Similarity Model of Spatial Spectra of Random Anisoscale Inhomogeneities of Atmosphere Permittivity and Its Application to Wave Propagation Problems

Abstract

The similarity model of anisoscale fluctuations of permittivity in a turbulent atmosphere is suggested. Correlation properties of fluctuations in different spatial directions are shown to be similar and to differ only in the direction-dependent scaling factor. The power-law similarity model of fluctuations is proposed. It is shown that in the geometric optics approximation the correlation properties of the phase of plane (and spherical) wave are similar and differ only in the scaling factor, depending on wave-propagation directions and the base (the coordinate difference at probe points). Correlation characteristics (dispersion, coefficient of anisomery, and a form of correlation coefficient) are calculated for a number of typical parameters of the problem.

Appendices

FORMULAS OF CORRELATION OF PERMITTIVITY FLUCTUATIONS

We show the explicit form of the correlation coefficient $R_{\varepsilon }^{0}(p,n,\delta )$, defined in (3.4), for particular cases $n = 0;1$ and spectra $p = {{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0em} 6};$$p = {5 \mathord{\left/ {\vphantom {5 {2{\kern 1pt} }}} \right. \kern-0em} {2{\kern 1pt} }}{\kern 1pt} :$

$$R_{\varepsilon }^{0}(p,0,\delta ) = \frac{{2{{{\left| \delta \right|}}^{{p - 3/2}}}{{K}_{{p - 3/2}}}(\left| \delta \right|)}}{{{{2}^{{p - 3/2}}}\Gamma (p - {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2})}};$$

$$R_{\varepsilon }^{0}({{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0em} 6},1,\delta ) = \frac{{{{2}^{{2/3}}}}}{{3\Gamma ({1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3})}}{{\left| \delta \right|}^{{1/3}}}\left( {3{{K}_{{1/3}}}(\left| \delta \right|) - \left| \delta \right|{{K}_{{2/3}}}(\left| \delta \right|)} \right);$$

$$R_{\varepsilon }^{0}({5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2},1,\delta ) = \frac{2}{9}\left( {5G_{{1,3}}^{{2,1}}\left( {\frac{{{{{\left| \delta \right|}}^{2}}}}{4}\left| \begin{gathered} {{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0em} 2} \hfill \\ 0,1,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2} \hfill \\ \end{gathered} \right.} \right) + {{{\left| \delta \right|}}^{2}}{{K}_{2}}(\left| \delta \right|)} \right),$$

${{K}_{q}}(x)$ is the modified Bessel function and $G_{{p,q}}^{{m,n}}\left( {z\left| \begin{gathered} {{a}_{1}},...,{{a}_{p}} \hfill \\ {{b}_{1}},...,{{b}_{q}} \hfill \\ \end{gathered} \right.} \right)$ is the Meijer’s G-function.

For coefficient (3.6), it is easy to derive the recurrent formula ${{C}_{0}}(p,n + 1)$ = ${{C}_{0}}(p,n){{(p + n)} \mathord{\left/ {\vphantom {{(p + n)} {(n + 1)}}} \right. \kern-0em} {(n + 1)}}$, allowing, e.g., it to be simplified significantly for integer $n$:

$$\begin{gathered} {{C}_{0}}(p,n) = p(p + 1)(p + 2)...{{(p + n - 1)} \mathord{\left/ {\vphantom {{(p + n - 1)} n}} \right. \kern-0em} n}!; \\ {{C}_{0}}(p,0) = 1. \\ \end{gathered} $$

