -- Standing waves on cord (UCSB Department of Physics)>[Standing waves on cord, interference explanation and sum-of-sines derivation]{ref}
For an original ideal-string example, set $L=1\,\mathrm m$, $c=20\,\mathrm{m\,s^{-1}}$, and choose the third <normal mode>. Then $k=3\pi\,\mathrm{m^{-1}}$, $\omega=60\pi\,\mathrm{s^{-1}}$, and $f=30\,\mathrm{Hz}$ by <fixed-end standing wave frequencies>. With each constituent <amplitude> $a=5\,\mathrm{mm}$,
$$
y(x,t)=0.010\,\mathrm m\;\sin\left(\frac{3\pi x}{1\,\mathrm m}\right)\cos\left(\frac{60\pi t}{1\,\mathrm s}\right).
$$
Its <nodes of a standing wave> are at $x=0,1/3,2/3,1\,\mathrm m$ and its <antinodes> at $x=1/6,1/2,5/6\,\mathrm m$. These are constructed model parameters, not experimental readings.
\Image[/standing-wave-phase-snapshots.png]
{title=Opposite travelling waves add to a standing third mode; permanent nodes survive every phase snapshot}
At $t=1/120\,\mathrm s$, every point has zero <displacement>, but most have nonzero transverse <velocity>: the string is crossing its straight configuration. The black dots alone mark permanent <nodes of a standing wave>. At an <antinode> the greatest transverse <speed> is $2a\omega=0.6\pi\,\mathrm{m\,s^{-1}}$, much smaller than the chosen $20\,\mathrm{m\,s^{-1}}$ <wave speed>. One is motion of string material; the other is propagation of the constituent profiles.
The beauty of a <standing wave> is that motion reveals geometry: fixed <nodes of a standing wave> turn a moving disturbance into a spatial ruler. Their separation measures <wavelength>, and <boundary conditions> turn the continuous choice of <wavenumber> into a discrete set of <normal modes>. This is useful both for interpreting string vibrations and for understanding how a bounded <wave equation> selects its <eigenfrequencies>.
Back to article page
