音波の散乱に関する計算．
運動方程式
次の記事から，運動方程式は\begin{aligned}
\rho_{1} \frac{\partial^{2} \bm{u}}{\partial t^{2}}
&= (\lambda + 2\mu) \mathrm{grad\,} \mathrm{div\,} \bm{u}
- \mu \mathrm{\,rot\,} (2 \tilde{\bm{\omega}})
\tag{1}
\end{aligned}
\rho_{1} \frac{\partial^{2} \bm{u}}{\partial t^{2}}
&= (\lambda + 2\mu) \mathrm{grad\,} \mathrm{div\,} \bm{u}
- \mu \mathrm{\,rot\,} (2 \tilde{\bm{\omega}})
\tag{1}
\end{aligned}
\begin{aligned}
2 \tilde{\bm{\omega}}
&= \mathrm{\,rot\,} \bm{u}
\tag{2}
\end{aligned}
となる．2 \tilde{\bm{\omega}}
&= \mathrm{\,rot\,} \bm{u}
\tag{2}
\end{aligned}
式(1)の$\mathrm{div}$をとると
\begin{aligned}
\rho_{1} \frac{\partial^{2} }{\partial t^{2}} (\mathrm{div\,} \bm{u})
&= (\lambda + 2\mu) \bm{\nabla}^{2} (\mathrm{div\,} \bm{u})
\tag{3}
\end{aligned}
が得られる．\rho_{1} \frac{\partial^{2} }{\partial t^{2}} (\mathrm{div\,} \bm{u})
&= (\lambda + 2\mu) \bm{\nabla}^{2} (\mathrm{div\,} \bm{u})
\tag{3}
\end{aligned}
式(1)の$\mathrm{rot}$をとると
\begin{aligned}
\rho_{1} \frac{\partial^{2}}{\partial t^{2}} (2 \tilde{\bm{\omega}})
&= \mu \bm{\nabla}^{2} (2 \tilde{\bm{\omega}})
\tag{4}
\end{aligned}
が得られる（ベクトル解析の公式 - Notes_JP）．\rho_{1} \frac{\partial^{2}}{\partial t^{2}} (2 \tilde{\bm{\omega}})
&= \mu \bm{\nabla}^{2} (2 \tilde{\bm{\omega}})
\tag{4}
\end{aligned}
よって，
\begin{aligned}
&\biggl(\frac{1}{c_{1}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) (\mathrm{div\,} \bm{u}) = 0 \\
&c_{1}
= \sqrt{\frac{\lambda + 2\mu}{\rho_{1}}}
=\sqrt{\frac{1-\nu}{(1+\nu)(1-2\nu)}} \sqrt{\frac{E}{\rho_{1}}}
\end{aligned}
および&\biggl(\frac{1}{c_{1}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) (\mathrm{div\,} \bm{u}) = 0 \\
&c_{1}
= \sqrt{\frac{\lambda + 2\mu}{\rho_{1}}}
=\sqrt{\frac{1-\nu}{(1+\nu)(1-2\nu)}} \sqrt{\frac{E}{\rho_{1}}}
\end{aligned}
\begin{aligned}
&\biggl(\frac{1}{c_{2}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) (2 \tilde{\bm{\omega}}) = 0 \\
&c_{2}
= \sqrt{\frac{\mu}{\rho_{1}}}
=\frac{1}{\sqrt{2(1+\nu)}} \sqrt{\frac{E}{\rho_{1}}}
\end{aligned}
が得られる（ラメ定数 - Notes_JP）．&\biggl(\frac{1}{c_{2}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) (2 \tilde{\bm{\omega}}) = 0 \\
&c_{2}
= \sqrt{\frac{\mu}{\rho_{1}}}
=\frac{1}{\sqrt{2(1+\nu)}} \sqrt{\frac{E}{\rho_{1}}}
\end{aligned}
$\bm{u}$はスカラーポテンシャル$\Psi$とベクトルポテンシャル$\bm{A}$を用いて
\begin{aligned}
\bm{u}
&=-\mathrm{grad\,} \Psi + \mathrm{rot\,}\bm{A}
\tag{7}
\end{aligned}
と表せる（ヘルムホルツの定理（ヘルムホルツ分解） - Notes_JP）．このとき\bm{u}
&=-\mathrm{grad\,} \Psi + \mathrm{rot\,}\bm{A}
\tag{7}
\end{aligned}
\begin{aligned}
&\mathrm{div\,}\bm{u}
=-\bm{\nabla}^{2} \Psi \\
&2 \tilde{\bm{\omega}}
= \mathrm{rot\,}\bm{u}
= \mathrm{rot\,}\mathrm{rot\,}\bm{A}
\end{aligned}
なので，以下の方程式の解を式(7)に代入して得た$\bm{u}$は式(3), (4)の解になる．&\mathrm{div\,}\bm{u}
=-\bm{\nabla}^{2} \Psi \\
&2 \tilde{\bm{\omega}}
= \mathrm{rot\,}\bm{u}
= \mathrm{rot\,}\mathrm{rot\,}\bm{A}
\end{aligned}
\begin{aligned}
&\biggl(\frac{1}{c_{1}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) \Psi = 0 \\
&\biggl(\frac{1}{c_{2}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) \bm{A} = 0
\end{aligned}
&\biggl(\frac{1}{c_{1}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) \Psi = 0 \\
&\biggl(\frac{1}{c_{2}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) \bm{A} = 0
\end{aligned}
ここで
\begin{aligned}
\begin{cases}
\, \Psi(\bm{x}, t) = e^{i\omega t} \Psi(\bm{x}) \\
\, \bm{A}(\bm{x}, t) = e^{i\omega t} \bm{A}(\bm{x}) \\
\end{cases}
\end{aligned}
の形の解を考えると，解くべき方程式は\begin{cases}
\, \Psi(\bm{x}, t) = e^{i\omega t} \Psi(\bm{x}) \\
\, \bm{A}(\bm{x}, t) = e^{i\omega t} \bm{A}(\bm{x}) \\
\end{cases}
\end{aligned}
\begin{aligned}
\begin{cases}
\, (\bm{\nabla}^{2} + k_{1}^{2}) \Psi(\bm{x}) = 0,
\quad (k_{1} = \omega / c_{1}) \\
\, (\bm{\nabla}^{2} + k_{2}^{2}) \bm{A}(\bm{x}) = 0,
\quad (k_{2} = \omega / c_{2})
\end{cases}
\end{aligned}
となる．\begin{cases}
\, (\bm{\nabla}^{2} + k_{1}^{2}) \Psi(\bm{x}) = 0,
\quad (k_{1} = \omega / c_{1}) \\
\, (\bm{\nabla}^{2} + k_{2}^{2}) \bm{A}(\bm{x}) = 0,
\quad (k_{2} = \omega / c_{2})
\end{cases}
\end{aligned}
また，
\begin{aligned}
\mathrm{div\,}\bm{u}(\bm{x})
&= -\bm{\nabla}^{2} \Psi(\bm{x}) \\
& = k_{1}^{2} \Psi(\bm{x}) \\
&=\frac{1}{r^{2}} (k_{1} r)^{2} \Psi(\bm{x})
\end{aligned}
が成り立つ．\mathrm{div\,}\bm{u}(\bm{x})
&= -\bm{\nabla}^{2} \Psi(\bm{x}) \\
& = k_{1}^{2} \Psi(\bm{x}) \\
&=\frac{1}{r^{2}} (k_{1} r)^{2} \Psi(\bm{x})
\end{aligned}
非粘性流体の音波の方程式は
\begin{aligned}
