**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)

(59) |

To inverse transform the *z*-axis we
only need to recognize that *i k*_{z} corresponds
to .Getting into the *x*-domain, however,
is not simply a matter of substituting
a second *x* derivative for *k*_{x}^{2}.
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 and 45 methods were
just truncations of a continued fraction expansion.
To see this, define

(60) |

(61) |

(62) |

(63) |

(64) | ||

For various historical reasons, the equations in the above equations are often referred to as the 5, 15, and 45 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 equation. It then turns out that a slightly wider range of angles can be accommodated if the recurrence is begun with something like 45.Figure 9 shows some plots.

Figure 9

12/26/2000