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On the reflection, diffraction and refraction of a spherical wave of parallel polarization by a sphere of electrically long radius

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Abstract

A novel mathematical approach is proposed for the analysis of the reflection, diffraction and refraction of spherical waves from curved material interfaces. As a paradigm, this approach is applied to the radiation of a vertical electric dipole over an electrically homogenous sphere. The material of the sphere can be dielectric or conducting. The radius of the sphere is assumed to be much greater than the wavelength in its less dense exterior. The formulation is developed for any height of the dipole above the sphere and for any radial distance of the point of observation from the center of the sphere. Novel expansions are applied to the integral representations of the resulting fields. It is shown that contributions to the value of the fields can be produced either from stationary-phase points of the integrands that are generated by such expansions or from any of the two strings of poles of these integrands. Contributions from rapidly converging residue series, as opposed to those coming from stationary-phase points, correspond to wave trajectories containing a curved path in addition to rectilinear paths. For the sake of demonstration, approximate analytical formulas for computation are derived for a number of select cases. The method yields leading-order approximations for the reflected and refracted fields. The affinity with the reflection from a planar interface is analysed.

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References

  1. 1.

    G. Mie, Ann. Phys. 25, 377 (1908)

  2. 2.

    G.N. Watson, Proc. R. Soc. Lond. A 95, 83 (1918)

  3. 3.

    B. van der Pol, H. Bremmer, Lond. Edinb. Dublin Philos. Mag. J. Sci. (7) Part I 24, 141 (1937)

  4. 4.

    B. van der Pol, H. Bremmer, Lond. Edinb. Dublin Philos. Mag. J. Sci. (7) Part II 24, 825 (1937)

  5. 5.

    B. van der Pol, H. Bremmer, Lond. Edinb. Dublin Philos. Mag. J. Sci. (7) Part III 25, 817 (1938)

  6. 6.

    B. van der Pol, H. Bremmer, Lond. Edinb. Dublin Philos. Mag. J. Sci. (7) Part IV 27, 261 (1939)

  7. 7.

    H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981)

  8. 8.

    M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic Press, New York, 1969)

  9. 9.

    D.A. Hill, J.R. Wait, Radio Sci. 15, 637 (1980)

  10. 10.

    D. Margetis, J. Math. Phys. 43, 3162 (2002)

  11. 11.

    H.M. Nussenzveig, J. Math. Phys. 10, 82 (1969)

  12. 12.

    P. Debye, Phys. Z. 9, 775 (1908)

  13. 13.

    T.T. Wu, Phys. Rev. 104, 1201 (1956)

  14. 14.

    V.A. Houdzoumis, Part II of Ph.D. Thesis (Harvard University, Cambridge, Mass., 1994)

  15. 15.

    V.A. Houdzoumis, J. Appl. Phys. 86, 3939 (1999)

  16. 16.

    M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions (Dover, New York, 1972)

  17. 17.

    R.W.P. King, M. Owens, T.T. Wu, Lateral Electromagnetic Waves (Springer, New York, 1992)

  18. 18.

    H. Bremmer, Applications of operational calculus to ground-wave propagation, particularly for long waves. IRE Trans. Antennas Propag. AP–6, 267–272 (1958)

  19. 19.

    T.T. Wu, Part II of Ph.D. Thesis (Harvard University, Cambridge, Mass., 1956)

  20. 20.

    Bateman Manuscript Project, Higher Transcendental Functions, vol. II (McGraw-Hill Book Company, New York, 1953)

  21. 21.

    F.W.J. Olver, Philos. Trans. R. Soc. Lond. A 247, 328 (1954)

  22. 22.

    Bateman Manuscript Project, Higher Transcendental Functions, vol. I (McGraw-Hill Book Company, New York, 1953)

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Acknowledgements

Viewing the long course that led eventually to the production of this paper, the author wishes ardently to acknowledge the encouragement and motivation to scientific endeavour inspired to him by his family, the late Professor John G. Fikioris and Professor Tai T. Wu.

Author information

Correspondence to Vassilios Athanassiou Houdzoumis.

Additional information

This paper is dedicated to the memory of my father Athanassios Houdzoumis.

