Dispersion relation ($\omega (k)$)

A dispersion relation gives angular frequency $\omega$ as a function of wavenumber $k$ for monochromatic waves admitted by a linear wave equation.

The response to a localized perturbation must be followed in space as well as time. Growth at a fixed position is absolute wave-packet instability; growth transported away while the original position decays is convective wave-packet instability. The Briggs-Bers criterion evaluates the relevant complex spatial branches of a dispersion relation, while a real-wavenumber temporal mode test alone does not generally distinguish those behaviors.

A criterion for absolute wave-packet instability based on analytic continuation of the spatial roots of a dispersion relation. Start the temporal inversion contour above the spectrum, then lower it while deforming the spatial contour. A spatial pinch point occurs when branches originating in opposite spatial half-planes obstruct that deformation. A double root $D=D_k=0$ is only an algebraic candidate. The opposite-half-plane condition is stated in the primary study doi.org/10.1017/jfm.2016.195, section 4.2.

A collision of spatial branches which pinches the spatial inversion contour as the temporal contour is lowered from above all singularities. The branches must originate on opposite sides of the spatial contour. A collision of two upper-half-plane branches or two lower-half-plane branches can solve the double-root equations without causing absolute wave-packet instability.

The real dispersion above is temporally stable for every real $k$, but its analytic continuation has stationary points at $k=\pm i$ with $\omega =i/2$. Writing $\omega =i\Omega$ gives $k^2=-1\pm \sqrt{1/2-\Omega }$; as $\Omega$ decreases to $1/2$, both branches at $+i$ approach from the upper half-plane and both at $-i$ from the lower. These are not spatial pinch points. This provides a concrete counterexample to treating growing algebraic double roots as sufficient for absolute wave-packet instability.

A localized disturbance grows along some moving rays but decays at its original fixed location. This wave-packet meaning differs from buoyancy-driven stellar convection and is therefore given a distinct canonical title. Changing the observation frame can change the absolute/convective classification. The Briggs-Bers criterion uses the laboratory-frame impulse response to distinguish this case from absolute wave-packet instability.

A localized perturbation has positive asymptotic exponential growth at a fixed observation point in the chosen reference frame. The relevant spatial pinch point of the Briggs-Bers criterion has positive imaginary frequency. Merely finding a growing temporal mode or a formal complex saddle is not sufficient; the saddle must contribute to the localized impulse response.

With normal modes $e^{ikx-i\omega t}$, a finite upper bound on $\mathrm{Im}\omega$ allows the inverse temporal Laplace transform contour to start above all temporal singularities. An unbounded growth rate at high real wavenumber prevents the standard causal Green function construction used in the Briggs-Bers criterion. For $\mathrm{Im}\omega =\alpha k^2-\beta k^4-\gamma$ with nonzero real $\beta$, the bound holds exactly when $\beta >0$.

Wave dispersion means that phase velocity depends on wavenumber or, equivalently, wavelength. Wave packets generally travel with group velocity, which need not equal phase velocity. The linear rotating shallow-water dispersion relation ${\omega }^2=f^2+c^2{k}^2$ has faster long-wave phase motion but slower long-wave packet motion when $f\ne 0$.

A nondispersive wave has constant phase velocity on a propagation branch. For $\omega =ck$, its group velocity is also $c$. A frequency relation with a nonzero offset, such as $\omega =ck+b$, has constant group velocity but a phase velocity $c+b/k$ depending on wavenumber; constant group velocity alone is therefore not sufficient for nondispersion.

A dispersion diagram plots the branches of a dispersion relation, usually with wavenumber on the horizontal axis and angular frequency on the vertical axis. Its slopes, intersections, and gaps display group velocities, mode conversion, and forbidden frequency ranges.

Ray tracing follows the paths along which a slowly modulated wave packet propagates. For a local dispersion relation $\omega =\Omega (\mathbf{k};\mathbf{x},t)$, the ray velocity is the group velocity ${\nabla }_{\mathbf{k}}\Omega$.

A local dispersion relation $\omega =\Omega (\mathbf{k};\mathbf{X},T)$ yieldsThe spatial and temporal derivatives on the right hold the wavevector fixed. Translation symmetries conserve the corresponding wavevector components, while time independence conserves frequency.