&\biggl(\frac{1}{c_{3}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) p = 0
\end{aligned}
となる（音波の方程式 - Notes_JP）．&\biggl(\frac{1}{c_{3}^{2}} \frac{\partial^{2}}{\partial t^{2}} - \bm{\nabla}^{2} \biggr) p = 0
\end{aligned}
連立方程式の形
境界条件から次の連立方程式が得られる．\begin{aligned}
\begin{pmatrix}
M_{11} & M_{12} & M_{13} \\
M_{21} & M_{22} & M_{23} \\
M_{31} & M_{32} & 0 \\
\end{pmatrix}
\begin{pmatrix}
a_{n} \\
b_{n} \\
c_{n}
\end{pmatrix}
=
\begin{pmatrix}
\xi \\
\eta \\
0
\end{pmatrix}
\end{aligned}
\begin{pmatrix}
M_{11} & M_{12} & M_{13} \\
M_{21} & M_{22} & M_{23} \\
M_{31} & M_{32} & 0 \\
\end{pmatrix}
\begin{pmatrix}
a_{n} \\
b_{n} \\
c_{n}
\end{pmatrix}
=
\begin{pmatrix}
\xi \\
\eta \\
0
\end{pmatrix}
\end{aligned}
\begin{aligned}
&
\left(
\begin{array}{ccc|c}
M_{11} & M_{12} & M_{13} & \xi \\
M_{21} & M_{22} & M_{23} & \eta \\
M_{31} & M_{32} & 0 & 0\\
\end{array}
\right)
\begin{array}{l}
\times M_{31} \\
\times M_{31} \\
\\
\end{array} \\
\rightarrow &
\left(
\begin{array}{ccc|c}
M_{11}M_{31} & M_{12}M_{31} & M_{13}M_{31} & M_{31} \xi \\
M_{21}M_{31} & M_{22}M_{31} & M_{23}M_{31} & M_{31} \eta \\
M_{31} & M_{32} & 0 & 0\\
\end{array}
\right) \\
\rightarrow &
\left(
\begin{array}{ccc|c}
0 & \underbrace{M_{12}M_{31} - M_{11}M_{32}}_{=A} & M_{13}M_{31} & M_{31} \xi \\
0 & \underbrace{M_{22}M_{31} - M_{21}M_{32}}_{=B} & M_{23}M_{31} & M_{31} \eta \\
M_{31} & M_{32} & 0 & 0\\
\end{array}
\right)
\end{aligned}
&
\left(
\begin{array}{ccc|c}
M_{11} & M_{12} & M_{13} & \xi \\
M_{21} & M_{22} & M_{23} & \eta \\
M_{31} & M_{32} & 0 & 0\\
\end{array}
\right)
\begin{array}{l}
\times M_{31} \\
\times M_{31} \\
\\
\end{array} \\
\rightarrow &
\left(
\begin{array}{ccc|c}
M_{11}M_{31} & M_{12}M_{31} & M_{13}M_{31} & M_{31} \xi \\
M_{21}M_{31} & M_{22}M_{31} & M_{23}M_{31} & M_{31} \eta \\
M_{31} & M_{32} & 0 & 0\\
\end{array}
\right) \\
\rightarrow &
\left(
\begin{array}{ccc|c}
0 & \underbrace{M_{12}M_{31} - M_{11}M_{32}}_{=A} & M_{13}M_{31} & M_{31} \xi \\
0 & \underbrace{M_{22}M_{31} - M_{21}M_{32}}_{=B} & M_{23}M_{31} & M_{31} \eta \\
M_{31} & M_{32} & 0 & 0\\
\end{array}
\right)
\end{aligned}
\begin{aligned}
[(-B/A)M_{13}M_{31} + M_{23}M_{31}]c_{n}
= (-B/A) M_{31} \xi + M_{31} \eta
\end{aligned}
よって，散乱波を定める係数は[(-B/A)M_{13}M_{31} + M_{23}M_{31}]c_{n}
= (-B/A) M_{31} \xi + M_{31} \eta
\end{aligned}
\begin{aligned}
c_{n}
&=\frac{(B/A) \xi - \eta}{(B/A)M_{13} - M_{23}} \\
\frac{B}{A}
&= \frac{M_{21} / M_{31} - M_{22} / M_{32}}
{M_{11} / M_{31} - M_{12} / M_{32}}
\end{aligned}
となる．c_{n}
&=\frac{(B/A) \xi - \eta}{(B/A)M_{13} - M_{23}} \\
\frac{B}{A}
&= \frac{M_{21} / M_{31} - M_{22} / M_{32}}
{M_{11} / M_{31} - M_{12} / M_{32}}
\end{aligned}
さらに，条件
\begin{aligned}
\begin{cases}
\, \displaystyle
\frac{\xi}{\mathrm{Re\,} M_{13}}
= \frac{\eta}{\mathrm{Re\,} M_{23}}
= X \\
\, B/A \in \mathbb{R}
\end{cases}
\end{aligned}
を満たす場合，\begin{cases}
\, \displaystyle
\frac{\xi}{\mathrm{Re\,} M_{13}}
= \frac{\eta}{\mathrm{Re\,} M_{23}}
= X \\
\, B/A \in \mathbb{R}
\end{cases}
\end{aligned}
\begin{aligned}
c_{n}
&= X \frac{\mathrm{Re}[(B/A)M_{13} - M_{23}] }{(B/A)M_{13} - M_{23}}
= X \frac{\mathrm{Re\,}Y}{Y} \frac{\bar{Y}}{\bar{Y}} \\
&= -iX \frac{\mathrm{Re\,}Y}{|Y|} \frac{i \bar{Y}}{|Y|}
= -iX \frac{\mathrm{Im\,}Z}{|Z|} \frac{Z}{|Z|} \\
&= -iX \sin\theta e^{i\theta} \\
&
\begin{cases}
Y = (B/A)M_{13} - M_{23} \\
Z = i\bar{Y} = \mathrm{Im\,}Y + i \mathrm{Re\,} Y = |Z|e^{i\theta}
\end{cases}
\end{aligned}
c_{n}
&= X \frac{\mathrm{Re}[(B/A)M_{13} - M_{23}] }{(B/A)M_{13} - M_{23}}
= X \frac{\mathrm{Re\,}Y}{Y} \frac{\bar{Y}}{\bar{Y}} \\
&= -iX \frac{\mathrm{Re\,}Y}{|Y|} \frac{i \bar{Y}}{|Y|}
= -iX \frac{\mathrm{Im\,}Z}{|Z|} \frac{Z}{|Z|} \\
&= -iX \sin\theta e^{i\theta} \\
&
\begin{cases}
Y = (B/A)M_{13} - M_{23} \\
Z = i\bar{Y} = \mathrm{Im\,}Y + i \mathrm{Re\,} Y = |Z|e^{i\theta}
\end{cases}
\end{aligned}
\begin{aligned}
\tan\theta
&= \frac{\mathrm{Re\,} Y}{\mathrm{Im\,}Y} \\
&= \frac{(B/A)\mathrm{Re\,} M_{13} - \mathrm{Re\,} M_{23}}
{(B/A)\mathrm{Im\,} M_{13} - \mathrm{Im\,} M_{23}} \\
&= \frac{\mathrm{Re\,} M_{13}}{\mathrm{Im\,} M_{13}}
\frac{B/A - \mathrm{Re\,} M_{23} / \mathrm{Re\,} M_{13}}
{B/A - \mathrm{Im\,} M_{23} / \mathrm{Im\,} M_{13}}
\end{aligned}
\tan\theta
&= \frac{\mathrm{Re\,} Y}{\mathrm{Im\,}Y} \\
&= \frac{(B/A)\mathrm{Re\,} M_{13} - \mathrm{Re\,} M_{23}}
{(B/A)\mathrm{Im\,} M_{13} - \mathrm{Im\,} M_{23}} \\
