Appendix A: Spherical coordinates system
The spherical coordinates system used in this work is shown in Fig. 19. The three coordinates are: the radial distance r (where \(0\leqslant r<\infty \)), the polar angle \(\theta \) (where \(0\leqslant \theta \leqslant \pi \)) and the azimut angle \(\varphi \) (where \(0\leqslant \varphi < 2 \pi \)).
The gradient, divergent and curl operators are
$$\begin{aligned} \varvec{\nabla }= & {} \begin{pmatrix} \frac{\partial }{\partial r} \\ \frac{1}{r} \frac{\partial }{\partial \theta } \\ \frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } \end{pmatrix} \end{aligned}$$
(78)
$$\begin{aligned} \varvec{\nabla }^*= & {} \begin{pmatrix} \frac{1}{r^2} \frac{\partial }{\partial r} r^2&\frac{1}{r \sin \theta } \frac{\partial }{\partial \theta } \sin \theta&\frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } \end{pmatrix} \end{aligned}$$
(79)
$$\begin{aligned} \varvec{{\tilde{\nabla }}}= & {} \begin{pmatrix} 0 &{} -\frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } &{} \frac{1}{r \sin \theta } \frac{\partial }{\partial \theta } \sin \theta \\ \frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } &{} 0 &{} -\frac{1}{r} \frac{\partial }{\partial r} r \\ -\frac{1}{r} \frac{\partial }{\partial \theta } &{} \frac{1}{r} \frac{\partial }{\partial r} r &{} 0 \end{pmatrix} \end{aligned}$$
(80)
and the scalar and vector Laplacian operators are
$$\begin{aligned} \nabla ^2= & {} \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial }{\partial r}\right) + \frac{1}{r^2 \sin \theta } \frac{\partial }{\partial \theta } \left( \sin \theta \frac{\partial }{\partial \theta }\right) \nonumber \\&+ \frac{1}{r^2 \sin ^2 \theta } \frac{\partial ^2}{\partial \varphi ^2} \end{aligned}$$
(81)
$$\begin{aligned} \varvec{\nabla }^2= & {} \begin{pmatrix} \nabla ^2 - \frac{2}{r^2} &{} - \frac{2}{r^2 \sin \theta } \frac{\partial }{\partial \theta } \sin \theta &{} -\frac{2}{r^2 \sin \theta } \frac{\partial }{\partial \varphi } \\ \frac{2}{r^2} \frac{\partial }{\partial \theta } &{} \nabla ^2 - \frac{1}{r^2 \sin ^2 \theta } &{} -\frac{2 \cos \theta }{r^2 \sin ^2 \theta } \frac{\partial }{\partial \varphi } \\ \frac{2}{r^2 \sin \theta } \frac{\partial }{\partial \varphi } &{} \frac{2 \cos \theta }{r^2 \sin ^2 \theta } \frac{\partial }{\partial \varphi } &{} \nabla ^2 - \frac{1}{r^2 \sin ^2 \theta } \end{pmatrix} \end{aligned}$$
(82)
Operators \(\varvec{{\mathcal {D}}}\) and \(\varvec{{\mathcal {D}}}^*\) are
$$\begin{aligned} \varvec{{\mathcal {D}}}= & {} \begin{pmatrix} \frac{2}{r} + \frac{\partial }{\partial r} &{} -\frac{1}{r} &{} -\frac{1}{r} &{} \frac{\cot \theta }{r} + \frac{1}{r} \frac{\partial }{\partial \theta } &{} \frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } &{} 0 \\ 0 &{} \frac{\cot \theta }{r} + \frac{1}{r}\frac{\partial }{\partial \theta } &{} -\frac{\cot \theta }{r} &{} \frac{3}{r} + \frac{\partial }{\partial r} &{} 0 &{} \frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } \\ 0 &{} 0 &{} \frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } &{} 0 &{} \frac{3}{r} + \frac{\partial }{\partial r} &{} \frac{2 \cot \theta }{r} + \frac{1}{r} \frac{\partial }{\partial \theta } \end{pmatrix} \nonumber \\ \end{aligned}$$
