WKBJ and its first order approximation

In this section, we demonstrate that the first order approximation of WKBJ is the same as the amplitude preserving one-way wave equation for v=v(z).

The one-way wave equation with the WKBJ amplitude correction is  

$$P(z+\Delta z)=\sqrt{\frac{k_z(z+\Delta z)}{k_z(z)}}e^{ik_z\Delta z}P(z).$$ (7)

The WKBJ amplitude correction term $\sqrt{\frac{k_z(z+\Delta z)}{k_z(z)}}$ can be rewritten as  

$$\sqrt{\frac{k_z(z+\Delta z)}{k_z(z)}}=e^{\frac{1}{2}\ln \frac{k_z(z+\Delta z)}{k_z(z)}}.$$ (8)

Then $k_z(z+\Delta z)$ in equation (8) can be linearized to $k_z(z)+\frac{\partial k_z}{\partial z}\Delta z$, so we have

$$\frac{1}{2}\ln \frac{k_z(z+\Delta z)}{k_z(z)} \approx\frac{... ...}\ln (1+\frac{1}{k_z}\frac{\partial k_z}{\partial z}\Delta z).$$ (9)

From

$$\ln (1+x)=x-x^2+\cdots,$$

and

$$\ln(1+x)\approx x,$$

we have

$$\frac{1}{2}\ln (1+\frac{1}{k_z}\frac{\partial k_z}{\partial ... ...frac{1}{2}\frac{1}{k_z}\frac{\partial k_z}{\partial z}\Delta z.$$ (10)

Because

$$\frac{1}{k_z}\frac{\partial k_z}{\partial z} =\frac{\partial}{\partial z}\ln k_z,$$ (11)

and from the dispersion relation $k_z=\sqrt{\frac{\omega^2}{v^2}-k_x^2}$, we have

$$\frac{1}{2}\frac{1}{k_z}\frac{\partial k_z}{\partial z}\Delt... ...partial z}\ln\left( \frac{\omega^2}{v^2}-k^2_x \right)\Delta z,$$ (12)

and  

$$\frac{1}{4}\frac{\partial }{\partial z}\ln\left( \frac{\omeg... ...frac{\partial v}{\partial z}\frac{\omega^2}{\omega^2-(k_xv)^2}.$$ (13)

From equation (8) to equation (13), we have

$$\sqrt{\frac{k_z(z+\Delta z)}{k_z(z)}}\approx e^{-\frac{1}{... ...tial v}{\partial z}\frac{\omega^2}{\omega^2-(k_xv)^2}}\Delta z.$$ (14)

So equation (7) can be rewritten as

$$\begin{eqnarray} P(z+\Delta z)&=&\sqrt{\frac{k_z(z+\Delta z)}{k_z(z)}}e^{ik_z\De... ...ial v}{\partial z}\frac{\omega^2}{\omega^2-(k_xv)^2}\Delta z}P(z).\end{eqnarray}$$ (15) (16)

Comparing the amplitude preserving one-way wave equation (equation (4)) with first order approximation of WKBJ (equation (16)), we find they are same. So we demonstrate theoretically that the amplitude preserving one-way wave equation is equivalent to the first order approximation of WKBJ.