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Numerical dispersion relation in 2D (1)
\tilde\omega = \frac{2}{\Delta t}\sin^{-1}\left(
\left( C_x^2\sin^2 p_x + C_y^2\sin^ p_y\right)^\half\right)
For visualization, introduce \theta :
k_x = k\sin\theta,\quad k_y=k\cos\theta,
\quad p_x=\half kh\cos\theta,\quad p_y=\half kh\sin\theta
Also: \Delta x=\Delta y=h . Then C_x=C_y=c\Delta t/h\equiv C .
Now \tilde\omega depends on
- C reflecting the number cells a wave is displaced during a time step
- kh reflecting the number of cells per wave length in space
- \theta expressing the direction of the wave