(83)
$$\begin{aligned} \varvec{{\mathcal {D}}}^*= & {} \begin{pmatrix} \frac{\partial }{\partial r} &{} 0 &{} 0 \\ \frac{1}{r} &{} \frac{1}{r} \frac{\partial }{\partial \theta } &{} 0 \\ \frac{1}{r} &{} \frac{\cot \theta }{r} &{} \frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } \\ \frac{1}{r} \frac{\partial }{\partial \theta } &{} -\frac{1}{r} + \frac{\partial }{\partial r} &{} 0 \\ \frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } &{} 0 &{} -\frac{1}{r} + \frac{\partial }{\partial r} \\ 0 &{} \frac{1}{r \sin \theta } \frac{\partial }{\partial \varphi } &{} -\frac{\cot \theta }{r}+\frac{1}{r} \frac{\partial }{\partial \theta } \end{pmatrix} \end{aligned}$$
(84)
Appendix B: Domain equations in spherical coordinates
In the spherical coordinates system defined in Appendix A, the spectral equilibrium Eq. (15) yields,
$$\begin{aligned}&\frac{\partial \sigma _{rr}}{\partial r} + \frac{1}{r} \frac{\partial \sigma _{r \theta }}{\partial \theta } + \frac{1}{r \sin \theta } \frac{\partial \sigma _{r \varphi }}{\partial \varphi }\nonumber \\&\quad + \frac{1}{r} \left( 2 \sigma _{rr} - \sigma _{\theta \theta } - \sigma _{\varphi \varphi } + \cot \theta \sigma _{r \theta } \right) + \omega ^2 \rho u_r = 0 \end{aligned}$$
(85)
$$\begin{aligned}&\frac{\partial \sigma _{r \theta }}{\partial r} + \frac{1}{r} \frac{\partial \sigma _{\theta \theta }}{\partial \theta } + \frac{1}{r \sin \theta } \frac{\partial \sigma _{\theta \varphi }}{\partial \varphi } \nonumber \\&\quad + \frac{1}{r} \left[ \left( \sigma _{\theta \theta } - \sigma _{\varphi \varphi } \right) \cot \theta + 3 \sigma _{r \theta } \right] + \omega ^2 \rho u_{\theta } = 0 \end{aligned}$$
(86)
$$\begin{aligned}&\frac{\partial \sigma _{r \varphi }}{\partial r} + \frac{1}{r} \frac{\partial \sigma _{\theta \varphi }}{\partial \theta } + \frac{1}{r \sin \theta } \frac{\partial \sigma _{\varphi \varphi }}{\partial \varphi }\nonumber \\&\quad + \frac{1}{r} \left( 2 \sigma _{\theta \varphi } \cot \theta + 3 \sigma _{r \varphi } \right) + \omega ^2 \rho u_{\varphi } = 0 \end{aligned}$$
(87)
Moreover, the spectral compatibility Eq. (16) yields,
$$\begin{aligned} \epsilon _{rr}= & {} \frac{\partial u_r}{\partial r} \end{aligned}$$
(88)
$$\begin{aligned} \epsilon _{\theta \theta }= & {} \frac{1}{r} \left( \frac{\partial u_{\theta }}{\partial \theta } + u_r \right) \end{aligned}$$
(89)
$$\begin{aligned} \epsilon _{\varphi \varphi }= & {} \frac{1}{r \sin \theta } \left( \frac{\partial u_{\varphi }}{\partial \varphi } + u_r \sin \theta + u_{\theta } \cos \theta \right) \end{aligned}$$
(90)
$$\begin{aligned} \epsilon _{r \theta }= & {} \frac{1}{2} \left( \frac{1}{r} \frac{\partial u_r}{\partial \theta } + \frac{\partial u_{\theta }}{\partial r} - \frac{u_{\theta }}{r} \right) \end{aligned}$$
(91)
$$\begin{aligned} \epsilon _{r \varphi }= & {} \frac{1}{2} \left( \frac{1}{r \sin \theta } \frac{\partial u_r}{\partial \varphi } + \frac{\partial u_{\varphi }}{\partial r} - \frac{u_{\varphi }}{r} \right) \end{aligned}$$
