= Phase-group orthogonality for degree-zero dispersion
{title2=$\mathbf c_p\cdot\mathbf c_g=0$}
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.
Back to article page
