Phase-group orthogonality for degree-zero dispersion (${\mathbf{c}}_p\cdot {\mathbf{c}}_g=0$)

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.