= Dispersion relation
{title2=$\omega(k)$}
{wiki}
A <dispersion relation> specifies which <angular frequencies> and <wavevectors> a linear wave model admits. Fix the <plane wave> convention
$$
u(\mathbf x,t)=\operatorname{Re}\!\left\{a e^{i(\mathbf k\cdot\mathbf x-\omega t)}\right\},\qquad D(\omega,\mathbf k)=0.
$$
Here $a$ is a constant <complex amplitude>, $\mathbf k$ is the <wavevector>, and $\omega$ is the <angular frequency>. Substitution into a constant-coefficient <linear partial differential equation> replaces $\partial_t$ by $-i\omega$ and $\nabla$ by $i\mathbf k$, turning the wave problem into an algebraic condition. A system can have several roots; $\omega=\omega(\mathbf k)$ means one chosen branch, rather than a promise that the full <dispersion relation> is single-valued. For real $\mathbf k$, $\operatorname{Im}\omega>0$ means temporal growth in this convention, while $\operatorname{Im}\omega<0$ means <damping>. The real-branch discussion of <phase velocity> and <group velocity> below assumes negligible loss.
For the one-dimensional <wave equation> $u_{tt}=c^2u_{xx}$, the <dispersion relation> is $\omega^2=c^2k^2$. The branch $\omega=ck$ carries right-moving <travelling waves> for $c>0$; both its <phase velocity> and <group velocity> are $c$. One may instead use the positive-<angular frequency> branch $\omega=c|k|$, with right-moving components at $k>0$ and left-moving components at $k<0$. A real wave pairs each $(k,\omega)$ with $(-k,-\omega)$; that conjugate pair represents the same travelling pattern. Branch and sign choices therefore matter when interpreting a <dispersion diagram>.
A particularly tangible <dispersive> example is a small-amplitude <surface gravity wave> on still water of uniform depth $h$. Neglect <surface tension> and <viscosity>, assume <incompressible flow> and <irrotational flow>, and impose a flat impermeable bottom. With $k>0$ and positive <angular frequency>,
$$
\omega^2=gk\tanh(kh),\qquad g>0.
$$
Here $g$ is <gravitational acceleration> and $\tanh$ is the <hyperbolic tangent>. Deep water means $kh\gg1$: $\omega\simeq\sqrt{gk}$ and longer <wavelengths> have larger <phase speed>. Shallow water means $kh\ll1$: $\omega\simeq\sqrt{gh}\,k$, an approximately <nondispersive wave>. Thus depth relative to <wavelength>, rather than depth alone, determines whether crests and <wave packets> travel together.<Free Surface Water Waves (13.012 Hydrodynamics for Ocean Engineers, Reading 5) by A. H. Techet>[equation 3.2]{ref}
The <dispersion diagram> below also includes the ideal deep-water <capillary wave> limit $\omega=\sqrt{\gamma/\rho}\,k^{3/2}$, where $\gamma$ is <surface tension> and $\rho$ is <mass density>. This limit requires the <surface tension> restoring term to dominate <gravity>; it reverses the trend of the <surface gravity wave>, giving larger <phase speed> at shorter <wavelength>. The combined <gravity-capillary wave> has $\omega^2=gk+(\gamma/\rho)k^3$.<Pulse Propagation by Richard Fitzpatrick>[equation 9.7]{ref}
\Image[/phase-group-dispersion-slopes.png]
{title=Analytical dispersion diagrams: origin-to-point slope gives phase velocity, tangent slope gives group velocity}
The curves use dimensionless normalizations chosen separately for each model; they are analytical diagrams, not experimental data. Their beauty is that a single <dispersion relation> encodes several observable behaviours: its value gives the <angular frequency>, its origin-to-point slope gives the one-dimensional <phase velocity>, its tangent slope gives the <group velocity>, and variation of that tangent slope predicts distortion of a <wave packet>. A straight line through the origin describes a <nondispersive wave>; an affine branch $\omega=ck+b$ with $b\ne0$ has constant <group velocity> but <phase velocity> $c+b/k$, so those properties must be distinguished.
Back to article page