For integral (3.8), it is likewise easy to obtain recurrent formula ${{I}_{0}}(p,n,\lambda )$ = $[{{(1 - \lambda )}^{n}} +$${{n{{I}_{0}}(p,n - 1,\lambda )]} \mathord{\left/ {\vphantom {{n{{I}_{0}}(p,n - 1,\lambda )]} {(p + n - 1)}}} \right. \kern-0em} {(p + n - 1)}}$, which makes it possible, e.g., to significantly simplify this integral for integer $n$:

$$\begin{gathered} {{I}_{0}}(p,0,\lambda ) = \frac{1}{{p - 1}};\,\,\,\,{{I}_{0}}(p,1,\lambda ) = \frac{1}{{p - 1}} - \frac{\lambda }{p}; \\ {{I}_{0}}(p,2,\lambda ) = \frac{1}{{p - 1}} - \frac{{2\lambda }}{p} + \frac{{{{\lambda }^{2}}}}{{p + 1}}; \\ {{I}_{0}}(p,3,\lambda ) = \frac{1}{{p - 1}} - \frac{{3\lambda }}{p} + \frac{{3{{\lambda }^{2}}}}{{p + 1}} - \frac{{{{\lambda }^{3}}}}{{p + 2}};... \\ \end{gathered} $$

Taking into account that $0 < \lambda (k) \leqslant 1,$ we find the estimate $C_{0}^{{ - 1}}(p,n) \leqslant (p - 1){{I}_{0}}(p,n,\lambda ) \leqslant 1.$

CALCULATION OF THE COEFFICIENT OF SCALABILITY OF THE WAVE EIKONAL ANISOMERY

To calculate the scalability coefficient $C({{{\mathbf{e}}}_{s}},{{{\mathbf{e}}}_{ \bot }}) =$$C({{\theta }_{s}},{{\varphi }_{s}},{{\psi }_{\rho }})$ (here $({{\theta }_{s}},{{\varphi }_{s}})$ are the angles of vector ${{{\mathbf{e}}}_{s}}$ in the spherical coordinate system and ${{\psi }_{\rho }}$ is the polar angle of vector ${{{\mathbf{e}}}_{ \bot }}$ in the polar coordinate system on the plane ${{\mathbb{S}}_{s}} \bot {{{\mathbf{e}}}_{s}}$), we express components of vectors ${{{\mathbf{e}}}_{s}},{{{\mathbf{e}}}_{ \bot }}$ (as well as ${{{\mathbf{E}}}_{s}},{{{\mathbf{E}}}_{ \bot }}$) in the initial Cartesian coordinate system $(x,y,z)$:

$$\begin{gathered} {{{\mathbf{e}}}_{s}}({{\theta }_{s}},{{\varphi }_{s}}) = ({{\begin{array}{*{20}{c}} {\sin {{\theta }_{s}}\cos {{\varphi }_{s}},}&{\sin {{\theta }_{s}}\sin {{\varphi }_{s}},}&{\cos \theta } \end{array}}_{s}}), \\ {{{\mathbf{E}}}_{s}}({{\theta }_{s}},{{\varphi }_{s}}) \\ = (\begin{array}{*{20}{c}} {{{\beta }_{x}}\sin {{\theta }_{s}}\cos {{\varphi }_{s}},}&{{{\beta }_{y}}\sin {{\theta }_{s}}\sin {{\varphi }_{s}},}&{{{\beta }_{z}}\cos {{\theta }_{s}}} \end{array}), \\ \end{gathered} $$

