\(\boxed{\frac{\partial^2}{\partial t^2} \left( \frac{\partial^2 w}{\partial z^2} \right) = -\nabla_H^2 \left( \frac{\partial^2 w}{\partial t^2} + N^2 w \right) \text{ or } \frac{\partial^2}{\partial t^2} \left( \nabla^2 w \right) + N^2 \nabla_H^2 w = 0}\) is fundamentally different from that of interface waves because there is no obvious direction of propagation. For interface waves constrained to follow a horizontal surface with the \(x\)-axis chosen along the direction of wave propagation, a dispersion relation \(\omega(k)\) was obtained that is independent of the wave direction. Furthermore, wave crests and wave groups propagate in the same direction

The fluid is continuously stratified and internal waves might propagate in any direction and at any angle to the vertical. In such a case the direction of the wave number vector \( \mathbf{K} = (k, l, m) \) becomes important and the dispersion relationship is anisotropic and depends on the wave number components \[ \omega = \omega(k, l, m) = \omega(\mathbf{K}) \]

The propagation of internal waves is a baroclinic process, in which the surfaces of constant pressure do not coincide with the surfaces of constant density BarotropicBaroclinicMode To reveal the structure of the situation described by \(\boxed{\frac{\partial^2}{\partial t^2} \left( \frac{\partial^2 w}{\partial z^2} \right) = -\nabla_H^2 \left( \frac{\partial^2 w}{\partial t^2} + N^2 w \right) \text{ or } \frac{\partial^2}{\partial t^2} \left( \nabla^2 w \right) + N^2 \nabla_H^2 w = 0}\) and \(\boxed{\omega = \omega(k, l, m) = \omega(\mathbf{K})}\), consider the complex version of \(\eta = a \cos\left( kx + ly + mz - \omega t \right) = a \cos\left( \mathbf{K} \cdot \mathbf{x} - \omega t \right)\) with wave number vector \(\mathbf{K} = (k, l, m)\) for the vertical velocity in a fluid medium having a constant buoyancy frequency \[ w = w_0 e^{i(kx + ly + mz - \omega t)} = w_0 e^{i(\mathbf{K} \cdot \mathbf{x} - \omega t)} \] Substituting it into \(\boxed{\frac{\partial^2}{\partial t^2} \left( \frac{\partial^2 w}{\partial z^2} \right) = -\nabla_H^2 \left( \frac{\partial^2 w}{\partial t^2} + N^2 w \right) \text{ or } \frac{\partial^2}{\partial t^2} \left( \nabla^2 w \right) + N^2 \nabla_H^2 w = 0}\) with constant \(N\) leads to the dispersion relation \[ \omega^2 = \frac{k^2 + l^2}{k^2 + l^2 + m^2} N^2 \]