(92)
$$\begin{aligned} \epsilon _{\theta \varphi }= & {} \frac{1}{2 r} \left( \frac{1}{\sin \theta } \frac{\partial u_{\theta }}{\partial \varphi } + \frac{\partial u_{\varphi }}{\partial \theta } - u_{\varphi } \cot \theta \right) \end{aligned}$$
(93)
Finally, the spectral elasticity Eq. (17) yields,
$$\begin{aligned} \sigma _{rr}= & {} \left( \lambda + 2 \mu + \alpha ^2 M \right) \epsilon _{rr}\nonumber \\&+ \left( \lambda + \alpha ^2 M \right) \left( \epsilon _{\theta \theta } + \epsilon _{\varphi \varphi } \right) \end{aligned}$$
(94)
$$\begin{aligned} \sigma _{\theta \theta }= & {} \left( \lambda + 2 \mu + \alpha ^2 M \right) \epsilon _{\theta \theta }\nonumber \\&+ \left( \lambda + \alpha ^2 M \right) \left( \epsilon _{rr} + \epsilon _{\varphi \varphi } \right) \end{aligned}$$
(95)
$$\begin{aligned} \sigma _{\varphi \varphi }= & {} \left( \lambda + 2 \mu + \alpha ^2 M \right) \epsilon _{\varphi \varphi } \nonumber \\&+ \left( \lambda + \alpha ^2 M \right) \left( \epsilon _{rr} + \epsilon _{\theta \theta } \right) \end{aligned}$$
(96)
$$\begin{aligned} \sigma _{r \theta }= & {} 2 \mu \epsilon _{r \theta } \end{aligned}$$
(97)
$$\begin{aligned} \sigma _{r \varphi }= & {} 2 \mu \epsilon _{r \varphi } \end{aligned}$$
(98)
$$\begin{aligned} \sigma _{\theta \varphi }= & {} 2 \mu \epsilon _{\theta \varphi } \end{aligned}$$
(99)
Appendix C: P-waves functions
Substituting the compression waves potential (37) into Eq. (30), the three components of the P-wave displacements in spherical coordinates are
$$\begin{aligned} u_{r_{p}}= & {} \frac{Y_n^m \left( \theta , \varphi \right) }{r} \left[ n j_n \left( \beta _{p} r \right) - \beta _{p} r j_{n+1} \left( \beta _{p} r \right) \right] \end{aligned}$$
(100)
$$\begin{aligned} u_{\theta _{p}}= & {} \frac{- j_n \left( \beta _{p} r \right) }{r \sin \theta }\nonumber \\&\left[ \left( n+1 \right) \cos \theta \ Y_n^m \left( \theta , \varphi \right) - \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \right] \end{aligned}$$
(101)
$$\begin{aligned} u_{\varphi _{p}}= & {} \frac{ {\hat{\imath }} m}{r \sin \theta } j_n \left( \beta _{p} r \right) Y_n^m \left( \theta , \varphi \right) \end{aligned}$$
(102)
where
$$\begin{aligned} \Lambda = \sqrt{ \frac{ \left( 2n+1 \right) \left( n+m+1 \right) \left( n-m+1 \right) }{2n+3}} \end{aligned}$$
(103)
The stresses of the P-wave are obtained substituting Eqs. (100)–(102) into Eqs. (88)–(99)
$$\begin{aligned} \sigma _{rr_p}= & {} \frac{Y_n^m \left( \theta , \varphi \right) }{\left( 1-\nu \right) r^2} \Big [\left( 2\mu n \left( n-1 \right) \left( 1-\nu \right) \right. \nonumber \\&\left. - \beta _p^2 r^2 \left[ \lambda + 2\mu -2\nu \left( \lambda +\mu \right) \right] \right) j_n \left( \beta _p r \right) \nonumber \\&+ 4\mu \left( 1-\nu \right) \beta _p r j_{n+1} \left( \beta _p r \right) \Big ] \end{aligned}$$
(104)
