Plane waves

Drag the arrow's tip (or click anywhere) to set the wavevector k; the white dot rides one crest

Experiments

Start here

The picture is a plane wave: the colour is its phase (red where it is real and positive, cyan where negative) and the stripes are its crests, one wavelength apart, sliding across the screen. The white arrow is the wavevector k: drag its tip (or click anywhere) and the stripes turn to stay square to it, and get finer as it gets longer. The white dot rides one crest; the dial at the bottom left is the wave at the centre of the picture going round the complex plane.

Try the four buttons under The wave: the same stripes, travelling along the arrow or against it. Then Two waves (standing waves, fringes, beats) and A packet: dispersion, where the red dot rides a crest and the blue triangle marks the packet, and the two go at different speeds.

1. Which way does it go?

Choose each of the four forms in turn and watch the crests (the white dot rides one). $e^{i(\vec k\cdot\vec r-\omega t)}$ and $e^{-i(\vec k\cdot\vec r-\omega t)}$ both travel along $\vec k$: they are complex conjugates, the same wave with its colours turning the other way (physicists write the first; electrical engineers write $e^{j(\omega t-\vec k\cdot\vec r)}$, which is the second). When $\vec k\cdot\vec r$ and $\omega t$ have the same sign, the wave travels against $\vec k$. What matters is the relative sign: a crest is where the phase is constant, $\vec k\cdot\vec r=\omega t$, and that moves along $\vec k$ as $t$ grows.

2. At one point, the phase goes round

The little complex plane shows $\psi$ at the centre of the picture: an arrow going round the circle, $\omega/2\pi$ times a second (at speed 1). With $e^{-i\omega t}$ it turns clockwise: that is the sign quantum mechanics fixes, $e^{-iEt/\hbar}$ for energy $E>0$; the two forms with $e^{+i\omega t}$ turn it anticlockwise. Change the phase of $A$ (the slider $\arg A_1$) and every crest shifts sideways: a phase is a position.

3. The real part is what a ripple tank shows

Only $\mathrm{Re}\,\psi=|A|\cos(\vec k\cdot\vec r-\omega t+\varphi)$, in grey: crests light, troughs dark. A real wave needs both $e^{i(\vec k\cdot\vec r-\omega t)}$ and its conjugate. A snapshot of the real part cannot tell you which way the wave is going; a snapshot of the complex wave can, once the sign convention is agreed: with $e^{-i\omega t}$ the wave moves towards the side where its phase increases (red, yellow-green, cyan, violet, red again).

4. Standing waves

Two equal waves travelling in opposite directions, $\vec k_2=-\vec k_1$. The sum $2\cos(\vec k\cdot\vec r)\,e^{-i\omega t}$ does not travel at all: its nodes (black lines, half a wavelength apart: $\lambda/2=2.09$ here) stay put, and between them the whole region changes colour together. This is a string, an organ pipe, a particle in a box.

5. Crossed beams make fringes

Two waves of the same wavelength crossing at an angle (60° here). Where they are in step the sum is bright, out of step dark: fringes perpendicular to $\vec k_1-\vec k_2$ (along $\vec k_1+\vec k_2$), spaced by $2\pi/|\vec k_1-\vec k_2|$, which is 4.05 here: at 60° the fringes are as far apart as the crests ($\lambda=4.05$). Drag either arrow to change the angle: the wider it is, the finer the fringes. This is how a hologram is recorded.

6. Beats

Two waves in the same direction with different $|\vec k|$ (1.20 and 1.74 here) and so, with $\omega=c|\vec k|$ as for light, different $\omega$. The sum is a fast wave inside a slowly beating envelope, whose dark bands are $2\pi/\Delta k=11.6$ apart. The envelope moves at $\Delta\omega/\Delta k$ and the crests at $\omega/k$; for light both are $c$, so here the whole pattern slides along together. That is the first step towards a wave packet, where the next mode lets the two speeds differ.

