Phase velocity and group velocity

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 $c_g=U+k^8$.