$$\begin{aligned} \sigma _{\theta \theta _p}= & {} \frac{1}{\left( 1-\nu \right) r^2 \sin ^2 \theta } \Bigg (\Bigg [\Big (2\mu \left( 1-\nu \right) \left( m^2+n+1 \right) \nonumber \\&+ \left[ \lambda \left( 2\nu -1 \right) \beta _p^2 r^2 \right. \nonumber \\&\left. - 2\mu \left( 1-\nu \right) \left( n^2+n+1 \right) \right] \sin ^2 \theta \Big ) j_n \left( \beta _p r \right) \nonumber \\&- 2\mu \left( 1-\nu \right) \beta _p r \sin ^2\theta j_{n+1} \left( \beta _p r \right) \Bigg ] Y_n^m \left( \theta , \varphi \right) \nonumber \\&-2\mu \left( 1-\nu \right) \cos \theta \Lambda j_n \left( \beta _p r \right) Y_{n+1}^m \left( \theta , \varphi \right) \Bigg ) \end{aligned}$$
(105)
$$\begin{aligned} \sigma _{\varphi \varphi _p}= & {} \frac{1}{2 \left( 1-\nu \right) r^2 \sin ^2 \theta } \Bigg [\Bigg (\Big [ \lambda \left( 2\nu -1 \right) \beta _p^2 r^2\nonumber \\&- 2\mu \left( 1-\nu \right) \left( 2 m^2+1 \right) \nonumber \\&+ \left[ \lambda \left( 1-2\nu \right) \beta _p^2 r^2 \right. \nonumber \\&\left. -2\mu \left( 1-\nu \right) \left( 2n+1 \right) \right] \cos \left( 2\theta \right) \Big ] j_n \left( \beta _p r \right) \nonumber \\&- 4 \mu \left( 1-\nu \right) \beta _p r \sin ^2 \theta j_{n+1} \left( \beta _p r \right) \Bigg ) Y_n^m \left( \theta , \varphi \right) \nonumber \\&+ 4\mu \left( 1-\nu \right) \cos \theta \Lambda j_n \left( \beta _p r \right) Y_{n+1}^m \left( \theta , \varphi \right) \Bigg ] \end{aligned}$$
(106)
$$\begin{aligned} \sigma _{r \theta _{p}}= & {} \frac{2\mu }{r^2 \sin \theta } \left[ \left( 1-n \right) j_n \left( \beta _{p} r \right) + \beta _{p} r j_{n+1} \left( \beta _{p} r \right) \right] \nonumber \\&\left[ \left( n+1 \right) \cos \theta Y_n^m \left( \theta , \varphi \right) - \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \right] \end{aligned}$$
(107)
$$\begin{aligned} \sigma _{r \varphi _{p}}= & {} \frac{2 {\hat{\imath }} m \mu }{r^2 \sin \theta } Y_n^m \left( \theta , \varphi \right) \nonumber \\&\left[ \left( n-1 \right) j_n \left( \beta _{p} r \right) - \beta _{p} r j_{n+1} \left( \beta _{p} r \right) \right] \end{aligned}$$
(108)
$$\begin{aligned} \sigma _{\theta \varphi _{p}}= & {} \frac{-2 {\hat{\imath }} m \mu }{r^2 \sin ^2 \theta } j_n \left( \beta _{p} r \right) \nonumber \\&\left[ \left( n+2 \right) \cos \theta Y_n^m \left( \theta , \varphi \right) - \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \right] \end{aligned}$$
(109)
Appendix D: S-waves functions
The components of the S1 and S2-waves displacements are obtained substituting the potentials (42)–(47) into Eq. (31)
$$\begin{aligned} u_{r_{s_1}}= & {} \frac{{\hat{\imath }} n \left( n+1 \right) }{2mr \sin ^2 \theta } \left[ \cos \left( 2 \theta \right) -1 \right] j_n \left( \beta _s r \right) Y_n^m \left( \theta , \varphi \right) \end{aligned}$$
(110)
$$\begin{aligned} u_{\theta _{s_1}}= & {} \frac{{\hat{\imath }}}{mr \sin \theta } \left[ \left( n+1 \right) j_n \left( \beta _s r \right) - \beta _s r j_{n+1} \left( \beta _s r \right) \right] \nonumber \\&\left[ \left( n+1 \right) \cos \theta Y_n^m \left( \theta , \varphi \right) - \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \right] \end{aligned}$$
(111)
$$\begin{aligned} u_{\varphi _{s_1}}= & {} \frac{Y_n^m \left( \theta , \varphi \right) }{r \sin \theta } \left[ \left( n+1 \right) j_n \left( \beta _s r \right) - \beta _s r j_{n+1} \left( \beta _s r \right) \right] \end{aligned}$$