Appendices

Appendix A: Approximations for Bessel functions

The behaviour of Bessel functions is in general different according as the order is less than, nearly equal to or larger than the variable. The variable of a Bessel function is more commonly called argument; in the present exposition however we reserve the term argument for the angle of a complex number. Since the integrations in our problem are over the entire positive real axis, all these modes of behaviour come into play. The variable of the Bessel functions is assumed below much greater than unity.

When the order \(a\lambda \) of the Bessel functions is nearly equal to the variable z with \(|z|\gg 1\), the following set of leading-term approximations can be used, deriving from Schöbe formulas [19]:

$$\begin{aligned} J_{a\lambda }(z)&\approx (z/2)^{-1/3} Ai(\xi ),\nonumber \\ H^{(1)}_{a\lambda }(z)&\approx -2e^{i2\pi /3}(z/2)^{-1/3} Ai\left( e^{i2\pi /3}\xi \right) ,\\ H^{(2)}_{a\lambda }(z)&\approx -2e^{-i2\pi /3}(z/2)^{-1/3} Ai\left( e^{-i2\pi /3}\xi \right) ,\nonumber \end{aligned}$$
(A.1)
$$\begin{aligned} J_{a\lambda }'(z)&\approx -(z/2)^{-2/3} Ai'(\xi ),\nonumber \\ H^{(1)'}_{a\lambda }(z)&\approx -2e^{i\pi /3}(z/2)^{-2/3} Ai'\left( e^{i2\pi /3}\xi \right) ,\\ H^{(2)'}_{a\lambda }(z)&\approx -2e^{-i\pi /3}(z/2)^{-2/3} Ai'\left( e^{-i2\pi /3}\xi \right) ,\nonumber \end{aligned}$$
(A.2)

where

$$\begin{aligned} \xi =(a\lambda -z)(z/2)^{-1/3}=2(z/2)^{2/3}(a\lambda /z-1). \end{aligned}$$
(A.3)

The relative error of these approximations is of order \(|z|^{-2/3}\) for \(|\xi |\le O(1)\). The condition \(a\lambda \approx z\) is meant equivalent to \(|\xi |\le O(1)\), specifying the so-called “transition region” of the Bessel functions. As a consequence of the above approximations, we obtain:

$$\begin{aligned} {J_{a\lambda }'(z)\over J_{a\lambda }(z)}\,\,&\approx -(z/2)^{-1/3}\, {Ai'(\xi )\over Ai(\xi )}, \nonumber \\ {H_{a\lambda }^{(1)'}(z)\over H_{a\lambda }^{(1)}(z)}&\approx -e^{i2\pi /3}\,(z/2)^ {-1/3}\,{Ai'\left( e^{i2\pi /3}\xi \right) \over Ai\left( e^{i2\pi /3} \xi \right) },\\ {H_{a\lambda }^{(2)'}(z)\over H_{a\lambda }^{(2)}(z)}&\approx -e^{-i2\pi /3}\, (z/2)^{-1/3}\, {Ai'\left( e^{-i2\pi /3}\xi \right) \over Ai\left( e^{-i2\pi /3}\xi \right) }.\nonumber \end{aligned}$$
(A.4)

Outside the transition region, i.e. for \(|\xi |\gg 1\), the following approximations [20] apply for \(H^{(1)}_{a\lambda } (z)\) and \(H ^{(2)}_{a\lambda } (z)\), always under the assumption that \(|z|\gg 1\):

$$\begin{aligned} H^{(1)}_{a\lambda }(z)&\approx \ \sqrt{{2\over \pi }}\left( z^2-(a\lambda )^2\right) ^{-1/4}e^{-i\pi /4}\exp \left[ i\sqrt{z^2-(a\lambda )^2}-ia\lambda \cos ^{-1} (a\lambda /z)\right] , \end{aligned}$$
(A.5)
$$\begin{aligned} {\hbox {where}}\;\;&\sqrt{z^2-(a\lambda )^2}= i\sqrt{(a\lambda )^2-z^2},\ \cos ^{-1} (a\lambda /z)=i\cosh ^{-1} (a\lambda /z), \nonumber \\ {H_{a\lambda }^{(1)'}(z)\over H_{a\lambda }^{(1)}(z)}&\approx \ i\sqrt{1-(a\lambda / z)^2}= -\sqrt{(a\lambda / z)^2-1}, \end{aligned}$$
(A.6)
$$\begin{aligned} H^{(2)}_{a\lambda }(z)&\approx \ \sqrt{{2\over \pi }}\left( z^2-(a\lambda )^2\right) ^{-1/4} e^{i\pi /4}\exp \left[ -i\sqrt{z^2-(a\lambda )^2}+ia\lambda \cos ^{-1} (a\lambda /z)\right] , \end{aligned}$$
(A.7)
$$\begin{aligned} {\hbox {where}}&\ \sqrt{z^2-(a\lambda )^2}= -i\sqrt{(a\lambda )^2-z^2},\ \cos ^{-1} (a\lambda /z)=-i\cosh ^{-1} (a\lambda /z), \nonumber \\ {H_{a\lambda }^{(2)'}(z)\over H_{a\lambda }^{(2)}(z)}&\approx -i\sqrt{1-(a\lambda / z)^2}= -\sqrt{(a\lambda / z)^2-1}. \end{aligned}$$
(A.8)

