Internal Waves
Assume a complex exponential traveling-wave solution of the form
$$
(u, v, w) = (\hat{u}(z), \hat{v}(z), \hat{w}(z)) \exp \left\{ i(kx + ly - \omega t) \right\}
$$
Substitution of \(\boxed{(u, v, w) = (\hat{u}(z), \hat{v}(z), \hat{w}(z)) \exp \left\{ i(kx + ly - \omega t) \right\}
}\) into \(\boxed{\frac{\partial^2}{\partial t^2} \nabla^2 w + N^2 \nabla_H^2 w + f^2 \frac{\partial^2 w}{\partial z^2} = 0
}\) leads to an ordinary differential equation
$$
(-i\omega)^2 \left[ (ik)^2 + (il)^2 + \frac{d^2}{dz^2} \right] \hat{w}
+ N^2 \left[ (ik)^2 + (il)^2 \right] \hat{w} + f^2 \frac{d^2 \hat{w}}{dz^2} = 0
$$
Simplified to
$$
\frac{d^2 \hat{w}}{dz^2} + m^2(z) \hat{w} = 0 \quad \text{where} \quad
m^2(z) \equiv \frac{(k^2 + l^2)(N^2(z) - \omega^2)}{\omega^2 - f^2}
$$
- For $m^2 < 0$, the solutions must be exponentially decaying (evanescent) with increasing depth, signifying that the resulting wave motion is surface-trapped and corresponds to a surface wave propagating horizontally
- For $m^2 > 0$, the solutions are trigonometric in $z$ and correspond to internal waves propagating vertically as well as horizontally
- From \(m^2(z) \equiv \frac{(k^2 + l^2)(N^2(z) - \omega^2)}{\omega^2 - f^2}\), internal waves are only possible in the frequency range lies between $f < \omega < N$, provided that $N > f$, which is typically the case in both the atmosphere and the ocean
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