&= \frac{\mathrm{Re\,} M_{13}}{\mathrm{Im\,} M_{13}}
\frac{B/A - \mathrm{Re\,} M_{23} / \mathrm{Re\,} M_{13}}
{B/A - \mathrm{Im\,} M_{23} / \mathrm{Im\,} M_{13}}
\end{aligned}
円筒座標
$z$軸上に中心軸をもつ円筒の弾性体が流体中に置かれており，$-x$方向から平面波が入射する問題を考える．弾性体（円筒内部）
$z$依存性がなく，原点で発散せず，$\theta = 0$に対して対称である場合には\begin{aligned}
\Psi(r, \theta)
&=\sum_{n = 0}^{\infty} a_{n} J_{n} (k_{1} r) \cos(n\theta)
\end{aligned}
となる．\Psi(r, \theta)
&=\sum_{n = 0}^{\infty} a_{n} J_{n} (k_{1} r) \cos(n\theta)
\end{aligned}
$A_{r} = A_{\theta} = 0$としてよく，$\mathrm{rot\,}\bm{A}$が$\theta = 0$に対して対称であることから
\begin{aligned}
A_{z}(r, \theta)
&=\sum_{n = 0}^{\infty} b_{n} J_{n} (k_{2} r) \sin(n\theta)
\end{aligned}
となる．A_{z}(r, \theta)
&=\sum_{n = 0}^{\infty} b_{n} J_{n} (k_{2} r) \sin(n\theta)
\end{aligned}
円筒座標の$\mathrm{grad},\mathrm{rot}$（Del in cylindrical and spherical coordinates - Wikipedia）から
\begin{aligned}
u_{r}
&= - \frac{\partial \Psi}{\partial r}
+ \biggl(\frac{1}{r} \frac{\partial A_{z}}{\partial\theta} - \frac{\partial A_{\theta}}{\partial z}\biggr) \\
&=\sum_{n = 0}^{\infty}
\biggl[-a_{n} k_{1} J_{n}^{\prime}(k_{1}r)
+ \frac{b_{n} n}{r} J_{n}(k_{2}r) \biggr] \cos (n\theta) \\
&=\sum_{n = 0}^{\infty} \frac{1}{r}
\biggl[-a_{n} k_{1} r J_{n}^{\prime}(k_{1}r)
+ b_{n} nJ_{n}(k_{2}r) \biggr] \cos (n\theta)
\end{aligned}
u_{r}
&= - \frac{\partial \Psi}{\partial r}
+ \biggl(\frac{1}{r} \frac{\partial A_{z}}{\partial\theta} - \frac{\partial A_{\theta}}{\partial z}\biggr) \\
&=\sum_{n = 0}^{\infty}
\biggl[-a_{n} k_{1} J_{n}^{\prime}(k_{1}r)
+ \frac{b_{n} n}{r} J_{n}(k_{2}r) \biggr] \cos (n\theta) \\
&=\sum_{n = 0}^{\infty} \frac{1}{r}
\biggl[-a_{n} k_{1} r J_{n}^{\prime}(k_{1}r)
+ b_{n} nJ_{n}(k_{2}r) \biggr] \cos (n\theta)
\end{aligned}
\begin{aligned}
u_{\theta}
&= - \frac{1}{r} \frac{\partial \Psi}{\partial \theta}
+ \biggl(\frac{\partial A_{r}}{\partial z} - \frac{\partial A_{z}}{\partial r}\biggr) \\
&=\sum_{n = 0}^{\infty}
\biggl[\frac{a_{n} n}{r} J_{n}(k_{1}r)
- b_{n} k_{2} J_{n}^{\prime}(k_{2}r) \biggr] \sin (n\theta) \\
&=\sum_{n = 0}^{\infty} \frac{1}{r}
\biggl[a_{n} n J_{n}(k_{1}r)
- b_{n} k_{2} r J_{n}^{\prime}(k_{2}r) \biggr] \sin (n\theta)
\end{aligned}
u_{\theta}
&= - \frac{1}{r} \frac{\partial \Psi}{\partial \theta}
+ \biggl(\frac{\partial A_{r}}{\partial z} - \frac{\partial A_{z}}{\partial r}\biggr) \\
&=\sum_{n = 0}^{\infty}
\biggl[\frac{a_{n} n}{r} J_{n}(k_{1}r)
- b_{n} k_{2} J_{n}^{\prime}(k_{2}r) \biggr] \sin (n\theta) \\
&=\sum_{n = 0}^{\infty} \frac{1}{r}
\biggl[a_{n} n J_{n}(k_{1}r)
- b_{n} k_{2} r J_{n}^{\prime}(k_{2}r) \biggr] \sin (n\theta)
\end{aligned}
\begin{aligned}
& \frac{\partial u_{r}}{\partial r} \\
&=\sum_{n = 0}^{\infty}
\biggl[-a_{n} k_{1}^{2} J_{n}^{\prime\prime}(k_{1}r)
- \frac{b_{n} n}{r^{2}} J_{n}(k_{2}r) \\
& \qquad\qquad
+ \frac{b_{n} n}{r} k_{2} J_{n}^{\prime}(k_{2}r) \biggr] \cos (n\theta) \\
&=\sum_{n = 0}^{\infty} \frac{1}{r^{2}}
\biggl\{-a_{n} (k_{1}r)^{2} J_{n}^{\prime\prime}(k_{1}r) \\
& \qquad
+ b_{n} n [ - J_{n}(k_{2}r) + k_{2}r J_{n}^{\prime}(k_{2}r) ]
\biggr\} \cos (n\theta)
\end{aligned}
& \frac{\partial u_{r}}{\partial r} \\
&=\sum_{n = 0}^{\infty}
\biggl[-a_{n} k_{1}^{2} J_{n}^{\prime\prime}(k_{1}r)
- \frac{b_{n} n}{r^{2}} J_{n}(k_{2}r) \\
& \qquad\qquad
+ \frac{b_{n} n}{r} k_{2} J_{n}^{\prime}(k_{2}r) \biggr] \cos (n\theta) \\
&=\sum_{n = 0}^{\infty} \frac{1}{r^{2}}
\biggl\{-a_{n} (k_{1}r)^{2} J_{n}^{\prime\prime}(k_{1}r) \\
& \qquad
+ b_{n} n [ - J_{n}(k_{2}r) + k_{2}r J_{n}^{\prime}(k_{2}r) ]
\biggr\} \cos (n\theta)
\end{aligned}
\begin{aligned}
& \frac{\partial u_{\theta}}{\partial r} \\
&=\sum_{n = 0}^{\infty} \frac{1}{r^{2}}
\biggl\{ [- a_{n} n J_{n} (k_{1}r)
+ a_{n} n k_{1}r J_{n}^{\prime}(k_{1}r)] \\
& \qquad\qquad\qquad
- b_{n} (k_{2} r)^{2} J_{n}^{\prime\prime}(k_{2}r) \biggr]\biggr\} \sin (n\theta)
\end{aligned}
& \frac{\partial u_{\theta}}{\partial r} \\
&=\sum_{n = 0}^{\infty} \frac{1}{r^{2}}
\biggl\{ [- a_{n} n J_{n} (k_{1}r)
+ a_{n} n k_{1}r J_{n}^{\prime}(k_{1}r)] \\
& \qquad\qquad\qquad
- b_{n} (k_{2} r)^{2} J_{n}^{\prime\prime}(k_{2}r) \biggr]\biggr\} \sin (n\theta)
\end{aligned}
流体（円筒外部）
入射平面波は\begin{aligned}
p_{\mathrm{i}}
&= P_{0} e^{-ik_{3}x} \\
&= P_{0} e^{-ik_{3} r \cos\theta} \\
&= P_{0} \sum_{n = 0}^{\infty} \epsilon_{n} (-i)^{n} J_{n}(k_{3}r) \cos(n\theta)
\end{aligned}
と表せる（ベッセル関数の関係式 - Notes_JP）．p_{\mathrm{i}}
&= P_{0} e^{-ik_{3}x} \\
