Muir square-root expansion

Muir's method of finding wave extrapolators seeks polynomial ratio approximations to a square-root dispersion relation. Then fractions are cleared and the approximate dispersion relation is inverse transformed into a differential equation.

Recall equation (1)  

$$k_z \ \ \ =\ \ \ {\omega \over v }\ \sqrt { 1 \ - \ { v^2 \, k_x^2 \over \omega^2 }}$$ (59)

To inverse transform the z-axis we only need to recognize that i k_z corresponds to $\partial / \partial z$.Getting into the x-domain, however, is not simply a matter of substituting a second x derivative for k_x². The problem is the meaning of the square root of a differential operator. The square root of a differential operator is not defined in undergraduate calculus courses and there is no straightforward finite difference representation. The square root becomes meaningful only when it is regarded as some kind of truncated series expansion. It is shown in IEI that the Taylor series is a poor choice. Francis Muir showed that my original 15$^\circ$ and 45$^\circ$ methods were just truncations of a continued fraction expansion. To see this, define  

$$X \eq {v k_x \over\omega} \quad\quad {\rm and} \quad\quad R \eq {v k_z \over\omega}$$ (60)

With the definitions (60) equation  (59) is more compactly written as  

$$R \eq \sqrt { 1 \ - \ X^2 }$$ (61)

which you recognize as meaning that cosine is the square root of one minus sine squared. The desired polynomial ratio of order n will be denoted R_n, and it will be determined by the recurrence  

$$R_{n+1} \eq 1 \ -\ { X^2 \over 1 \ +\ R_n }$$ (62)

The recurrence is a guess that we verify by seeing what it converges to (if it converges). Set $n=\infty$ in (62) and solve

$$\begin{eqnarray} R_{\infty} \ \ &=& \ \ 1 \ -\ { X^2 \over 1 \ +\ R_{\infty} } \... ...1 \ +\ R_{\infty} \ -\ X^2 \nonumber \\ R^2 \ \ &=& \ \ 1 \ -\ X^2\end{eqnarray}$$ (63)

The square root of (63) gives the required expression (61). Geometrically, (63) says that the cosine squared of the incident angle equals one minus the sine squared and truncating the expansion leads to angle errors. Muir said, and you can verify, that his recurrence relationship formalizes what I was doing by re-estimating the $\partial_{zz}$ term. Although it is pleasing to think of large values of n, in real life only the low-order terms in the expansion are used. The first four truncations of Muir's continued fraction expansion beginning from R₀ = 1 are

$$\begin{eqnarray} 5^\circ : \quad\quad & R_0 &\eq 1 \\ 15^\circ : \quad\quad &... ...rut X^2 \over \displaystyle 2 - {\strut X^2\over 2}}} \nonumber\end{eqnarray}$$ (64)

For various historical reasons, the equations in the above equations are often referred to as the 5$^\circ$, 15$^\circ$, and 45$^\circ$ equations, respectively, the names giving a reasonable qualitative (but poor quantitative) guide to the range of angles that are adequately handled. A trade-off between complexity and accuracy frequently dictates choice of the 45$^\circ$ equation. It then turns out that a slightly wider range of angles can be accommodated if the recurrence is begun with something like $R_0 = \cos$ 45$^\circ$.Figure 9 shows some plots.

 

disper
Figure 9 Dispersion relation of equation (65). The curve labeled $45^\circ_+$ was constructed with $R_0 = \cos 45^\circ$. It fits exactly at 0$^\circ$ and 45$^\circ$.