The above sets of approximations are consistent with the next one [21], which is uniform with respect to the order:

$$\begin{aligned} J_{a\lambda }(z)&\approx \left( {4w\over 1-(z/a\lambda )^2} \right) ^{1/4}(a\lambda )^{-1/3} Ai\left( (a\lambda )^{2/3}w\right) ,\nonumber \\ H^{(1)}_{a\lambda }(z)&\approx 2e^{-i\pi /3} \left( {4w\over 1-(z/a\lambda )^2} \right) ^{1/4}(a\lambda )^{-1/3} Ai\left( e^{i2\pi /3}(a\lambda )^{2/3}w\right) , \\ H^{(2)}_{a\lambda }(z)&\approx 2e^{i\pi /3} \left( {4w\over 1-(z/a\lambda )^2} \right) ^{1/4}(a\lambda )^{-1/3} Ai\left( e^{-i2\pi /3}(a\lambda )^{2/3}w\right) ,\nonumber \end{aligned}$$
(A.9)
$$\begin{aligned} J_{a\lambda }'(z)&\approx -{2a\lambda \over z}\left( {4w\over 1-(z/a\lambda )^2} \right) ^{-1/4}(a\lambda )^{-2/3} Ai'\left( (a\lambda )^{2/3}w\right) ,\nonumber \\ H^{(1)'}_{a\lambda }(z)&\approx -2e^{i\pi /3}{2a\lambda \over z} \left( {4w\over 1-(z/a\lambda )^2} \right) ^{-1/4}(a\lambda )^{-2/3} Ai'\left( e^{i2\pi /3}(a\lambda )^{2/3}w\right) ,\\ H^{(2)'}_{a\lambda }(z)&\approx -2e^{-i\pi /3}{2a\lambda \over z}\left( {4w\over 1-(z/a\lambda )^2} \right) ^{-1/4}(a\lambda )^{-2/3} Ai'\left( e^{-i2\pi /3}(a\lambda )^{2/3}w\right) ,\nonumber \end{aligned}$$
(A.10)

where

$$\begin{aligned} w=w(\lambda ,z)&=\left[ {3\over 2}\left( \ln {1+\sqrt{1-(z/a\lambda )^2}\over z/a\lambda }-\sqrt{1-(z/a\lambda )^2}\right) \right] ^{2/3}\nonumber \\&= \left[ {3\over 2}\left( \cosh ^{-1}(a\lambda /z)-\sqrt{1-(z/a\lambda )^2}\ \right) \right] ^{2/3}. \end{aligned}$$
(A.11)

It is noted that \(w(\lambda )\) is an analytic function of \(\lambda \) at \(\lambda =z/a\) and that in the limit \(|a\lambda /z|\rightarrow \infty \) the function \(w(\lambda )\) takes on the asymptotic form:

$$\begin{aligned} w(\lambda )\approx w_\infty (\lambda )&=\left[ {3\over 2}\left( \ln {2 a\lambda \over z}-1 \right) \right] ^{2/3} =\left[ {3\over 2}\ln {2 a\lambda \over ez} \right] ^{2/3}. \end{aligned}$$
(A.12)

Yet another set of approximations, consistent with the approximations (A.5)–(A.10) but simpler, is appropriate for \(|a\lambda /z|\rightarrow \infty \) (large-order approximations):

  1. (a)