&= P_{0} e^{-ik_{3} r \cos\theta} \\
&= P_{0} \sum_{n = 0}^{\infty} \epsilon_{n} (-i)^{n} J_{n}(k_{3}r) \cos(n\theta)
\end{aligned}
$\rho_{3} \ddot{u}_{\mathrm{i}, r} = - \partial p_{\mathrm{i}} / \partial r$より
\begin{aligned}
u_{\mathrm{i}, r}
&= \frac{1}{\rho_{3} \omega^{2}}
\frac{\partial p_{\mathrm{i}}}{\partial r} \\
&= \frac{1}{r} \frac{P_{0} k_{3} r}{\rho_{3} \omega^{2}}
\sum_{n = 0}^{\infty} \epsilon_{n} (-i)^{n} J_{n}^{\prime} (k_{3}r) \cos(n\theta)
\end{aligned}
u_{\mathrm{i}, r}
&= \frac{1}{\rho_{3} \omega^{2}}
\frac{\partial p_{\mathrm{i}}}{\partial r} \\
&= \frac{1}{r} \frac{P_{0} k_{3} r}{\rho_{3} \omega^{2}}
\sum_{n = 0}^{\infty} \epsilon_{n} (-i)^{n} J_{n}^{\prime} (k_{3}r) \cos(n\theta)
\end{aligned}
散乱波は$\theta = 0$に対して対称な外向きの波だから，第2種Hankel関数$H_{n}^{(2)}(x) = J_{n}(x) - iN_{n}(x)$を使って
\begin{aligned}
p_{\mathrm{s}}
&= \sum_{n = 0}^{\infty}
c_{n}[J_{n}(k_{3}r) - iN(k_{3}r)] \cos(n\theta)
\end{aligned}
と表せる．このとき，p_{\mathrm{s}}
&= \sum_{n = 0}^{\infty}
c_{n}[J_{n}(k_{3}r) - iN(k_{3}r)] \cos(n\theta)
\end{aligned}
\begin{aligned}
u_{\mathrm{s}, r}
&= \frac{1}{\rho_{3} \omega^{2}}
\frac{\partial p_{\mathrm{s}}}{\partial r} \\
&= \frac{1}{r} \frac{k_{3}r}{\rho_{3} \omega^{2}}
\sum_{n = 0}^{\infty} c_{n} [J_{n}^{\prime} (k_{3}r) - iN_{n}^{\prime} (k_{3}r)] \cos(n\theta)
\end{aligned}
u_{\mathrm{s}, r}
&= \frac{1}{\rho_{3} \omega^{2}}
\frac{\partial p_{\mathrm{s}}}{\partial r} \\
&= \frac{1}{r} \frac{k_{3}r}{\rho_{3} \omega^{2}}
\sum_{n = 0}^{\infty} c_{n} [J_{n}^{\prime} (k_{3}r) - iN_{n}^{\prime} (k_{3}r)] \cos(n\theta)
\end{aligned}
応力テンソル（弾性体）
フックの法則/ひずみテンソルの座標変換（極座標・円筒座標） - Notes_JPラメ定数 - Notes_JP
弾性率 - Wikipedia
\begin{aligned}
\mu & = \rho_{1} c_{2}^{2} = \rho_{1} (\omega / k_{2})^{2} \\
\lambda &= \frac{2\mu\nu}{1 - 2\nu} = 2\rho_{1} c_{2}^{2} \frac{\nu}{1 - 2\nu}
\end{aligned}
\mu & = \rho_{1} c_{2}^{2} = \rho_{1} (\omega / k_{2})^{2} \\
\lambda &= \frac{2\mu\nu}{1 - 2\nu} = 2\rho_{1} c_{2}^{2} \frac{\nu}{1 - 2\nu}
\end{aligned}
$\sigma_{rr}$
\begin{aligned}
&\mu
\Biggl(\frac{\partial u_r}{\partial r} + \frac{\partial u_r}{\partial r} \Biggr)
+ \lambda \mathrm{\,div\,}\boldsymbol{u} \\
&=2\mu \frac{\partial u_r}{\partial r} + \lambda \mathrm{\,div\,}\boldsymbol{u} \\
&=2\rho_{1} (\omega / k_{2})^{2}
\biggl(\frac{\partial u_r}{\partial r} + \frac{\nu}{1 - 2\nu} \mathrm{\,div\,}\boldsymbol{u} \biggr) \\
&= \frac{2\rho_{1} \omega^{2}}{(k_{2} r)^{2}} \sum_{n = 0}^{\infty} \cos(n\theta) \\
&\qquad
\biggl\{
a_{n} (k_{1}r)^{2} \biggl[\frac{\nu}{1 - 2\nu} J_{n} (k_{1} r) - J_{n}^{\prime\prime}(k_{1}r)\biggr] \\
& \qquad\qquad
+ b_{n} n [ - J_{n}(k_{2}r) + k_{2}r J_{n}^{\prime}(k_{2}r) ]
\biggr\}
\end{aligned}
&\mu
\Biggl(\frac{\partial u_r}{\partial r} + \frac{\partial u_r}{\partial r} \Biggr)
+ \lambda \mathrm{\,div\,}\boldsymbol{u} \\
&=2\mu \frac{\partial u_r}{\partial r} + \lambda \mathrm{\,div\,}\boldsymbol{u} \\
&=2\rho_{1} (\omega / k_{2})^{2}
\biggl(\frac{\partial u_r}{\partial r} + \frac{\nu}{1 - 2\nu} \mathrm{\,div\,}\boldsymbol{u} \biggr) \\
&= \frac{2\rho_{1} \omega^{2}}{(k_{2} r)^{2}} \sum_{n = 0}^{\infty} \cos(n\theta) \\
&\qquad
\biggl\{
a_{n} (k_{1}r)^{2} \biggl[\frac{\nu}{1 - 2\nu} J_{n} (k_{1} r) - J_{n}^{\prime\prime}(k_{1}r)\biggr] \\
& \qquad\qquad
+ b_{n} n [ - J_{n}(k_{2}r) + k_{2}r J_{n}^{\prime}(k_{2}r) ]
\biggr\}
\end{aligned}
$\sigma_{r\theta}$
\begin{aligned}
&\mu \frac{1}{r}
\Biggl[\frac{\partial u_r}{\partial \theta}
+ r^2 \frac{\partial }{\partial r} \biggl( \frac{u_\theta }{r} \biggr) \Biggr] \\
&=\mu \biggl(\frac{1}{r} \frac{\partial u_r}{\partial \theta}
+\frac{\partial u_\theta}{\partial r}
-\frac{u_\theta}{r}
\biggr) \\
&= \mu \frac{1}{r^{2}} \sum_{n=0}^{\infty} \sin(n\theta) \\
& \quad \times
\biggl[a_{n} n k_{1} r J_{n}^{\prime}(k_{1}r)
- b_{n} n^{2} J_{n}(k_{2}r) \\
& \qquad
- a_{n} n J_{n} (k_{1}r)
+ a_{n} n k_{1}r J_{n}^{\prime}(k_{1}r)
- b_{n} (k_{2} r)^{2} J_{n}^{\prime\prime}(k_{2}r) \\
& \qquad
- a_{n} n J_{n}(k_{1}r)
+ b_{n} k_{2} r J_{n}^{\prime}(k_{2}r)
\biggr] \\
&= \mu \frac{1}{r^{2}} \sum_{n=0}^{\infty} \sin(n\theta) \\
& \quad \times
\biggl\{
2n[k_{1} r J_{n}^{\prime}(k_{1}r) - J_{n} (k_{1}r) ] a_{n} \\
& \qquad
- [(k_{2} r)^{2} J_{n}^{\prime\prime}(k_{2}r)
- k_{2} r J_{n}^{\prime}(k_{2}r)
+ n^{2} J_{n}(k_{2}r)
] b_{n}
\biggr\}
\end{aligned}
&\mu \frac{1}{r}
\Biggl[\frac{\partial u_r}{\partial \theta}
