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The radial displacement  \begin{equation}  \label{eqn:dr}  k_r^2=\bigg( k_r^2=\Bigg(  1 - \frac{N^2}{\omega^2}\bigg) \bigg(1 \frac{N^2}{\omega^2}\Bigg) \Bigg(1  - \frac{S_\ell^2}{\omega^2}\bigg)\,\frac{\omega^2}{c_s^2} \frac{S_\ell^2}{\omega^2}\Bigg)\,\frac{\omega^2}{c_s^2}  \end{equation}  Here, $\omega$ is the angular frequency of the wave, $N$ is the Brunt-Vaisala frequency, $S_\ell$ is the Lamb frequency for waves with harmonic degree $\ell$ and $c_s$ is the sound speed.