WiRoot.cpp File Reference

Implements a function to find roots of equations involving Airy functions. More...

Functions
std::complex< double >	ITS::Propagation::LFMF::WiRoot (const int i, std::complex< double > &DWi, const std::complex< double > q, std::complex< double > &Wi, const AiryKind kind, const AiryScaling scaling)
	Finds the roots to the equation \( Wi'(ti) - q*Wi(ti) = 0 \).

Detailed Description

Implements a function to find roots of equations involving Airy functions.

Function Documentation

WiRoot()

std::complex< double > ITS::Propagation::LFMF::WiRoot	(	const int	i,
		std::complex< double > &	DWi,
		const std::complex< double >	q,
		std::complex< double > &	Wi,
		const AiryKind	kind,
		const AiryScaling	scaling )

Finds the roots to the equation \( Wi'(ti) - q*Wi(ti) = 0 \).

The parameter i selects the \(i\)-th root of the equation. The function \( Wi(ti) \) is the "Airy function of the third kind" as defined by Hufford [1] and Wait. The root is found by iteration starting from a real root.

Note

Although roots that are found for \( w_1 \) (Wait) and \( \mathrm{Wi}^{(2)} \) (Hufford) will be equal, and the roots found for \( w_2 \) (Wait) and \( \mathrm{Wi}^{(1)} \) (Hufford) will be equal, the return values for Wi and DWi will not be the same. The input parameters for kind and scale are used here as they are in Airy() for consistency.

Parameters

[in]	i	The \( i \)-th complex root of \( Wi'^{(2)}(ti) - q*Wi^{(2)}(ti) \), starting with 1.
[in]	q	Intermediate value: \( -j \nu \delta \)
[in]	kind	Kind of Airy function to use, either WONE or WTWO
[in]	scaling	Type of scaling to use, either HUFFORD or WAIT
[out]	DWi	Derivative of "Airy function of the third kind" \( Wi'^{(2)}(ti) \)
[out]	Wi	"Airy function of the third kind" \( Wi^{(2)}(ti) \)

Returns

The \( i \)-th complex root of the "Airy function of the third kind"

Exceptions

std::invalid_argument	If the values provided for i, kind, or scaling are not valid for this function.
std::runtime_error	If the root finding algorithm fails to converge.

References

• "Airy Functions of the third kind" are found in equation 38 of NTIA Report 87-219 "A General Theory of Radio Propagation through a Stratified Atmosphere", George Hufford, July 1987.

See also

ITS::Propagation::LFMF::AiryKind

ITS::Propagation::LFMF::AiryScaling

ITS::Propagation::LFMF::Airy