+ r^2 \frac{\partial }{\partial r} \biggl( \frac{u_\theta }{r} \biggr) \Biggr] \\
&=\mu \biggl(\frac{1}{r} \frac{\partial u_r}{\partial \theta}
+\frac{\partial u_\theta}{\partial r}
-\frac{u_\theta}{r}
\biggr) \\
&= \mu \frac{1}{r^{2}} \sum_{n=0}^{\infty} \sin(n\theta) \\
& \quad \times
\biggl[a_{n} n k_{1} r J_{n}^{\prime}(k_{1}r)
- b_{n} n^{2} J_{n}(k_{2}r) \\
& \qquad
- a_{n} n J_{n} (k_{1}r)
+ a_{n} n k_{1}r J_{n}^{\prime}(k_{1}r)
- b_{n} (k_{2} r)^{2} J_{n}^{\prime\prime}(k_{2}r) \\
& \qquad
- a_{n} n J_{n}(k_{1}r)
+ b_{n} k_{2} r J_{n}^{\prime}(k_{2}r)
\biggr] \\
&= \mu \frac{1}{r^{2}} \sum_{n=0}^{\infty} \sin(n\theta) \\
& \quad \times
\biggl\{
2n[k_{1} r J_{n}^{\prime}(k_{1}r) - J_{n} (k_{1}r) ] a_{n} \\
& \qquad
- [(k_{2} r)^{2} J_{n}^{\prime\prime}(k_{2}r)
- k_{2} r J_{n}^{\prime}(k_{2}r)
+ n^{2} J_{n}(k_{2}r)
] b_{n}
\biggr\}
\end{aligned}
$\sigma_{rz}$
\begin{aligned}
&\mu \biggl(\frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \biggr) \\
&=0
\end{aligned}
&\mu \biggl(\frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \biggr) \\
&=0
\end{aligned}
境界条件
$p_{\mathrm{i}} + p_{\mathrm{s}} = - \sigma_{rr}$
$x_{i} = k_{i} a$とすると\begin{aligned}
& \sum_{n = 0}^{\infty} \cos(n\theta)
\biggl\{
P_{0}\epsilon_{n} (-i)^{n} J_{n}(x_{3}) \\
&\qquad
+ c_{n}[J_{n}(x_{3}) - iN(x_{3})] \\
&\qquad
+ \frac{2\rho_{1} \omega^{2}}{x_{2}^{2}}
a_{n} x_{1}^{2} \biggl[\frac{\nu}{1 - 2\nu} J_{n} (x_{1}) - J_{n}^{\prime\prime}(x_{1})\biggr] \\
&\qquad
+ \frac{2\rho_{1} \omega^{2}}{x_{2}^{2}}
b_{n} n [ - J_{n}(x_{2}) + x_{2} J_{n}^{\prime}(x_{2}) ]
\biggr\}
= 0
\end{aligned}
& \sum_{n = 0}^{\infty} \cos(n\theta)
\biggl\{
P_{0}\epsilon_{n} (-i)^{n} J_{n}(x_{3}) \\
&\qquad
+ c_{n}[J_{n}(x_{3}) - iN(x_{3})] \\
&\qquad
+ \frac{2\rho_{1} \omega^{2}}{x_{2}^{2}}
a_{n} x_{1}^{2} \biggl[\frac{\nu}{1 - 2\nu} J_{n} (x_{1}) - J_{n}^{\prime\prime}(x_{1})\biggr] \\
&\qquad
+ \frac{2\rho_{1} \omega^{2}}{x_{2}^{2}}
b_{n} n [ - J_{n}(x_{2}) + x_{2} J_{n}^{\prime}(x_{2}) ]
\biggr\}
= 0
\end{aligned}
よって
\begin{aligned}
& x_{1}^{2} \biggl[\frac{\nu}{1 - 2\nu} J_{n}(x_{1}) - J_{n}^{\prime\prime}(x_{1})\biggr] a_{n} \\
& + n \biggl[x_{2} J_{n}^{\prime}(x_{2}) - J_{n}(x_{2})\biggr] b_{n} \\
& + \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} \biggl[J_{n} (x_{3}) - i N_{n}(x_{3})\biggr] c_{n} \\
&= \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}}
\biggl[-P_{0} \epsilon_{n} (- i)^{n} J_{n}(x_{3}) \biggr]
\end{aligned}
& x_{1}^{2} \biggl[\frac{\nu}{1 - 2\nu} J_{n}(x_{1}) - J_{n}^{\prime\prime}(x_{1})\biggr] a_{n} \\
& + n \biggl[x_{2} J_{n}^{\prime}(x_{2}) - J_{n}(x_{2})\biggr] b_{n} \\
& + \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} \biggl[J_{n} (x_{3}) - i N_{n}(x_{3})\biggr] c_{n} \\
&= \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}}
\biggl[-P_{0} \epsilon_{n} (- i)^{n} J_{n}(x_{3}) \biggr]
\end{aligned}
$u_{\mathrm{i},r} + u_{\mathrm{s},r} = - u_{r}$
\begin{aligned}
& x_{1} J_{n}^{\prime}(x_{1}) a_{n} \\
& -nJ_{n}(x_{2}) b_{n} \\
& + \frac{x_{3}}{\rho_{3} \omega^{2}}
\biggl[J_{n}^{\prime} (x_{3}) - i N_{n}^{\prime} (x_{3})\biggr] c_{n} \\
&= \frac{x_{3}}{\rho_{3} \omega^{2}}
\biggl[-P_{0} \epsilon_{n} (- i)^{n} J_{n}^{\prime} (x_{3}) \biggr]
\end{aligned}
& x_{1} J_{n}^{\prime}(x_{1}) a_{n} \\
& -nJ_{n}(x_{2}) b_{n} \\
& + \frac{x_{3}}{\rho_{3} \omega^{2}}
\biggl[J_{n}^{\prime} (x_{3}) - i N_{n}^{\prime} (x_{3})\biggr] c_{n} \\
&= \frac{x_{3}}{\rho_{3} \omega^{2}}
\biggl[-P_{0} \epsilon_{n} (- i)^{n} J_{n}^{\prime} (x_{3}) \biggr]
\end{aligned}
$\sigma_{r\theta} = 0$
\begin{aligned}
& 2n\biggl[x_{1} J_{n}^{\prime}(x_{1}) - J_{n}(x_{1}) \biggr] a_{n} \\
& - \biggl[n^{2} J_{n} (x_{2}) - x_{2} J_{n}^{\prime}(x_{2}) + x_{2}^{2} J_{n}^{\prime\prime}(x_{2}) \biggr] b_{n} \\
&= 0
\end{aligned}
& 2n\biggl[x_{1} J_{n}^{\prime}(x_{1}) - J_{n}(x_{1}) \biggr] a_{n} \\
& - \biggl[n^{2} J_{n} (x_{2}) - x_{2} J_{n}^{\prime}(x_{2}) + x_{2}^{2} J_{n}^{\prime\prime}(x_{2}) \biggr] b_{n} \\
&= 0
\end{aligned}
散乱波の計算
\begin{aligned}
\frac{\xi}{\mathrm{Re\,} M_{13}}
= \frac{\eta}{\mathrm{Re\,} M_{23}}
= X
= - P_{0} \epsilon_{n} (- i)^{n}
\end{aligned}
\frac{\xi}{\mathrm{Re\,} M_{13}}
= \frac{\eta}{\mathrm{Re\,} M_{23}}
= X
= - P_{0} \epsilon_{n} (- i)^{n}
\end{aligned}
\begin{aligned}
c_{n}
&= - P_{0} \epsilon_{n} (- i)^{n + 1} \sin\eta_{n} e^{i\eta_{n}} \\
\frac{B}{A}
&= \frac{M_{21} / M_{31} - M_{22} / M_{32}}
