Internal Wave

Phase speed is perpendicular to group speed

Let us assume that a fluid is inviscid, linear, non-diffusive, Boussinesq, and consists of motion independent of the $y$ coordinate. The Navier-Stokes equations can therefore be written using the stream-function, $\psi$, as

$(\nabla ^{2}\psi )_{tt}=-N^{2}(z)\psi _{xx}$.

Making the wave ansatz, $\psi =\exp {\left({\vec {k}}\cdot {\vec {x}}-\omega t\right)}$, where ${\vec {k}}=(k_{x},k_{z})$ and ${\vec {x}}=(x,z)$, the dispersion relation is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \frac{N k_x}{\lvert\vec{k}\rvert} }

where Failed to parse (unknown function "\lvert"): {\displaystyle \lvert\vec{k}\rvert = \sqrt{k_x^2 + k_z^2} } .

The phase speed is thus

Failed to parse (unknown function "\lvert"): {\displaystyle \vec{c}_p = \frac{\omega}{\lvert\vec{k}\rvert} \hat{k} = \frac{\omega}{\lvert\vec{k}\rvert^2} \vec{k} = \frac{Nk_x}{\lvert\vec{k}\rvert^3}\vec{k} = \frac{Nk_x}{\lvert\vec{k}\rvert^3}\left( k_x, k_z \right)} .

The group speed is

Failed to parse (unknown function "\lvert"): {\displaystyle \begin{align} \vec{c}_g &= \left( \frac{\partial\omega}{\partial k_x}, \frac{\partial\omega}{\partial k_z} \right)\\ &= \left( \frac{N}{\lvert\vec{k}\rvert} - \frac{N k_x^2}{\lvert\vec{k}\rvert^3}, - \frac{N k_x k_z}{\lvert\vec{k}\rvert^3} \right)\\ &= \left( \frac{N\left( k_x^2 + k_z^2\right)}{\lvert\vec{k}\rvert^3} - \frac{N k_x^2}{\lvert\vec{k}\rvert^3}, - \frac{N k_x k_z}{\lvert\vec{k}\rvert^3} \right)\\ &= \frac{N k_z}{\lvert\vec{k}\rvert^3}\left( k_z, - k_x \right)\\ \end{align}}

Therefore we find that the group velocity is perpendicular to the phase velocity!

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{c}_p \cdot \vec{c}_g = \frac{N^2 k_x k_z}{\lvert\vec{k}\rvert^6}\left( k_x, k_z \right) \cdot \left( k_z, -k_x \right) = 0}

This is completely counterintuitive from our experiences with surface waves, sound waves, and in fact most other waves, where the energy propagates in the same direction as the phase. The Saint Andrew's Cross experiment is the classic example demonstrating this. See Kundu for pictures and further elaboration.