    Over the domain limited by \(\arg \left( (a\lambda )^{2/3}w\right) =-\pi /3\) and \( \arg (\lambda )=\pi /2\):

    $$\begin{aligned} J_{a\lambda }(z)&\approx {1\over \sqrt{2\pi a\lambda }}\left( {2a\lambda \over ez} \right) ^{-a\lambda },\nonumber \\ H^{(1)}_{a\lambda }(z)&\approx -{2i\over \sqrt{2\pi a\lambda }}\left[ \left( {2a\lambda \over ez}\right) ^{a\lambda }+i\left( {2a\lambda \over ez}\right) ^{-a\lambda }\right] , \\ H^{(2)}_{a\lambda }(z)&\approx {2i\over \sqrt{2\pi a\lambda }}\left( {2a\lambda \over ez} \right) ^{a\lambda };\nonumber \end{aligned}$$
    (A.13a)
  2. (b)

    Over the domain limited by \(\arg (\lambda )=-\pi /2\) and \(\arg \left( (a\lambda )^{2/3}w\right) =\pi /3\):

    $$\begin{aligned} J_{a\lambda }(z)&\approx {1\over \sqrt{2\pi a\lambda }}\left( {2a\lambda \over ez} \right) ^{-a\lambda },\nonumber \\ H^{(1)}_{a\lambda }(z)&\approx -{2i\over \sqrt{2\pi a\lambda }} \left( {2a\lambda \over ez} \right) ^{a\lambda }, \\ H^{(2)}_{a\lambda }(z)&\approx {2i\over \sqrt{2\pi a\lambda }} \left[ \left( {2a\lambda \over ez}\right) ^{a\lambda }-i\left( {2a\lambda \over ez}\right) ^{-a\lambda }\right] .\nonumber \end{aligned}$$
    (A.13b)

The above formulas are particularly useful for checking the convergence of integrals at infinity.

Referring to approximations (A.9), the region of the right half of the \(\lambda \) plane in which \(J_{a\lambda }(z)\) exhibits a decaying behaviour with increasing the magnitude of \((a\lambda )^{2/3}w(\lambda )\) is demarcated by the contours over which the argument of \((a\lambda )^{2/3}w(\lambda )\) equals \(\pm \pi /3\), or equivalently, the argument of \((a\lambda )(w(\lambda ))^{3/2}\) equals \(\pm \pi /2\). These contours, being symmetric with respect to the real \(\lambda \) axis for z real, start from \(\lambda =z/a\) with an inclination of \(\pm \pi /3\) and bend upward/downward very slowly (because of the logarithm in formula (A.12)), their angle with the vertical axis tending to zero as \(|a\lambda /z|\rightarrow \infty \). By virtue of formula (A.12), the condition \(\arg \left( (a\lambda )^{2/3}w\right) =\pi /3\) can be simplified asymptotically as follows in the limit as \(|a\lambda /z|\rightarrow \infty \):

$$\begin{aligned} \arg (a\lambda )&={\pi \over 2}-\arg \left( (w(\lambda ))^{3/2}\right) ,\nonumber \\&\approx {\pi \over 2}-\arg \left( \ln {2 a\lambda \over ez} \right) ,\nonumber \\&\approx {\pi \over 2}-{\pi /2-\arg z\over \ln |2a \lambda /ez|} . \end{aligned}$$
(A.14)
Fig. 28
figure28

The regions \({A}_0, {A}_1\) and \({A}_2\)

On the basis of the above asymptotic condition, it is instructive to demonstrate below the afore-mentioned asymptotic behaviour of \(J_{a\lambda }(z)\) on either side of the contour satisfying \(\arg \left( (a\lambda )^{2/3}w\right) \)\(= \pi /3\):

$$\begin{aligned} \left( {2a \lambda \over e z} \right) ^{-a\lambda }&=\exp \left( -a \lambda \ln {2a \lambda \over ez} \right) , \end{aligned}$$
(A.15)
$$\begin{aligned} \mathrm{Re} \left\{ a \lambda \ln {2a\lambda \over ez} \right\}&=\mathrm{Re} \{a \lambda \}\ln \left| {2a \lambda \over ez} \right| -\mathrm{Im} \{a \lambda \}\arg (a \lambda /z), \nonumber \\&=|a \lambda | \left( \cos (\arg a \lambda ) \ln \left| {2a \lambda \over ez} \right| -\sin (\arg a \lambda )\arg (a \lambda /z) \right) , \\&\approx |a \lambda | \left( (\pi /2-\arg a \lambda )\ln \left| {2a \lambda \over ez} \right| -\left( \pi /2-\arg z\right) \right) .\nonumber \end{aligned}$$
(A.16)

