One-way wave equation

For the one-way wave equation

$$\frac{\partial p}{\partial z}= \pm \frac{i\omega}{c}\sqrt{1+\frac{c^2}{\omega^2}\frac{\partial^2}{\partial x^2}} p,$$

we can write its (2n+1)th order approximation Zhang (1985) in time domain

$$\begin{eqnarray} \left(\frac{\partial}{\partial z}\pm\frac{1}{c}\frac{\partial}{... ... x^2}\right)q(s_k,t,x,z) =\frac{\partial^2}{\partial x^2}p(t,x,z),\end{eqnarray}$$ (1) (2)

where q is the auxiliary wavefield, c is the velocity, and

$$s_k=\cos(\frac{k\pi}{n+1}),\ \ \ \ a_k=\frac{1}{n+1}\sin^2(\frac{k\pi}{n+1}),\ \ \ \ k=0,1,\cdots,n+1.$$

When n=0, we obtain the 5^o one-way equation

$$\left(\frac{\partial}{\partial z}\pm\frac{1}{c}\frac{\partial}{\partial t}\right)p=0.$$ (3)

When n=1, we obtain the 15^o one-way equation in Claerbout (1999)

$$\begin{eqnarray} \left(\frac{\partial}{\partial z}\mp\frac{1}{c}\frac{\partial}{... ...al^2}{\partial t^2}q=\frac{1}{2}\frac{\partial^2}{\partial x^2}p .\end{eqnarray}$$ (4) (5)

When n=2, we obtain the 45^o one-way wave equation in Claerbout (1999)

$$\begin{eqnarray} \left(\frac{\partial}{\partial z}\mp\frac{1}{c}\frac{\partial}{... ...\partial t^2}\right)q=\frac{1}{2}\frac{\partial^2}{\partial x^2}p.\end{eqnarray}$$ (6) (7)