A numerical tour of wave propagation

Approximate the wave equation

The innovative work was done by John Claerbout, and is well-known as $15^{\circ}$ wave equation to separate the up-going and down-going waves (Claerbout, 1971,1986).

Eliminating the source term, the Fourier transform of the scalar wave equation (Eq. (5)) can be specified as:

$$\frac{\omega^2}{v^2}=k_x^2+k_z^2.$$ (17)

The down-going wave equation in Fourier domain is

$$k_z=\sqrt{\frac{\omega^2}{v^2}-k_x^2}=\frac{\omega}{v}\sqrt{1-\frac{v^2k_x^2}{\omega^2}}.$$ (18)

Using the different order Pade expansions, we have:

$$\left\{ \begin{split}\mathrm{1st-order:} &k_z=\frac{\omega}{v}\... ...omega^2}+\frac{v^4k_x^4}{16\omega^4}}\right)\\ \end{split}\right.$$ (19)

The corresponding time domain equations are:

$$\left\{ \begin{split}&\mathrm{1st-order:} \frac{\partial^2 p}{\... ...frac{\partial^5 p}{\partial t\partial x^4}=0\\ \end{split}\right.$$ (20)

A numerical tour of wave propagation