\[\underbrace{ \frac{\partial}{\partial t} \left( \rho_0 \frac{1}{2} (u^2 + v^2 + w^2) \right) }_{\text{Kinetic energy change}} + \underbrace{ g \rho' w }_{\text{Potential energy change}} + \underbrace{ \nabla \cdot (\mathbf{p'} \mathbf{u}) }_{\text{Work done by pressure forces (energy flux)}} = 0 \]

\[ \frac{\partial E_p}{\partial t} = \underbrace{g \rho' w}_{\text{Rate of change of potential energy}} = \frac{\partial}{\partial t} \left( \underbrace{ \frac{g^2 \rho'^2}{2 \rho_0 N^2} }_{\text{Depends quadratically on } \rho'} \right) \] \[ \frac{\partial \rho'}{\partial t} = \frac{N^2 \rho_0}{g} \frac{\partial \zeta}{\partial t} \Rightarrow E_p = \frac{1}{2} N^2 \rho_0 \zeta^2 \]

For two-layer systems: \(\frac{1}{4} (\rho_2 - \rho_1) g a^2\)
Sharp density jumps: \(N^2 = \frac{g}{\rho_0} (\rho_2 - \rho_1) \delta(z)\), \(\delta(z)\) is the Dirac delta function, it is valid in the integral sense

Assume periodic form \[ [u, w, p', \rho'] = [\hat{u}, \hat{w}, \hat{p}, \hat{\rho}] e^{i(kx+ly+mz-\omega t)} \] \[\Downarrow\] \[ p' = -\frac{\omega \rho_0}{k^2} m \hat{w} e^{i(kx+ly+mz-\omega t)} \quad , \quad \rho' = i \frac{N^2 \rho_0}{\omega g} \hat{w} e^{i(kx+ly+mz-\omega t)} \quad , \quad u = -\frac{m}{k} \hat{w} e^{i(kx+ly+mz-\omega t)} \]

Kinetic energy: \( E_k = \frac{1}{4} \rho_0 \left( \frac{m^2}{k^2} + 1 \right) \hat{w}^2 \)
Potential energy: \( E_p = \frac{N^2 \rho_0}{4 \omega^2} \hat{w}^2 \)
Using dispersion relation: \( \omega^2 = \frac{k^2 N^2}{k^2 + m^2} \Rightarrow E_k = E_p \)
\(\therefore\) the total wave energy \( E = E_k + E_p = \frac{1}{2} \rho_0 \left( \frac{m^2}{k^2} + 1 \right) \hat{w}^2 \)

\[ \mathbf{F} = -\frac{\rho_0 \omega m \hat{w}^2}{2k^2} \left( \mathbf{e}_x \frac{m}{k} - \mathbf{e}_z \right) \Rightarrow \mathbf{F} = \mathbf{c}_g E \] This shows that energy travels with group velocity