$$\begin{gathered} {{{\mathbf{e}}}_{ \bot }}({{\theta }_{s}},{{\varphi }_{s}},{{\psi }_{\rho }}) \equiv {{\Delta {{{\mathbf{r}}}_{ \bot }}} \mathord{\left/ {\vphantom {{\Delta {{{\mathbf{r}}}_{ \bot }}} {\Delta {{r}_{ \bot }}}}} \right. \kern-0em} {\Delta {{r}_{ \bot }}}} = \cos {{\psi }_{\rho }}{{{\mathbf{e}}}_{\theta }} + \sin {{\psi }_{\rho }}{{{\mathbf{e}}}_{\varphi }} \\ = \left( \begin{gathered} \cos {{\psi }_{\rho }}\cos {{\theta }_{s}}\cos {{\varphi }_{s}} - \sin {{\psi }_{\rho }}\sin {{\varphi }_{s}} \hfill \\ \cos {{\psi }_{\rho }}\cos {{\theta }_{s}}\sin {{\varphi }_{s}} + \sin {{\psi }_{\rho }}\cos {{\varphi }_{s}} \hfill \\ - \cos {{\psi }_{\rho }}\sin {{\theta }_{s}} \hfill \\ \end{gathered} \right), \\ {{{\mathbf{E}}}_{ \bot }}({{\theta }_{s}},{{\varphi }_{s}},{{\psi }_{\rho }}) = ({{\beta }_{x}}{{e}_{{ \bot x}}},{{\beta }_{y}}{{e}_{{ \bot y}}},{{\beta }_{z}}{{e}_{{ \bot z}}}) \\ = \left( {\begin{array}{*{20}{c}} {{{\beta }_{x}}(\cos {{\theta }_{s}}\cos {{\varphi }_{s}}\cos {{\psi }_{\rho }} - \sin {{\varphi }_{s}}\sin {{\psi }_{\rho }})} \\ {{{\beta }_{y}}(\cos {{\theta }_{s}}\sin {{\varphi }_{s}}\cos {{\psi }_{\rho }} + \cos {{\varphi }_{s}}\sin {{\psi }_{\rho }})} \\ {{{\beta }_{z}}( - \sin {{\theta }_{s}}\cos {{\psi }_{\rho }})} \end{array}} \right). \\ \end{gathered} $$

Here, ${{{\mathbf{e}}}_{\varphi }} = (\begin{array}{*{20}{c}} { - \sin {{\varphi }_{s}},}&{\cos {{\varphi }_{s}},}&0 \end{array}),$${{{\mathbf{e}}}_{s}} = (\sin {{\theta }_{s}}\cos {{\varphi }_{s}},$$\sin {{\theta }_{s}}\sin {{\varphi }_{s}},\cos {{\theta }_{s}})$ are the unit vectors on the plane ${{\mathbb{S}}_{s}} \bot {{{\mathbf{e}}}_{s}}$ in the local coordinate system. The calculation of cross product (4.10) yields

$$\begin{gathered} \left[ {{{{\mathbf{E}}}_{ \bot }}({{{\mathbf{e}}}_{s}},{{{\mathbf{e}}}_{ \bot }}) \times {{{\mathbf{E}}}_{s}}({{{\mathbf{e}}}_{s}})} \right] \\ = \left( {\begin{array}{*{20}{c}} {{{\beta }_{y}}{{\beta }_{z}}(\cos {{\theta }_{s}}\cos {{\varphi }_{s}}\sin {{\psi }_{\rho }} + \sin {{\varphi }_{s}}\cos {{\psi }_{\rho }})} \\ {{{\beta }_{x}}{{\beta }_{z}}(\cos {{\theta }_{s}}\sin {{\varphi }_{s}}\sin {{\psi }_{\rho }} - \cos {{\varphi }_{s}}\cos {{\psi }_{\rho }})} \\ {{{\beta }_{x}}{{\beta }_{y}}( - \sin {{\theta }_{s}}\sin {{\psi }_{\rho }})} \end{array}} \right). \\ \end{gathered} $$

To simplify calculations, we introduce for multipliers of horizontal anisomery on the plane $(x,y)$ the following designations:

$$\begin{gathered} {{{\rm B}}_{{xy}}} \equiv {{\beta }_{x}}{{\beta }_{y}},\,\,\,\,{{{\rm B}}_{z}} \equiv \beta _{z}^{2} = {1 \mathord{\left/ {\vphantom {1 {(\beta _{x}^{2}\beta _{y}^{2})}}} \right. \kern-0em} {(\beta _{x}^{2}\beta _{y}^{2})}} = {\rm B}_{{xy}}^{{ - 2}}, \\ \nu \equiv {{(\beta _{y}^{2} - \beta _{x}^{2})} \mathord{\left/ {\vphantom {{(\beta _{y}^{2} - \beta _{x}^{2})} {(2{{\beta }_{x}}{{\beta }_{y}})}}} \right. \kern-0em} {(2{{\beta }_{x}}{{\beta }_{y}})}}, \\ \end{gathered} $$