7. Phase velocity and group velocity

A packet of waves near $k_0$, for a free quantum particle, $\omega=\hbar k^2/2m$ (with $\hbar=m=1$). The crests (red dot) move at the phase velocity $\omega/k=k_0/2=0.75$; the packet (blue triangle) at the group velocity $d\omega/dk=k_0=1.5$, twice as fast: the packet overtakes its crests, so they are born at its front and die at its back. (The packet also spreads: each wavelength has its own speed.) On the $\omega(k)$ plot the two velocities are slopes: of the line from the origin to the dot (the red chord), and of the tangent there (the blue line).

8. Deep water waves

$\omega=\sqrt{gk}$: now the group moves at half the phase velocity (0.82 against 1.63), and crests run forward through the packet and vanish at its front, as you can see behind a boat or a duck.

9. Light in vacuum

$\omega=c|k|$: every wavelength moves at the same speed $c$, so the packet keeps its shape and crests ride along with it (both velocities read 1.000). No dispersion.

10. Faster than light, harmlessly

$\omega=\sqrt{c^2k^2+\mu^2}$ with $c=1$: a relativistic particle ($\mu=mc^2/\hbar$), or light in a waveguide or in a plasma ($\mu$ the cut-off or the plasma frequency). The crests move faster than light, at 1.41; the packet, which carries the energy and the information, slower, at 0.71; and $v_\text{phase}\,v_\text{group}=c^2$ (1.41 × 0.71 = 1).

How it works

A plane wave $\psi(\vec r,t)=A\,e^{i(\vec k\cdot\vec r-\omega t)}$ has the same value all along each line perpendicular to $\vec k$, and its phase advances by $2\pi$ every wavelength $\lambda=2\pi/|\vec k|$ along $\vec k$. The colours are its phase (red is positive real, yellow-green $+i$, cyan negative real, violet $-i$); the line under the picture gives $|\vec k|$, $\lambda$, $\omega$ (the angular frequency, $2\pi$ times the frequency) and the speed of the crests. The complex amplitude $A=|A|e^{i\varphi}$ sets the height of the wave and where its crests sit. This is the BrainFlux PlaneWave applet, with more to do.

The four forms. The phase is $\pm\vec k\cdot\vec r\pm\omega t$. A crest is where the phase stays constant, so it moves along $\vec k$ when the two signs differ and against $\vec k$ when they agree; $e^{i(\ldots)}$ and $e^{-i(\ldots)}$ are complex conjugates, the same wave with the colours turning the other way. Physicists write $e^{i(\vec k\cdot\vec r-\omega t)}$, whose phasor turns clockwise; electrical engineers $e^{j(\omega t-\vec k\cdot\vec r)}$.

Phase velocity, group velocity, dispersion. Crests move at the phase velocity $v_p=\omega/|\vec k|$. A packet, a sum of waves with wavenumbers near $k_0$, moves as a whole at the group velocity $v_g=d\omega/dk$ at $k_0$. On a plot of $\omega(k)$, $v_p$ is the slope of the chord from the origin to $(k_0,\omega_0)$ and $v_g$ the slope of the tangent there. They are equal only when $\omega$ is proportional to $k$ (light in vacuum); otherwise the waves are dispersive: each wavelength has its own speed and the packet spreads.

What is computed. With two waves both have $\omega=c|\vec k|$, as for light, so a standing wave needs equal and opposite $\vec k$. The packet is $\psi(x,t)=\int\tilde\psi(k)\,e^{i(kx-\omega(k)t)}\,dk$ with $\tilde\psi$ a Gaussian about $k_0$ ($|\tilde\psi|^2$ has standard deviation $\Delta k$), summed over 2048 wavenumbers by an FFT; each component runs at $\omega(|k|)$, so the part with $k<0$ goes left. Units: $\hbar=m=1$ for the Schrödinger relation, $c=1$ for light and the relativistic one.

Rewritten in 2026 from the BrainFlux PlaneWave Java applet. Source: github.com/gallicchio/brainflux, MIT licence.