It is thus readily verified that as \(|a \lambda /z|\rightarrow \infty \), the factor \((2a \lambda /ez)^{-a \lambda }\) is exponentially decreasing when

$$\begin{aligned} \arg a \lambda <{\pi \over 2}-{\pi /2-\arg z\over \ln |2a \lambda /ez|}, \end{aligned}$$
(A.17)

and exponentially increasing when:

$$\begin{aligned} \arg a \lambda >{\pi \over 2}-{\pi /2-\arg z\over \ln |2a \lambda /ez|}. \end{aligned}$$
(A.18)

Figure 28 shows the contours \(\arg \left( (a\lambda )^{2/3}w(\lambda )\right) =\pm \pi /3\) dividing the right half of the \(\lambda \) plane in three open regions. Region \(A_0\) is the afore-mentioned region in which \(J_{a\lambda }(z)\) exhibits a decaying behaviour with increasing \(|(a\lambda )^{2/3}w(\lambda )|\). Regions \(A_1\) (in the fourth quadrant) and \(A_2\) (in the first quadrant) are respectively the regions in which \(H^{(1)}_{a\lambda }(z)\) and \(H^{(2)}_{a\lambda }(z)\) exhibit decaying behaviour with increasing \(|(a\lambda )^{2/3}w(\lambda )|\).

It must be added that formulas (A.5)–(A.8) for \(H^{(1)}_{a\lambda } (z)\) and \(H ^{(2)}_{a\lambda } (z)\) are not valid close to the contours \(\arg \left( (a\lambda )^{2/3}w(\lambda )\right) =\pm \pi /3\) respectively. For this reason, these contours can be conveniently chosen respectively as branch cuts from \(\lambda =z/a\) for the multi-valued functions appearing on the right-hand side of the formulas (A.5)–(A.8).

Fig. 29
figure29

The values of the functions \(\hbox {cos}^{-1}(z)\) and \(\hbox {cosh}^{-1}(z)\) in the complex z-plane

In relation to the functions \(\cosh ^{-1}(z)\) and \(\cos ^{-1}(z)\), when used to describe the behaviour of \(H^{(1)}_{a\lambda } (z)\) in formula (A.5), Fig. 29 indicates the sign of their real and imaginary parts.

It is lastly noted that as \( t\rightarrow 1\):

$$\begin{aligned}&\cos ^{-1} t= -\int _1^t{du\over \sqrt{1-u^2}}=\sqrt{2(1-t)}+{1\over 3} \left( {1-t\over 2} \right) ^{3/2}+O\left( (1-t)^{5/2}\right) , \end{aligned}$$
(A.19)
$$\begin{aligned}&\cos ^{-1} t-\sqrt{t^{-2}-1}= -{\left( 2(1-t)\right) ^{3/2}\over 3}+O\left( (1-t)^{5/2}\right) . \end{aligned}$$
(A.20)

Appendix B: Approximations for Legendre functions

The following approximations [22] are valid for \(|a\lambda \theta |,|a\lambda (\pi -\theta )|\gg 1\) (which implies that the order is already large: \(|a\lambda |\gg 1\)):

$$\begin{aligned} \mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}^1(\cos \theta )&\approx \sqrt{{2a\lambda \over \pi \sin \theta }} \cos (a\lambda \theta +\pi /4), \end{aligned}$$
(B.1)
$$\begin{aligned} \mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )&\approx \sqrt{{2\over \pi a\lambda \sin \theta }} \cos (a\lambda \theta -\pi /4). \end{aligned}$$
(B.2)

Although the above condition (\(|a\lambda \theta |,|a\lambda (\pi -\theta )|\gg 1\)) represents the commonest case for the calculation of the Legendre functions, the need may arise of calculating Legendre functions with \(|a\lambda |\gg 1\) but with the angle \(\theta \) so small (\(\theta \ll 1\)) that \(|a\lambda \theta |\le O(1)\). In such a case, we can use approximations:

$$\begin{aligned} \mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}^1(\cos \theta )&\approx -a\lambda J_1(a\lambda \theta ), \end{aligned}$$
(B.3)
$$\begin{aligned} \mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )&\approx J_0(a\lambda \theta ). \end{aligned}$$
(B.4)