(112)
$$\begin{aligned} u_{r_{s_2}}= & {} \frac{{\hat{\imath }}}{2m \beta _s r^2 \sin ^2 \theta } \Bigg (\Big [\left( 2n^2-n-1 \right) \left( \cos \left( 2 \theta \right) -1 \right) j_n \left( \beta _s r \right) \nonumber \\&+ 2 \left( n-1 \right) \beta _s r \sin ^2 \theta j_{n+1} \left( \beta _{s_{s}} r \right) \Big ] \cos \theta Y_n^m \left( \theta , \varphi \right) \nonumber \\&+ 2 \sin ^2 \theta \left[ \left( n-1 \right) j_n \left( \beta _s r \right) \right. \nonumber \\&\left. - \beta _s r j_{n+1} \left( \beta _s r \right) \right] \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \Bigg ) \end{aligned}$$
(113)
$$\begin{aligned} u_{\theta _{s_2}}= & {} \frac{{\hat{\imath }}}{2m \beta _s r^2 \sin \theta } \Bigg [\Big ([2m^2+n+\beta _s^2 r^2 \nonumber \\&+ \left( 2n^2+n- \beta _s^2 r^2 \right) \cos \left( 2\theta \right) ] j_n \left( \beta _s r \right) \nonumber \\&- \beta _s r \left( n+2+n \cos \left( 2\theta \right) \right) j_{n+1} \left( \beta _s r \right) \Big ) Y_n^m \left( \theta , \varphi \right) \nonumber \\&- 2 \cos \theta \left( n j_n \left( \beta _s r \right) \right. \nonumber \\&\left. - \beta _s r j_{n+1} \left( \beta _s r \right) \right) \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \Bigg ] \end{aligned}$$
(114)
$$\begin{aligned} u_{\varphi _{s_2}}= & {} \frac{1}{\beta _sr^2 \sin \theta } \Bigg (\Big [\left( 2n+1 \right) j_n \left( \beta _s r \right) \nonumber \\&- \beta _s r j_{n+1} \left( \beta _s r \right) \Big ] \cos \theta Y_n^m \left( \theta , \varphi \right) \nonumber \\&- \Lambda j_n \left( \beta _s r \right) Y_{n+1}^m \left( \theta , \varphi \right) \Bigg ) \end{aligned}$$
(115)
The stresses of the S1 and S2-waves are calculated substituting Eqs. (110)–(115) into Eqs. (88)–(99)
$$\begin{aligned} \sigma _{rr_{s_1}}= & {} \frac{2\mu {\hat{\imath }} n \left( n+1 \right) }{mr^2} Y_n^m \left( \theta , \varphi \right) \left[ \left( 1-n \right) j_n \left( \beta _s r \right) \right. \nonumber \\&\left. + \beta _s r j_{n+1} \left( \beta _s r \right) \right] \end{aligned}$$
(116)
$$\begin{aligned} \sigma _{\theta \theta _{s_1}}= & {} \frac{\mu {\hat{\imath }}}{mr^2 \sin ^2 \theta } \Bigg [\Big (\left( n+1 \right) [n^2-2m^2-n\nonumber \\&-1- \left( n^2+n+1 \right) \cos \left( 2\theta \right) ] j_n \left( \beta _s r \right) \nonumber \\&+ \beta _s r \left[ 2m^2-n^2+1\right. \nonumber \\&\left. + \left( n+1 \right) ^2 \cos \left( 2\theta \right) \right] j_{n+1} \left( \beta _s r \right) \Big ) Y_n^m \left( \theta , \varphi \right) \nonumber \\&+ 2 \cos \theta \left[ \left( n+1 \right) j_n \left( \beta _s r \right) \right. \nonumber \\&\left. - \beta _s r j_{n+1} \left( \beta _s r \right) \right] \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \Bigg ] \end{aligned}$$
(117)
$$\begin{aligned} \sigma _{\varphi \varphi _{s_1}}= & {} \frac{\mu {\hat{\imath }}}{mr^2 \sin ^2 \theta } \Bigg [\Big (\left( n+1 \right) \left[ 2m^2+1\right. \nonumber \\&\left. + \left( 2n+1 \right) \cos \left( 2 \theta \right) \right] j_n \left( \beta _s r \right) \nonumber \\&- \beta _s r \left[ 2m^2+n+1\right. \nonumber \\&\left. + \left( n+1 \right) \cos \left( 2\theta \right) \right] j_{n+1} \left( \beta _s r \right) \Big ) Y_n^m \left( \theta , \varphi \right) \nonumber \\&- 2 \cos \theta \left[ \left( n+1 \right) j_n \left( \beta _s r \right) \right. \nonumber \\&\left. - \beta _s r j_{n+1} \left( \beta _s r \right) \right] \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \Bigg ] \end{aligned}$$