A simple turning point for a stratified wavespeed occurs at $c(Y_s)=\omega /k$, where $Q={\omega }^2/c^2-k^2$ has a nonzero derivative. If $c_Y(Y_s)>0$, write $q=-Q^{\prime }(Y_s)>0$; then $\ell \sim [q(Y_s-Y){]}^{1/2}$ below it. The vertical group velocity vanishes and the stationary wave amplitude in a stratified wavespeed diverges as $(Y_s-Y{)}^{-1/4}$.

In ${\epsilon }^2{\Psi }^{\prime \prime }+2{\epsilon }^2(c^{\prime }/c){\Psi }^{\prime }+Q(Y)\Psi =0$, with $Q\sim -q(Y-Y_s)$ and $q>0$, set $Y-Y_s=({\epsilon }^2/q{)}^{1/3}\eta$. The leading equation is ${\Psi }_{\eta \eta }-\eta \Psi =0$, the Airy ordinary differential equation. The first derivative and nonlinear coefficient corrections are smaller by $O({\epsilon }^{2/3})$.

The local inner solution is a combination of Airy functions $\mathrm{Ai},\mathrm{Bi}$. Decay in the forbidden region removes $\mathrm{Bi}$. The remaining Airy turning-point connection formula is oscillatory on the allowed side and represents an incident/reflected pair, not a single upward branch. Without the forbidden-side condition the local differential equation alone leaves both coefficients free.

A simple ray turning point has outer amplitude proportional to $|Y-Y_s{|}^{-1/4}$. At the Airy scaling at a variable-speed wave turning point, $|Y-Y_s|=O({\epsilon }^{2/3})$, this gives an inner wavefield of size $O({\epsilon }^{-1/6})$ when the incident action normalization is order one. The enhancement is finite for nonzero $\epsilon$ and is resolved by the Airy function.

For the zero-frequency dispersion relationwrite $\mathbf{k}=\kappa (\cos \varphi ,\sin \varphi )$. Thenwhere $\psi$ is the direction of the group velocity. Maximizing the angular slope gives a downstream wake wedge of semi-angle

For a normal mode proportional to $e^{\sigma t}$, the real part $\mathrm{Re}\sigma$ is its exponential growth rate. A positive growth rate signals linear instability.

For a dispersion relation $\omega (k)$, the phase velocity is $c_p=\omega /k$ and the group velocity is $c_g=d\omega /dk$. For the Klein-Gordon relation at $k>0$,

If a differentiable dispersion relation $\omega (\mathbf{k})$ is homogeneous of degree zero at nonzero $\mathbf{k}$, the Euler homogeneous function theorem gives $\mathbf{k}\cdot {\nabla }_{\mathbf{k}}\omega =0$. Therefore phase velocity $\omega \mathbf{k}/k^2$ is orthogonal to group velocity ${\nabla }_{\mathbf{k}}\omega$. The result applies exactly to limiting inertial waves; finite compressibility can alter the dispersion.

Phase velocity is the velocity of a surface of constant phase. For a one-dimensional monochromatic wave it is $c_p=\omega /k$.

The magnitude of the phase velocity of a real-frequency plane Fourier mode is $|\omega |/|\mathbf{k}|$. In an anisotropic medium it depends on the direction of the wave vector and need not equal the group velocity magnitude.

Group velocity is the gradient of angular frequency with respect to wavevector, ${\mathbf{c}}_g={\nabla }_{\mathbf{k}}\omega$. It gives the leading propagation velocity of a narrow wave packet.

A wave crest is a point of constant phase. For a monochromatic wave it propagates at the phase velocity.

A wave packet is a localized superposition of nearby wavenumbers. Its envelope propagates at the group velocity to leading order.

A wave envelope is a slowly varying amplitude multiplying a rapidly oscillating carrier wave. Its variation describes localization, modulation, growth, or attenuation of the carrier.

Forthe dispersion relation is $\omega =Uk+k^9/9$, so $c_p=U+k^8/9$ and $c_g=U+k^8$.