For values of the angle \(\theta \) close to \(\pi \) (i.e. near the antipodes), we can apply the above results on \(\mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}^1(\cos (\pi -\theta ))\) and \(\mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}(\cos (\pi -\theta ))\). But prior to that it is necessary to express the fields in terms of Legendre functions with variable \(\cos (\pi -\theta )\). To this end, the first step is to apply the formulas below in the initial series expressions for the fields:

$$\begin{aligned} \mathrm{P}_m^1(\cos \theta )&=(-1)^{m+1}\mathrm{P}^1_m(\cos (\pi -\theta ))=-e^{im\pi }\,\mathrm{P}^1_m(\cos (\pi -\theta )), \end{aligned}$$
(B.5)
$$\begin{aligned} \mathrm{P}_m(\cos \theta )&=(-1)^m\mathrm{P}_m(\cos (\pi -\theta ))=e^{im\pi }\mathrm{P}_m(\cos (\pi -\theta )). \end{aligned}$$
(B.6)

The procedure of converting the initial series into series of integrals along with the change of variable \(m=a\lambda -{1\over 2}\) can then be applied. The only difference is that now instead of \(\mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}^1(\cos \theta )\) in the integrand we have the factor \(ie^{i\pi a\lambda }\mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}^1(\cos (\pi -\theta ))\) and instead of \(\mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )\), the factor \(-ie^{i\pi a\lambda }\mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}(\cos (\pi -\theta ))\). The functions \(\mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}^1(\cos (\pi -\theta ))\) and \(\mathrm{P}_{a\lambda -{\textstyle {1\over 2}}}(\cos (\pi -\theta ))\) can then be approximated for \(\theta \approx \pi \) by use of formulas (B.3) and (B.4) with \((\pi -\theta )\) in the place of \(\theta \).

Referring back to the approximations (B.1) and (B.2), it is observed that off the real axis of the \(\lambda \) plane the Legendre functions of the first kind are unbounded as \(|a\lambda |\rightarrow \infty \). It is desirable to decompose them into two parts, one of which is exponentially decreasing in the upper half plane and the other is exponentially decreasing in the lower half plane.

Let:

$$\begin{aligned} {{L}}^{1,+}_{a\lambda }(\theta )&={1\over 2} \left( {P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )-{2i\over \pi }\,{Q}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ) \right) , \end{aligned}$$
(B.7)
$$\begin{aligned} {{L}}^{1,-}_{a\lambda }(\theta )&={1\over 2} \left( {P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )+{2i\over \pi }\,{Q}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ) \right) ,\end{aligned}$$
(B.8)
$$\begin{aligned} {{L}}^{+}_{a\lambda }(\theta )&={1\over 2} \left( {P}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )-{2i\over \pi }\,{Q}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ) \right) ,\end{aligned}$$
(B.9)
$$\begin{aligned} {{L}}^{-}_{a\lambda }(\theta )&={1\over 2} \left( {P}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )+{2i\over \pi }\,{Q}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ) \right) , \end{aligned}$$
(B.10)

where \({Q}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )\) and \({Q}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )\) denote Legendre functions of the second kind. It is noted in passing that the above notation is not conventional. Clearly:

$$\begin{aligned} {P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )&= {{L}}^{1,+}_{a\lambda }(\theta )+{{L}}^{1,-}_{a\lambda }(\theta ), \end{aligned}$$
(B.11)
$$\begin{aligned} {P}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )&= {{L}}^{+}_{a\lambda }(\theta )+{{L}}^{-}_{a\lambda }(\theta ). \end{aligned}$$
(B.12)

