Water Wave

Wave Synthesis

We want to create digital ocean waves. Let the height field be $\eta (x,t)$. The sum of different frequencies can be expressed as a sum of different frequencies:

$$\eta (x,t)=\sum _{k}h(k)\exp (i(kx-\omega t))$$

where

• $\lambda$ is the wavelength, and $p$ is the period.
• $k=2\pi /\lambda$
• $\omega =2\pi /p$.

There is a compatibility relation between $k$ and $\omega$ called the dispersion relation, $\omega =\omega (k)$.

In continuous 2D,

$$\eta (\mathbf{x},t)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }h(\mathbf{k})\exp (i(\mathbf{k}\cdot \mathbf{x}-{\omega }_{\mathbf{k}}t))d{k}_{x}d{k}_y$$

Let $\eta$ be discretized on a rectangle of size $L_x\times L_y$ with the number of grids $N_x\times N_y$. So,

$$h={\text{FFT}}_{\text{2D}}(\eta )\eta ={\text{IFFT}}_{\text{2D}}(h)$$

such that

$${\eta }_{n_x,n_y}=\frac{1}{N_x{N}_y}\sum _{m_x=0}^{N_x-1}\sum _{m_y=0}^{N_y-1}{h}_{m_x,m_y}\exp \left(2\pi i\left(\frac{m_x{n}_x}{N_x}+\frac{m_y{n}_y}{N_y}\right)\right)\exp \left(i{\omega }_{m_x,m_y}t\right)$$

so

$$\begin{array}{rl}\mathbf{k}& \approx \left(2\pi \frac{{\tilde{m}}_x}{L_x},2\pi \frac{{\tilde{m}}_y}{L_y}\right)\\ \tilde{m}& =\left(m+\frac{N}{2}\right)\mathrm{mod}N-\frac{N}{2}\end{array}$$

Wave synthesis boils down to

• modeling the dispersion relation ${\omega }_{\mathbf{k}}$,
• model the spectral energy density $h(\mathbf{k})$