{M_{11} / M_{31} - M_{12} / M_{32}} \\
&= \scriptsize
\frac{x_{1} J_{n}^{\prime}(x_{1}) / (M_{31} / 2n) - 2n \cdot nJ_{n}(x_{2}) / (-M_{32})}
{x_{1}^{2} [\frac{\nu}{1 - 2\nu} J_{n}(x_{1}) - J_{n}^{\prime\prime}(x_{1})] / (M_{31} / 2n) + 2n \cdot n [x_{2} J_{n}^{\prime}(x_{2}) - J_{n}(x_{2})] / (-M_{32})}
\end{aligned}
c_{n}
&= - P_{0} \epsilon_{n} (- i)^{n + 1} \sin\eta_{n} e^{i\eta_{n}} \\
\frac{B}{A}
&= \frac{M_{21} / M_{31} - M_{22} / M_{32}}
{M_{11} / M_{31} - M_{12} / M_{32}} \\
&= \scriptsize
\frac{x_{1} J_{n}^{\prime}(x_{1}) / (M_{31} / 2n) - 2n \cdot nJ_{n}(x_{2}) / (-M_{32})}
{x_{1}^{2} [\frac{\nu}{1 - 2\nu} J_{n}(x_{1}) - J_{n}^{\prime\prime}(x_{1})] / (M_{31} / 2n) + 2n \cdot n [x_{2} J_{n}^{\prime}(x_{2}) - J_{n}(x_{2})] / (-M_{32})}
\end{aligned}
\begin{aligned}
\tan\eta_{n}
&= \frac{\mathrm{Re\,} M_{13}}{\mathrm{Im\,} M_{13}}
\frac{B/A - \mathrm{Re\,} M_{23} / \mathrm{Re\,} M_{13}}
{B/A - \mathrm{Im\,} M_{23} / \mathrm{Im\,} M_{13}} \\
&= - \frac{J_{n} (x_{3})}{N_{n} (x_{3})}
\frac{\dfrac{B}{A} - \frac{\frac{x_{3}}{\rho_{3} \omega^{2}} J_{n}^{\prime} (x_{3})}{\frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} J_{n} (x_{3})}}
{\dfrac{B}{A} - \frac{- \frac{x_{3}}{\rho_{3} \omega^{2}} N_{n}^{\prime} (x_{3})}{- \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} N_{n} (x_{3})}} \\
&= - \frac{J_{n} (x_{3})}{N_{n} (x_{3})}
\frac{\dfrac{\rho_{3}}{\rho_{1}} \dfrac{x_{2}^{2}}{2} \dfrac{B}{A} - x_{3} \dfrac{J_{n}^{\prime} (x_{3})}{J_{n} (x_{3})}}
{\dfrac{\rho_{3}}{\rho_{1}} \dfrac{x_{2}^{2}}{2} \dfrac{B}{A} - x_{3} \dfrac{N_{n}^{\prime} (x_{3})}{N_{n} (x_{3})}}
\end{aligned}
\tan\eta_{n}
&= \frac{\mathrm{Re\,} M_{13}}{\mathrm{Im\,} M_{13}}
\frac{B/A - \mathrm{Re\,} M_{23} / \mathrm{Re\,} M_{13}}
{B/A - \mathrm{Im\,} M_{23} / \mathrm{Im\,} M_{13}} \\
&= - \frac{J_{n} (x_{3})}{N_{n} (x_{3})}
\frac{\dfrac{B}{A} - \frac{\frac{x_{3}}{\rho_{3} \omega^{2}} J_{n}^{\prime} (x_{3})}{\frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} J_{n} (x_{3})}}
{\dfrac{B}{A} - \frac{- \frac{x_{3}}{\rho_{3} \omega^{2}} N_{n}^{\prime} (x_{3})}{- \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} N_{n} (x_{3})}} \\
&= - \frac{J_{n} (x_{3})}{N_{n} (x_{3})}
\frac{\dfrac{\rho_{3}}{\rho_{1}} \dfrac{x_{2}^{2}}{2} \dfrac{B}{A} - x_{3} \dfrac{J_{n}^{\prime} (x_{3})}{J_{n} (x_{3})}}
{\dfrac{\rho_{3}}{\rho_{1}} \dfrac{x_{2}^{2}}{2} \dfrac{B}{A} - x_{3} \dfrac{N_{n}^{\prime} (x_{3})}{N_{n} (x_{3})}}
\end{aligned}
不要だが一応残しておく
【直接計算1（不要）】\begin{aligned}
& \mathrm{div\,}\bm{u} \\
&=\frac{1}{r}\frac{\partial}{\partial r}(ru_{r})
+ \frac{1}{r} \frac{\partial u_{\theta}}{\partial \theta}
+ \frac{\partial u_{z}}{\partial z} \\
&=\sum_{n = 0}^{\infty} \cos (n\theta) \\
&\quad \times
\biggl\{-a_{n} \biggl[ \frac{ k_{1}}{r} J_{n}^{\prime}(k_{1}r)
+ k_{1}^{2} J_{n}^{\prime\prime}(k_{1}r)
- \frac{n^{2}}{r^{2}} J_{n}(k_{1}r)
\biggr] \\
&\quad\qquad
+ b_{n} \biggl[ \frac{n k_{2}}{r} J_{n}^{\prime}(k_{2}r)
-\frac{n k_{2}}{r} J_{n}^{\prime}(k_{2}r)
\biggr]
\biggr\} \\
&= -k_{1}^{2} \sum_{n = 0}^{\infty} a_{n} \cos (n\theta) \\
&\qquad \times
\biggl[J_{n}^{\prime\prime}(k_{1}r)
+ \frac{1}{k_{1}r} J_{n}^{\prime}(k_{1}r)
- \frac{n^{2}}{(k_{1} r)^{2}} J_{n}(k_{1}r)
\biggr] \\
&= k_{1}^{2} \sum_{n = 0}^{\infty} a_{n} J_{n}(k_{1}r) \cos (n\theta)
\end{aligned}
（最後にBesselの微分方程式を使った）．& \mathrm{div\,}\bm{u} \\
&=\frac{1}{r}\frac{\partial}{\partial r}(ru_{r})
+ \frac{1}{r} \frac{\partial u_{\theta}}{\partial \theta}
+ \frac{\partial u_{z}}{\partial z} \\
&=\sum_{n = 0}^{\infty} \cos (n\theta) \\
&\quad \times
\biggl\{-a_{n} \biggl[ \frac{ k_{1}}{r} J_{n}^{\prime}(k_{1}r)
+ k_{1}^{2} J_{n}^{\prime\prime}(k_{1}r)
- \frac{n^{2}}{r^{2}} J_{n}(k_{1}r)
\biggr] \\
&\quad\qquad
+ b_{n} \biggl[ \frac{n k_{2}}{r} J_{n}^{\prime}(k_{2}r)
-\frac{n k_{2}}{r} J_{n}^{\prime}(k_{2}r)
\biggr]
\biggr\} \\
&= -k_{1}^{2} \sum_{n = 0}^{\infty} a_{n} \cos (n\theta) \\
&\qquad \times
\biggl[J_{n}^{\prime\prime}(k_{1}r)
+ \frac{1}{k_{1}r} J_{n}^{\prime}(k_{1}r)
- \frac{n^{2}}{(k_{1} r)^{2}} J_{n}(k_{1}r)
\biggr] \\
&= k_{1}^{2} \sum_{n = 0}^{\infty} a_{n} J_{n}(k_{1}r) \cos (n\theta)
\end{aligned}
【直接計算2（不要）】
\begin{aligned}
& \mathrm{div\,}\bm{u} = -\bm{\nabla}^{2} \Psi \\
&= -\biggl[ \frac{1}{r}\frac{\partial}{\partial r}\biggl(r \frac{\partial \Psi}{\partial r} \biggr)
+\frac{1}{r^2}\frac{\partial^2 \Psi}{\partial \theta^2}