Taking now into account that as \(|a\lambda |\rightarrow \infty \):

$$\begin{aligned} {Q}_{a\lambda -{\textstyle {1\over 2}}}^1(\cos \theta )&\approx -\sqrt{{a\lambda \pi \over 2\sin \theta }} \sin (a\lambda \theta +\pi /4), \end{aligned}$$
(B.13)
$$\begin{aligned} {Q}_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )&\approx -\sqrt{{\pi \over 2 a\lambda \sin \theta }} \sin (a\lambda \theta -\pi /4), \end{aligned}$$
(B.14)

it follows that as \(|a\lambda |\rightarrow \infty \):

$$\begin{aligned} {{L}}^{1,+}_{a\lambda }(\theta )\approx&\ \sqrt{{a\lambda \over 2\pi \sin \theta }} \exp (ia\lambda \theta +i\pi /4), \end{aligned}$$
(B.15)
$$\begin{aligned} {{L}}^{1,-}_{a\lambda }(\theta )\approx&\ \sqrt{{a\lambda \over 2\pi \sin \theta }} \exp (-ia\lambda \theta -i\pi /4),\end{aligned}$$
(B.16)
$$\begin{aligned} {{L}}^{+}_{a\lambda }(\theta )\approx&\ {1\over \sqrt{2\pi a\lambda \sin \theta }} \exp (ia\lambda \theta -i\pi /4),\end{aligned}$$
(B.17)
$$\begin{aligned} {{L}}^{-}_{a\lambda }(\theta )\approx&\ {1\over \sqrt{2\pi a\lambda \sin \theta }} \exp (-ia\lambda \theta +i\pi /4). \end{aligned}$$
(B.18)

A number of useful identities involving Legendre functions is cited below:

$$\begin{aligned} {P}^1_{-a\lambda -{\textstyle {1\over 2}}}(\cos \theta )&={P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ), \end{aligned}$$
(B.19)
$$\begin{aligned} {Q}^1_{-a\lambda -{\textstyle {1\over 2}}}(\cos \theta )&= {Q}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )+\pi \tan (\pi a\lambda ){P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ),\end{aligned}$$
(B.20)
$$\begin{aligned} {{L}}^{1,\pm }_{-a\lambda }(\theta )&={{L}}^{1,\pm }_{a\lambda }(\theta )\mp i\tan (\pi a\lambda ){P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ),\end{aligned}$$
(B.21)
$$\begin{aligned} {{L}}^{1,+}_{a\lambda }(\theta )+{{L}}^{1,-}_{-a\lambda }(\theta )&={P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )-{i\over \pi }{Q}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )+{i\over \pi }{Q}^1_{-a\lambda -{\textstyle {1\over 2}}}(\cos \theta ),\nonumber \\&={P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )+i\tan (\pi a\lambda ){P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ),\nonumber \\&={e^{i\pi a\lambda }\over \cos (\pi a\lambda )}{P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )={2e^{i2\pi a\lambda }\over 1+e^{i2\pi a\lambda }}{P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ), \end{aligned}$$
(B.22)
$$\begin{aligned} {{L}}^{1,+}_{a\lambda }(\theta )-{{L}}^{1,-}_{-a\lambda }(\theta )&=-{i\over \pi } \left[ {Q}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )+{Q}^1_{-a\lambda -{\textstyle {1\over 2}}}(\cos \theta ) \right] ,\nonumber \\&={1\over i} \left[ \tan (\pi a\lambda ){P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta )+{2\over \pi }{Q}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos \theta ) \right] ,\nonumber \\&={i\over \cos (\pi a\lambda )}{P}^1_{a\lambda -{\textstyle {1\over 2}}}(\cos (\pi -\theta )), \end{aligned}$$
(B.23)
$$\begin{aligned} {{L}}^{+}_{a\lambda }(\theta )-{{L}}^{-}_{-a\lambda }(\theta )&=-{i\over \cos (\pi a\lambda )}{P}_{a\lambda -{\textstyle {1\over 2}}}(\cos (\pi -\theta )). \end{aligned}$$
(B.24)

Appendix C: Approximations for the function \(N(\lambda \))

By applying formulas (A.4) and (A.6), the following approximation can be obtained for the function \(N(\lambda )\) in the vicinity of \(\lambda =k_2\) (\(|\xi _2|\le O(1)\)), subject to condition (2.22):

$$\begin{aligned} N(\lambda )\approx -(k_2a/2)^{-1/3} \left( F(\xi _2)-ig_2\right) , \end{aligned}$$
(C.1)

where

$$\begin{aligned} F(\xi )&=e^{i2\pi /3}\,{Ai'\left( e^{i2\pi /3}\xi \right) \over Ai\left( e^{i2\pi /3} \xi \right) }, \end{aligned}$$
(C.2)
$$\begin{aligned} \xi _2&=(a\lambda -k_2a)(k_2a/2)^{-1/3}=2^{1\over 3}(k_2a)^{2/3}(\lambda /k_2-1),\end{aligned}$$
(C.3)
$$\begin{aligned} g_2&={k_2\over k_1}(k_2a/2)^{1/3}\,\sqrt{1-(k_2/k_1)^2}. \end{aligned}$$
(C.4)