(118)
$$\begin{aligned} \sigma _{r \theta _{s_1}}= & {} \frac{\mu {\hat{\imath }}}{mr^2 \sin \theta } \left[ \left( 2n^2-2-\beta _s^2 r^2 \right) j_n \left( \beta _s \right) \right. \nonumber \\&\left. + 2 \beta _s r j_{n+1} \left( \beta _s r \right) \right] \nonumber \\&\left[ \left( n+1 \right) \cos \theta Y_n^m \left( \theta , \varphi \right) - \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \right] \end{aligned}$$
(119)
$$\begin{aligned} \sigma _{r \varphi _{s_1}}= & {} \frac{\mu }{r^2 \sin \theta } Y_n^m \left( \theta , \varphi \right) \left[ \left( 2n^2-2-\beta _s^2 r^2 \right) j_n \left( \beta _s r \right) \right. \nonumber \\&\left. + 2 \beta _s r j_{n+1} \left( \beta _s r \right) \right] \end{aligned}$$
(120)
$$\begin{aligned} \sigma _{\theta \varphi _{s_1}}= & {} \frac{-2\mu }{r^2 \sin ^2 \theta } \left[ \left( n+1 \right) j_n \left( \beta _s r \right) - \beta _s r j_{n+1} \left( \beta _s r \right) \right] \nonumber \\&\left[ \left( n+2 \right) \cos \theta Y_n^m \left( \theta , \varphi \right) - \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \right] \end{aligned}$$
(121)
$$\begin{aligned} \sigma _{rr_{s_2}}= & {} \frac{2\mu {\hat{\imath }}}{m\beta _s r^3} \Big [ \left( \left[ \beta _s^2 r^2 + n \left( 3-2n \right) +2 \right] j_n \left( \beta _s r \right) \right. \nonumber \\&\left. + \beta _s r \left( n-2 \right) j_{n+1} \left( \beta _s r \right) \right) \left( n-1 \right) \cos \theta Y_n^m \left( \theta ,\varphi \right) \nonumber \\&+ \left( \left[ n \left( n-3 \right) +2-\beta _s^2 r^2 \right] j_n \left( \beta _s r \right) \right. \nonumber \\&\left. + 4 \beta _s r j_{n+1} \left( \beta _s r \right) \right) \Lambda Y_{n+1}^m \left( \theta ,\varphi \right) \Big ] \end{aligned}$$
(122)
$$\begin{aligned} \sigma _{\theta \theta _{s_2}}= & {} \frac{\mu {\hat{\imath }}}{m \beta _s r^3 \sin ^2{\theta }} \Bigg (\Big [\Big ( 1-4m^2\left( n+1 \right) \nonumber \\&-n \left[ n \left( 3-2n \right) +1+\beta ^2 r^2 \right] \nonumber \\&- \left( 1+n \left[ 1+n\left( 2n-1 \right) \right. \right. \nonumber \\&\left. \left. -\beta ^2 r^2 \right] \right) \cos \left( 2\theta \right) ) j_n \left( \beta _s r \right) \nonumber \\&+ \beta _s r \left( 2m^2+1-n \left( n-3 \right) \right. \nonumber \\&\left. + \left[ 1+n \left( n-1 \right) \right] \cos \left( 2\theta \right) \right) j_{n+1} \left( \beta _s r \right) \Big ] \cos \theta Y_n^m \left( \theta ,\varphi \right) \nonumber \\&+ \Big (\left[ 2m^2-1+n\left( 3-n \right) + \beta ^2 r^2 \right. \nonumber \\&\left. + \left( 1-n+n^2-\beta ^2 r^2 \right) \cos \left( 2\theta \right) \right] j_n \left( \beta _s r \right) \nonumber \\&+ 2 \beta _s r \left( \cos \left( 2\theta \right) -2 \right) j_{n+1} \left( \beta _s r \right) \Big ) \Lambda Y_{n+1}^m \left( \theta ,\varphi \right) \Bigg ) \end{aligned}$$
