Mixed domain -- pseudo-screen

The pseudo-screen solution to equation (9) derives from a first-order expansion of the square-root around a₀ and b₀ which are reference values for the medium characterized by the parameters a and b:

$$k_\tau\approx {k_\tau}_0+ \left. \done{k_\tau}{a} \right\ver... ...\done{k_\tau}{b} \right\vert _{a_0,b_0} \left (b-b_0\right )\;.$$ (24)

The partial derivatives relative to a and b, respectively, are:

$$\begin{eqnarray} \left. \done{k_\tau}{a} \right\vert _{a_0,b_0} &=& \omega\fra... ...\frac{a_0}{b_0} \left (\frac{b_0k_\gamma}{a_0 \omega}\right )^2\;.\end{eqnarray}$$ (25) (26)

Therefore, the pseudo-screen equation becomes

$$k_\tau\approx {k_\tau}_0+ \omega\left (a-a_0\right )+ \ome... ...}{a_0}\right )^2\left (\frac{ k_\gamma}{ \omega}\right )^2} \;.$$ (27)

If we make the notations

$$\left\{ \begin{array} {l} \nu = a_0\left [c_1\left (\frac{a}... ... 1 \\ \rho=3b\left (\frac{b_0}{a_0}\right )^2\end{array}\right.$$ (28)

we obtain the mixed-domain pseudo-screen solution to the one-way wave equation in Riemannian coordinates:

$$k_\tau\approx {k_\tau}_0+ \omega\left (a-a_0\right )+ \ome... ... )^2} {\mu-\rho\left (\frac{ k_\gamma}{ \omega}\right )^2} \;.$$ (29)