Hence the poles \(\lambda _{2,j}\) of the function \(N^{-1}(\lambda )\) in the vicinity of \(\lambda =k_2\) satisfy approximately:

$$\begin{aligned} F(\xi _{2,j})=ig_2. \end{aligned}$$
(C.5)

It is noted in passing that the magnitude of \(g_2\) can be either large or small (if \(|k_1/k_2|\gg (k_2a/2)^{1/3}\)) compared to unity.

Likewise, in the vicinity of \(\lambda =k_1\) (\(|\xi _1|\le O(1)\)), subject to condition (2.22):

$$\begin{aligned} N(\lambda )\approx -{k_2\over k_1}(k_1a/2)^{-1/3}\left( F(\xi _1)+g_1\right) , \end{aligned}$$
(C.6)

where

$$\begin{aligned} \xi _1&=(a\lambda -k_1a)(k_1a/2)^{-1/3}=2^{1\over 3}(k_1a)^{2/3}(\lambda /k_1-1), \end{aligned}$$
(C.7)
$$\begin{aligned} g_1&={k_1\over k_2}(k_1a/2)^{1/3}\,\sqrt{(k_1/k_2)^2-1}. \end{aligned}$$
(C.8)

Hence the poles \(\lambda _{1,j}\) of the function \(N^{-1}(\lambda )\) in the vicinity of \(\lambda =k_1\) satisfy approximately:

$$\begin{aligned} F(\xi _{1,j})=-g_1. \end{aligned}$$
(C.9)

Because of the assumed condition (2.22), the magnitude of \(g_1\) is greater than unity.

As for the residues, the following formulas are a direct consequence of the above approximations:

$$\begin{aligned} \mathop {\mathrm{Res}}_{\lambda =\lambda _{2,\,j}}\left\{ {1\over N(\lambda )} \right\}&\approx -{(k_2a/2)^{1/3}\over F'(\xi _{2,j})}, \end{aligned}$$
(C.10)
$$\begin{aligned} \mathop {\mathrm{Res}}_{\lambda =\lambda _{1,\,j}}\left\{ {1\over N(\lambda )} \right\}&\approx -{k_1\over k_2}{(k_1a/2)^{1/3}\over F'(\xi _{1,j})}. \end{aligned}$$
(C.11)

With regard to the function \(F(\xi )\) it is noted that:

$$\begin{aligned} F'(\xi )&=\xi -F^2(\xi ), \end{aligned}$$
(C.12)
$$\begin{aligned} F(\xi )&\approx \sqrt{\xi }-{1\over 4\xi },\quad |\,\xi |\gg 1, \left| \arg \left( e^{i2\pi /3}\xi \right) \right| <\pi . \end{aligned}$$
(C.13)

Hence:

$$\begin{aligned} F'(\xi _{2,j})&=\xi _{2,j}+g_2^2, \end{aligned}$$
(C.14)
$$\begin{aligned} F'(\xi _{1,j})&=\xi _{1,j}-g_1^2. \end{aligned}$$
(C.15)

On the other hand, for \(\lambda \) not close to the contours \(C_{k_2}\) and \(C_{k_1}\):

$$\begin{aligned} {1\over N(\lambda )}&\approx -{i\over \sqrt{1-(\lambda /k_2)^2}+(k_2/k_1)\sqrt{1-(\lambda /k_1)^2} },\nonumber \\&=-{ik_2\over \sqrt{k_2^2-\lambda ^2}+(k_2/k_1)^2\sqrt{k_1^2-\lambda ^2} }. \end{aligned}$$
(C.16)

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Houdzoumis, V.A. On the reflection, diffraction and refraction of a spherical wave of parallel polarization by a sphere of electrically long radius. Eur. Phys. J. Plus 135, 240 (2020). https://doi.org/10.1140/epjp/s13360-020-00117-0

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