hence

$$\begin{gathered} \beta _{x}^{2} = {{{\rm B}}_{{xy}}}\left( {\sqrt {1 + {{\nu }^{2}}} - \nu } \right) = {{{\rm B}}_{{xy}}}\left( {q(\nu ) - \nu } \right), \\ \beta _{y}^{2} = {{{\rm B}}_{{xy}}}\left( {\sqrt {1 + {{\nu }^{2}}} + \nu } \right) = {{{\rm B}}_{{xy}}}\left( {q(\nu ) + \nu } \right), \\ q(\nu ) \equiv \sqrt {1 + {{\nu }^{2}}} = {{(\beta _{x}^{2} + \beta _{y}^{2})} \mathord{\left/ {\vphantom {{(\beta _{x}^{2} + \beta _{y}^{2})} {(2{{\beta }_{x}}{{\beta }_{y}})}}} \right. \kern-0em} {(2{{\beta }_{x}}{{\beta }_{y}})}};\,\,\,\left( {\nu \equiv \sqrt {{{q}^{2}} - 1} } \right), \\ \end{gathered} $$

and the horizontal anisomery multipliers depending on the polar angle ${{\varphi }_{s}}$

$$\begin{gathered} {{{\rm B}}_{ \bot }}({{\varphi }_{s}}) \equiv \beta _{x}^{2}{{\cos }^{2}}{{\varphi }_{s}} + \beta _{y}^{2}{{\sin }^{2}}{{\varphi }_{s}} \\ = {{{\rm B}}_{{xy}}}[q(\nu ) - \nu \cos (2{{\varphi }_{s}})], \\ {{{\rm B}}_{\parallel }}({{\varphi }_{s}}) \equiv \beta _{x}^{2}{{\sin }^{2}}{{\varphi }_{s}} + \beta _{y}^{2}{{\cos }^{2}}{{\varphi }_{s}} \\ = {{{\rm B}}_{{xy}}}[q(\nu ) + \nu \cos (2{{\varphi }_{s}})], \\ {{{\rm B}}_{\Delta }}({{\varphi }_{s}}) \equiv (\beta _{y}^{2} - \beta _{x}^{2})\sin {{\varphi }_{s}}\cos {{\varphi }_{s}} = {{{\rm B}}_{{xy}}}\nu \sin (2{{\varphi }_{s}}). \\ \end{gathered} $$

For the wavenumber in the direction of ${{{\mathbf{e}}}_{s}}$, we have

$${{\beta }_{0}}^{2}({{{\mathbf{e}}}_{s}}) = {{{\rm B}}_{ \bot }}({{\varphi }_{s}}){{\sin }^{2}}{{\theta }_{s}} + {{{\rm B}}_{z}}{{\cos }^{2}}{{\theta }_{s}}.$$

After calculations and simplification, we finally find an expression for the scalability coefficient of spatial correlation

$${{Q}^{2}}({{\theta }_{s}},{{\varphi }_{s}},{{\psi }_{\rho }}) \equiv {{C}^{2}}({{\theta }_{s}},{{\varphi }_{s}},{{\psi }_{\rho }}){{\beta }_{0}}^{2}({{\theta }_{s}},{{\varphi }_{s}}).$$