+\frac{\partial^2 \Psi}{\partial z^2} \biggr] \\
&= - k_{1}^{2} \sum_{n = 0}^{\infty} a_{n} \cos (n\theta) \\
&\quad \times
\biggl[ \frac{1}{k_{1}r}\frac{\partial}{\partial (k_{1}r)}\biggl(k_{1}r \frac{\partial J_{n}(k_{1}r)}{\partial (k_{1}r)} \biggr)
- \frac{n^{2}}{(k_{1} r)^2} J_{n}(k_{1}r) \biggr] \\
&= k_{1}^{2} \sum_{n = 0}^{\infty} a_{n} J_{n}(k_{1}r) \cos (n\theta)
\end{aligned}
（最後にBesselの微分方程式（ヘルムホルツ方程式 - Notes_JP）を使った）．& \mathrm{div\,}\bm{u} = -\bm{\nabla}^{2} \Psi \\
&= -\biggl[ \frac{1}{r}\frac{\partial}{\partial r}\biggl(r \frac{\partial \Psi}{\partial r} \biggr)
+\frac{1}{r^2}\frac{\partial^2 \Psi}{\partial \theta^2}
+\frac{\partial^2 \Psi}{\partial z^2} \biggr] \\
&= - k_{1}^{2} \sum_{n = 0}^{\infty} a_{n} \cos (n\theta) \\
&\quad \times
\biggl[ \frac{1}{k_{1}r}\frac{\partial}{\partial (k_{1}r)}\biggl(k_{1}r \frac{\partial J_{n}(k_{1}r)}{\partial (k_{1}r)} \biggr)
- \frac{n^{2}}{(k_{1} r)^2} J_{n}(k_{1}r) \biggr] \\
&= k_{1}^{2} \sum_{n = 0}^{\infty} a_{n} J_{n}(k_{1}r) \cos (n\theta)
\end{aligned}
球座標
弾性体（球内部）
$\Psi(\bm{x})$の方程式はヘルムホルツ方程式で，極座標で表した一般解は次の記事で与えられる：ヘルムホルツ方程式 - Notes_JP．$z$軸に関して対称な解（$\varphi$に依存しない解）は$m = 0$の項を考えればよく，原点で発散しないためには$n_{n}$は含まれないので，
\begin{aligned}
\Psi(r, \theta)
&= \sum_{n=0}^{\infty} a_{n} j_{n}(k_{1} r) P_{n}(\cos\theta)
\end{aligned}
\Psi(r, \theta)
&= \sum_{n=0}^{\infty} a_{n} j_{n}(k_{1} r) P_{n}(\cos\theta)
\end{aligned}
\begin{aligned}
A_{\varphi}(r, \theta)
&=\sum_{n=0}^{\infty} b_{n} j_{n}(k_{2} r)
\underbrace{\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta)}_{= - \sin\theta \cdot P^{\prime}_{n}(\cos\theta)}
\end{aligned}
A_{\varphi}(r, \theta)
&=\sum_{n=0}^{\infty} b_{n} j_{n}(k_{2} r)
\underbrace{\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta)}_{= - \sin\theta \cdot P^{\prime}_{n}(\cos\theta)}
\end{aligned}
変位$\bm{u}$が軸対称で，$\varphi$依存性がないとする．
軸対称な波動方程式（ベクトルポテンシャル） - Notes_JP
極座標の$\mathrm{grad},\mathrm{rot}$（Del in cylindrical and spherical coordinates - Wikipedia）から
\begin{aligned}
u_{r}
&= - \frac{\partial \Psi}{\partial r}
+ \frac{1}{r\sin\theta}
\biggl[\frac{\partial}{\partial \theta} (A_{\varphi} \sin\theta)
- \cancel{\frac{\partial A_{\theta}}{\partial \varphi}} \biggr] \\
&= \sum_{n=0}^{\infty} \frac{1}{r}
\biggl[-a_{n} k_{1}r j_{n}^{\prime} (k_{1} r)
-n(n + 1) b_{n} j_{n}(k_{2} r)
\biggr] P_{n} (\cos\theta)
\end{aligned}
（ここで，軸対称な波動方程式（ベクトルポテンシャル） - Notes_JPのように，$z=\cos\theta$とするとLegendreの微分方程式（Legendre polynomials - Wikipedia）よりu_{r}
&= - \frac{\partial \Psi}{\partial r}
+ \frac{1}{r\sin\theta}
\biggl[\frac{\partial}{\partial \theta} (A_{\varphi} \sin\theta)
- \cancel{\frac{\partial A_{\theta}}{\partial \varphi}} \biggr] \\
&= \sum_{n=0}^{\infty} \frac{1}{r}
\biggl[-a_{n} k_{1}r j_{n}^{\prime} (k_{1} r)
-n(n + 1) b_{n} j_{n}(k_{2} r)
\biggr] P_{n} (\cos\theta)
\end{aligned}
\begin{aligned}
& \frac{1}{\sin\theta} \frac{\mathrm{d}}{\mathrm{d} \theta}
\biggl[\sin\theta \frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta) \biggr] \\
&= -\frac{\mathrm{d}}{\mathrm{d} z}
\biggl[-(1 - z^{2}) P_{n}^{\prime}(z) \biggr] \\
&=(1 - z^{2}) P_{n}^{\prime\prime}(z) - 2z P_{n}^{\prime}(z) \\
&= -n(n + 1) P_{n}(z)
\end{aligned}
が成り立つことを用いた）．& \frac{1}{\sin\theta} \frac{\mathrm{d}}{\mathrm{d} \theta}
\biggl[\sin\theta \frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta) \biggr] \\
&= -\frac{\mathrm{d}}{\mathrm{d} z}
\biggl[-(1 - z^{2}) P_{n}^{\prime}(z) \biggr] \\
&=(1 - z^{2}) P_{n}^{\prime\prime}(z) - 2z P_{n}^{\prime}(z) \\
&= -n(n + 1) P_{n}(z)
\end{aligned}
\begin{aligned}
u_{\theta}
&= - \frac{1}{r} \frac{\partial \Psi}{\partial \theta}
+ \frac{1}{r}
\biggl[\cancel{\frac{1}{\sin\theta} \frac{\partial A_{r}}{\partial \varphi}}
- \frac{\partial}{\partial r} (r A_{\varphi}) \biggr] \\
&=\sum_{n=0}^{\infty} \frac{1}{r}
\biggl\{- a_{n} j_{n}(k_{1} r)
- b_{n} [j_{n}(k_{2} r) + k_{2} r j_{n}^{\prime}(k_{2} r)] \biggr\}
\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta)
\end{aligned}
u_{\theta}
&= - \frac{1}{r} \frac{\partial \Psi}{\partial \theta}
+ \frac{1}{r}
\biggl[\cancel{\frac{1}{\sin\theta} \frac{\partial A_{r}}{\partial \varphi}}
- \frac{\partial}{\partial r} (r A_{\varphi}) \biggr] \\
&=\sum_{n=0}^{\infty} \frac{1}{r}
\biggl\{- a_{n} j_{n}(k_{1} r)
- b_{n} [j_{n}(k_{2} r) + k_{2} r j_{n}^{\prime}(k_{2} r)] \biggr\}
\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta)
\end{aligned}
\begin{aligned}