(123)
$$\begin{aligned} \sigma _{\varphi \varphi _{s_2}}= & {} \frac{\mu {\hat{\imath }}}{m\beta _s r^3 \sin ^2{\theta }} \Bigg (\Big [\Big (2 \left[ 2m^2\left( n+1 \right) - n^2+n \right] \nonumber \\&+ 1 + \beta _s^2 r^2 + \left( 4n^2-1-\beta ^2 r^2 \right) \cos \left( 2\theta \right) \Big ) j_n \left( \beta _s r \right) \nonumber \\&+ \beta r \left[ \left( 1-2n \right) \cos \left( 2\theta \right) \right. \nonumber \\&\left. -2m^2 -3 \right] j_{n+1} \left( \beta _s r \right) \Big ] \cos \theta Y_n^m \left( \theta ,\varphi \right) \nonumber \\&+ \left( \left[ \left( 1-2n \right) \cos \left( 2 \theta \right) -2m^2-1 \right] j_n \left( \beta _s r \right) \right. \nonumber \\&\left. + 2 \beta _s r \cos \left( 2 \theta \right) j_{n+1} \left( \beta r \right) \right) \Lambda Y_{n+1}^m \left( \theta ,\varphi \right) \Bigg ) \end{aligned}$$
(124)
$$\begin{aligned} \sigma _{r \theta _{s_2}}= & {} \frac{\mu {\hat{\imath }}}{2m\beta _s r^3 \sin \theta } \Bigg (\Big [\Big [2 \left( n-2 \right) \left( 2m^2+n \right) - 3 \beta _s^2 r^2 \nonumber \\&+ \left[ 2n^2 \left( 2n-3 \right) +\beta ^2 r^2 \right. \nonumber \\&\left. -2n \left( \beta _s^2 r^2 +2 \right) \right] \cos \left( 2 \theta \right) \Big ] j_n \left( \beta _s r \right) \nonumber \\&+ \beta _s r (2n \left( n+4 \right) +4 \left( 3-m^2 \right) -\beta ^2 r^2 \nonumber \\&+ \left[ 2n \left( 2-n \right) + \beta ^2 r^2 \right] \cos \left( 2 \theta \right) ) j_{n+1} \left( \beta _s r \right) \Big ] Y_n^m \left( \theta , \varphi \right) \nonumber \\&+ 2 \cos \theta \left( \left[ 2n \left( 2-n \right) + \beta _s^2 r^2 \right] j_n \left( \beta _s r \right) \right. \nonumber \\&\left. -6 \beta _s r j_{n+1} \left( \beta _s r \right) \right) \Lambda Y_{n+1}^m \left( \theta ,\varphi \right) \Bigg ) \end{aligned}$$
(125)
$$\begin{aligned} \sigma _{r \varphi _{s_2}}= & {} \frac{\mu }{\beta _s r^3 \sin \theta } \Bigg [\Big (\left( 2 \left( 2n^2-3n-2 \right) - \beta ^2 r^2 \right] j_n \left( \beta _s r \right) \nonumber \\&+ 2 \left( 2-n \right) \beta _s r j_{n+1} \left( \beta _s r \right) \Big ) \cos \theta Y_n^m \left( \theta ,\varphi \right) \nonumber \\&+ 2 \left[ \left( 2-n \right) j_n \left( \beta _s r \right) \right. \nonumber \\&\left. + \beta _s r j_{n+1} \left( \beta _s r \right) \right] \Lambda Y_{n+1}^m \left( \theta ,\varphi \right) \Bigg ] \end{aligned}$$
(126)
$$\begin{aligned} \sigma _{\theta \varphi _{s_2}}= & {} \frac{\mu }{2\beta _sr^3\sin ^2 \theta } \Bigg (\Big [\Big (\left[ \beta _s^2 r^2 - 2 \left( 2n^2+3n+1 \right) \right] \cos \left( 2\theta \right) \nonumber \\&- \left[ 2 \left( 2m^2+3n+1 \right) + \beta _s^2r^2 \right] \Big ) j_n \left( \beta _s r \right) \nonumber \\&+ 2 \beta _s r \left[ n+3+ \left( n+1 \right) \cos \left( 2\theta \right) \right] j_{n+1} \left( \beta _s r \right) \Big ] Y_n^m \left( \theta ,\varphi \right) \nonumber \\&+ 4 \cos \theta \left[ \left( n+1 \right) j_n \left( \beta _s r \right) \right. \nonumber \\&\left. - \beta _s r j_{n+1} \left( \beta _s r \right) \right] \Lambda Y_{n+1}^m \left( \theta , \varphi \right) \Bigg ) \end{aligned}$$
(127)