As a function of parameter ${{\psi }_{\rho }}$, this is an ellipse on the plane ${{\mathbb{S}}_{s}} \bot {{{\mathbf{e}}}_{s}},$ the parameters of which—the semiaxes and angle of semiaxes inclination ${{\psi }_{0}}$—can be calculated as functions of coordinates of the wave-incidence vector ${{{\mathbf{e}}}_{s}}$ by reducing the quadratic form to a sum of squares by a turn through this angle:

$$\begin{gathered} {{Q}^{2}}({{{\mathbf{e}}}_{s}},{{\psi }_{\rho }}) = {{A}_{1}}({{{\mathbf{e}}}_{s}}){{\cos }^{2}}{{\psi }_{\rho }} + {{A}_{2}}({{{\mathbf{e}}}_{s}}){{\sin }^{2}}{{\psi }_{\rho }} \\ + \,\,2D({{{\mathbf{e}}}_{s}})\sin {{\psi }_{\rho }}\cos {{\psi }_{\rho }}. \\ \end{gathered} $$

Here,

$$\begin{gathered} {{A}_{1}}({{{\mathbf{e}}}_{s}}) = {{{\rm B}}_{z}}{{{\rm B}}_{ \bot }}({{\varphi }_{s}}), \\ {{A}_{2}}({{{\mathbf{e}}}_{s}}) = {{{\rm B}}_{z}}[{{{\rm B}}_{\parallel }}({{\varphi }_{s}}){{\cos }^{2}}{{\theta }_{s}} + {\rm B}_{{xy}}^{4}{{\sin }^{2}}{{\theta }_{s}}], \\ D({{{\mathbf{e}}}_{s}}) = {{{\rm B}}_{z}}{{{\rm B}}_{\Delta }}({{\varphi }_{s}})\cos {{\theta }_{s}}. \\ \end{gathered} $$

Determining the parameters $M({{{\mathbf{e}}}_{s}}) \geqslant 0$ and ${{\psi }_{0}}$ from the relations

$$\begin{gathered} {{A}_{1}}({{{\mathbf{e}}}_{s}}) - {{A}_{2}}({{{\mathbf{e}}}_{s}}) = M({{{\mathbf{e}}}_{s}})\cos 2{{\psi }_{0}}, \\ 2D({{{\mathbf{e}}}_{s}}) = M({{{\mathbf{e}}}_{s}})\sin 2{{\psi }_{0}}, \\ {{M}^{2}}({{{\mathbf{e}}}_{s}}) = {{[{{A}_{1}}({{{\mathbf{e}}}_{s}}) - {{A}_{2}}({{{\mathbf{e}}}_{s}})]}^{2}} + 4{{D}^{2}}({{{\mathbf{e}}}_{s}}), \\ \end{gathered} $$

we derive

$$\begin{gathered} 2{{Q}^{2}}({{{\mathbf{e}}}_{s}},{{\psi }_{\rho }}) = {{A}_{1}}({{{\mathbf{e}}}_{s}}) + {{A}_{2}}({{{\mathbf{e}}}_{s}}) \\ + \,\,M({{{\mathbf{e}}}_{s}})\cos (2{{\psi }_{\rho }} - 2{{\psi }_{0}}) \\ = ({{A}_{1}} + {{A}_{2}} + M){{\cos }^{2}}({{\psi }_{\rho }} - {{\psi }_{0}}) \\ + \,\,({{A}_{1}} + {{A}_{2}} - M){{\sin }^{2}}({{\psi }_{\rho }} - {{\psi }_{0}}). \\ \end{gathered} $$

From here it can be seen that

$$\begin{gathered} Q_{{\max }}^{2}({{{\mathbf{e}}}_{s}}) = \mathop {\max }\limits_{{{\psi }_{\rho }}} \{ {{Q}^{2}}({{{\mathbf{e}}}_{s}},{{\psi }_{\rho }})\} \\ = [{{A}_{1}}({{{\mathbf{e}}}_{s}}) + {{A}_{2}}({{{\mathbf{e}}}_{s}}) + {{M({{{\mathbf{e}}}_{s}})]} \mathord{\left/ {\vphantom {{M({{{\mathbf{e}}}_{s}})]} 2}} \right. \kern-0em} 2} \\ {\text{with}}\,\,\,\,{{\psi }_{\rho }} = {{\psi }_{0}} + \pi n, \\ \end{gathered} $$

$$\begin{gathered} Q_{{\min }}^{2}({{{\mathbf{e}}}_{s}}) = \mathop {\min }\limits_{{{\psi }_{\rho }}} \{ {{Q}^{2}}({{{\mathbf{e}}}_{s}},{{\psi }_{\rho }})\} \\ = [{{A}_{1}}({{{\mathbf{e}}}_{s}}) + {{A}_{2}}({{{\mathbf{e}}}_{s}}) - {{M({{{\mathbf{e}}}_{s}})]} \mathord{\left/ {\vphantom {{M({{{\mathbf{e}}}_{s}})]} 2}} \right. \kern-0em} 2} \\ {\text{with}}\,\,\,\,{{\psi }_{\rho }} = {{\psi }_{0}} + \pi n + {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}, \\ \end{gathered} $$

$${{C}_{{\max ,\min }}}({{{\mathbf{e}}}_{s}}) = {{{{Q}_{{\max ,\min }}}({{{\mathbf{e}}}_{s}})} \mathord{\left/ {\vphantom {{{{Q}_{{\max ,\min }}}({{{\mathbf{e}}}_{s}})} {{{\beta }_{0}}({{{\mathbf{e}}}_{s}})}}} \right. \kern-0em} {{{\beta }_{0}}({{{\mathbf{e}}}_{s}})}}.$$

FORMULAS FOR CORRELATION COEFFICIENT OF WAVE EIKONAL FLUCTUATIONS FOR THE POWER-LAW SIMILARITY MODEL

Substituting power-law similarity model (3.1) into (4.6), we derive the expression for the “normalized” correlation coefficient of plane wave eikonal:

$$\begin{gathered} {{R}_{{{\text{iso}}}}}(p,n,x) = \frac{{\int\limits_0^\infty {kdk\Phi _{\varepsilon }^{{\text{0}}}({{k}^{2}}){{J}_{0}}(xk)} }}{{\int\limits_0^\infty {kdk\Phi _{\varepsilon }^{{\text{0}}}({{k}^{2}})} }} \\ = \frac{{2\Gamma (p + n)}}{{\Gamma (n + 1)\Gamma (p - 1)}}\int\limits_0^\infty {\frac{{{{J}_{0}}(xk){{k}^{{2n + 1}}}}}{{{{{(1 + {{k}^{2}})}}^{{p + n}}}}}dk} . \\ \end{gathered} $$

We present the explicit form of the correlation coefficient of plane-wave eikonal in the case of the power-law model with parameter values of $n = 0,$$n = 1$, $p = {{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0em} 6}$ (Kolmogorov’s locally homogeneous turbulence), and $p = {5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}$ (ground stratified turbulence):

$$\begin{gathered} {{R}_{{{\text{iso}}}}}({5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2},0,x) = (1 + x)\exp ( - x); \\ {{R}_{{{\text{iso}}}}}({5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2},1,x) = (1 + x - 0.5{{x}^{2}})\exp ( - x); \\ {{R}_{{{\text{iso}}}}}({{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0em} 6},0,x) = [{{{{2}^{{1/6}}}} \mathord{\left/ {\vphantom {{{{2}^{{1/6}}}} \Gamma }} \right. \kern-0em} \Gamma }({5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6})]{{x}^{{5/6}}}{{K}_{{5/6}}}(x); \\ {{R}_{{{\text{iso}}}}}({{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0em} 6},1,x) = [{\pi \mathord{\left/ {\vphantom {\pi {\Gamma ({5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6})}}} \right. \kern-0em} {\Gamma ({5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6})}}]{{({x \mathord{\left/ {\vphantom {x 2}} \right. \kern-0em} 2})}^{{5/6}}} \\ \times \,\,\{ [x({{I}_{{1/6}}}(x) - {{I}_{{11/6}}}(x)) + 2{{I}_{{ - 5/6}}}(x)] - ({{11} \mathord{\left/ {\vphantom {{11} 3}} \right. \kern-0em} 3}){{I}_{{5/6}}}(x)\} . \\ \end{gathered} $$

Here, ${{I}_{\nu }}$ is the Bessel function and ${{K}_{\nu }}$ is the modified Bessel function.

For the normalized correlation coefficient of the spherical wave eikonal $R_{{{\text{iso}}}}^{{{\text{sph}}}}(p,n,x)$ = $\int_0^1 {{{R}_{{{\text{iso}}}}}(p,n,x\xi )d\xi }$, we find the expression

$$\begin{gathered} R_{{{\text{iso}}}}^{{{\text{sph}}}}(p,n,x) = \frac{{\Gamma (p + n)}}{{\Gamma (n + 1)\Gamma (p - 1)}} \\ \times \,\,\,\int\limits_0^\infty {\left\{ {\pi {{{\rm H}}_{0}}(xk){{J}_{1}}(xk) + \left[ {2 - \pi {{{\rm H}}_{1}}(xk){{J}_{0}}(xk)} \right]} \right\}} \\ \times \,\,\frac{{{{k}^{{2n + 1}}}dk}}{{{{{(1 + {{k}^{2}})}}^{{p + n}}}}}. \\ \end{gathered} $$

Here, ${{{\rm H}}_{\nu }}$ is the Struve function and ${{J}_{\nu }}$ is the Bessel function.

We present the explicit form of the normalized correlation coefficient of spherical wave eikonal with the parameter values of $n = 0,\,\,\,n = 1,$$p = {{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0em} 6},$ and $p = {5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}$:

$$\begin{gathered} R_{{{\text{iso}}}}^{{{\text{sph}}}}({5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2},0,x) = \frac{{2 - (2 + x){{e}^{{ - x}}}}}{x},\,\,\,\,R_{{{\text{iso}}}}^{{{\text{sph}}}}({5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2},1,x) = \frac{{2 + ({{x}^{2}} - 2){{e}^{{ - x}}}}}{{2x}}, \\ R_{{{\text{iso}}}}^{{{\text{sph}}}}\left( {\frac{{11}}{6},0,x} \right) = {}_{1}{{F}_{2}}\left( {\frac{1}{2};\frac{1}{6},\frac{3}{2};\frac{{{{x}^{2}}}}{4}} \right) + \frac{{3{{x}^{{5/3}}}\Gamma \left( { - \frac{5}{6}} \right){}_{1}{{F}_{2}}\left( {\frac{4}{3};\frac{{11}}{6},\frac{7}{3};\frac{{{{x}^{2}}}}{4}} \right)}}{{16 \times {{2}^{{2/3}}}\Gamma \left( {\frac{5}{6}} \right)}}, \\ R_{{{\text{iso}}}}^{{{\text{sph}}}}({{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0em} 6},1,x) = \frac{{{{x}^{2}}}}{2}{}_{1}{{F}_{2}}\left( {\frac{3}{2};\frac{7}{6},\frac{5}{2};\frac{{{{x}^{2}}}}{4}} \right) + {}_{1}{{F}_{2}}\left( {\frac{1}{2};\frac{1}{6},\frac{3}{2};\frac{{{{x}^{2}}}}{4}} \right) \\ - \,\,\frac{{3\pi {{x}^{{5/3}}}\left[ {36{{x}^{2}}{}_{1}{{F}_{2}}\left( {\frac{7}{3};\frac{{17}}{6},\frac{{10}}{3};\frac{{{{x}^{2}}}}{4}} \right) + 847{}_{1}{{F}_{2}}\left( {\frac{4}{3};\frac{{11}}{6},\frac{7}{3};\frac{{{{x}^{2}}}}{4}} \right)} \right]}}{{3080 \times {{2}^{{2/3}}}\Gamma {{{\left( {\frac{5}{6}} \right)}}^{2}}}}. \\ \end{gathered} $$

Here, ${}_{1}{{F}_{2}}$ is the hypergeometric function.