\frac{\partial u_{r}}{\partial r}
&= - \frac{\partial^{2} \Psi}{\partial r^{2}}
- \frac{1}{r^{2}}
\frac{1}{\sin\theta}
\biggl[
\frac{\partial}{\partial \theta} (A_{\varphi} \sin\theta)
- \frac{\partial}{\partial \theta}
\biggl(r\frac{\partial A_{\varphi}}{\partial r} \sin\theta \biggr)
\biggr]\\
&= \sum_{n=0}^{\infty} \frac{1}{r^{2}}
\biggl\{-a_{n} (k_{1}r)^{2} j_{n}^{\prime\prime} (k_{1} r) \\
&\qquad\qquad
+ n(n + 1) b_{n} [j_{n}(k_{2} r) - k_{2} r j_{n}^{\prime}(k_{2} r)]
\biggr\} P_{n} (\cos\theta)
\end{aligned}
\frac{\partial u_{r}}{\partial r}
&= - \frac{\partial^{2} \Psi}{\partial r^{2}}
- \frac{1}{r^{2}}
\frac{1}{\sin\theta}
\biggl[
\frac{\partial}{\partial \theta} (A_{\varphi} \sin\theta)
- \frac{\partial}{\partial \theta}
\biggl(r\frac{\partial A_{\varphi}}{\partial r} \sin\theta \biggr)
\biggr]\\
&= \sum_{n=0}^{\infty} \frac{1}{r^{2}}
\biggl\{-a_{n} (k_{1}r)^{2} j_{n}^{\prime\prime} (k_{1} r) \\
&\qquad\qquad
+ n(n + 1) b_{n} [j_{n}(k_{2} r) - k_{2} r j_{n}^{\prime}(k_{2} r)]
\biggr\} P_{n} (\cos\theta)
\end{aligned}
\begin{aligned}
\frac{\partial u_{\theta}}{\partial r}
&= - \frac{u_{\theta}}{r}
+ \sum_{n=0}^{\infty} \frac{1}{r^{2}}
\biggl\{- a_{n} k_{1} r j_{n}^{\prime}(k_{1} r)
- b_{n} [2 k_{2} r j_{n}^{\prime}(k_{2} r)
+ (k_{2} r)^{2} j_{n}^{\prime\prime}(k_{2} r)]
\biggr\}
\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta)
\end{aligned}
\frac{\partial u_{\theta}}{\partial r}
&= - \frac{u_{\theta}}{r}
+ \sum_{n=0}^{\infty} \frac{1}{r^{2}}
\biggl\{- a_{n} k_{1} r j_{n}^{\prime}(k_{1} r)
- b_{n} [2 k_{2} r j_{n}^{\prime}(k_{2} r)
+ (k_{2} r)^{2} j_{n}^{\prime\prime}(k_{2} r)]
\biggr\}
\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta)
\end{aligned}
流体（球外部）
入射平面波は\begin{aligned}
p_{\mathrm{i}}
&= P_{0} e^{-ik_{3}z} \\
&= P_{0} e^{-ik_{3} r \cos\theta} \\
&= P_{0} \sum_{n = 0}^{\infty} (2n + 1) (-i)^{n} j_{n}(k_{3}r) P_{n}(\cos\theta)
\end{aligned}
と表せる（）．p_{\mathrm{i}}
&= P_{0} e^{-ik_{3}z} \\
&= P_{0} e^{-ik_{3} r \cos\theta} \\
&= P_{0} \sum_{n = 0}^{\infty} (2n + 1) (-i)^{n} j_{n}(k_{3}r) P_{n}(\cos\theta)
\end{aligned}
$\rho_{3} \ddot{u}_{\mathrm{i}, r} = - \partial p_{\mathrm{i}} / \partial r$より
\begin{aligned}
u_{\mathrm{i}, r}
&= \frac{1}{\rho_{3} \omega^{2}}
\frac{\partial p_{\mathrm{i}}}{\partial r} \\
&= \frac{1}{r} \frac{P_{0} k_{3} r}{\rho_{3} \omega^{2}}
\sum_{n = 0}^{\infty} (2n + 1) (-i)^{n} j_{n}^{\prime} (k_{3}r) P_{n}(\cos\theta)
\end{aligned}
u_{\mathrm{i}, r}
&= \frac{1}{\rho_{3} \omega^{2}}
\frac{\partial p_{\mathrm{i}}}{\partial r} \\
&= \frac{1}{r} \frac{P_{0} k_{3} r}{\rho_{3} \omega^{2}}
\sum_{n = 0}^{\infty} (2n + 1) (-i)^{n} j_{n}^{\prime} (k_{3}r) P_{n}(\cos\theta)
\end{aligned}
散乱波は$\theta = 0$に対して対称な外向きの波だから，第2種球Hankel関数$h_{n}^{(2)}(x) = j_{n}(x) - in_{n}(x) \sim i^{n + 1} \dfrac{e^{-ix}}{x} \,(|x| \to \infty)$を使って
\begin{aligned}
p_{\mathrm{s}}
&= \sum_{n = 0}^{\infty}
c_{n}[j_{n}(k_{3}r) - in_{n}(k_{3}r)] P_{n}(\cos\theta)
\end{aligned}
と表せる．このとき，p_{\mathrm{s}}
&= \sum_{n = 0}^{\infty}
c_{n}[j_{n}(k_{3}r) - in_{n}(k_{3}r)] P_{n}(\cos\theta)
\end{aligned}
\begin{aligned}
u_{\mathrm{s}, r}
&= \frac{1}{\rho_{3} \omega^{2}}
\frac{\partial p_{\mathrm{s}}}{\partial r} \\
&= \frac{1}{r} \frac{k_{3}r}{\rho_{3} \omega^{2}}
\sum_{n = 0}^{\infty} c_{n} [j_{n}^{\prime} (k_{3}r) - in_{n}^{\prime} (k_{3}r)] P_{n}(\cos\theta)
\end{aligned}
u_{\mathrm{s}, r}
&= \frac{1}{\rho_{3} \omega^{2}}
\frac{\partial p_{\mathrm{s}}}{\partial r} \\
&= \frac{1}{r} \frac{k_{3}r}{\rho_{3} \omega^{2}}
\sum_{n = 0}^{\infty} c_{n} [j_{n}^{\prime} (k_{3}r) - in_{n}^{\prime} (k_{3}r)] P_{n}(\cos\theta)
\end{aligned}
応力テンソル（弾性体）
フックの法則/ひずみテンソルの座標変換（極座標・円筒座標） - Notes_JP$\sigma_{rr}$
\begin{aligned}
&\mu
\Biggl(\frac{\partial u_r}{\partial r} + \frac{\partial u_r}{\partial r} \Biggr)
+ \lambda \mathrm{\,div\,}\boldsymbol{u} \\
&=2\mu \frac{\partial u_r}{\partial r} + \lambda \mathrm{\,div\,}\boldsymbol{u} \\
&=2\rho_{1} (\omega / k_{2})^{2}
\biggl(\frac{\partial u_r}{\partial r} + \frac{\nu}{1 - 2\nu} \mathrm{\,div\,}\boldsymbol{u} \biggr) \\
&= \frac{2\rho_{1}\omega^{2} }{(k_{2}r)^{2}}
\sum_{n=0}^{\infty} P_{n}(\cos\theta) \\
&\qquad
\biggl\{
a_{n} (k_{1}r)^{2}
\biggl[\frac{\nu}{1 - 2\nu} j_{n}(k_{1} r)
- j_{n}^{\prime\prime} (k_{1} r) \biggr] \\
&\qquad\qquad
+ b_{n} n(n + 1) [j_{n}(k_{2} r) - k_{2} r j_{n}^{\prime}(k_{2} r)]
\biggr\}
\end{aligned}
&\mu
\Biggl(\frac{\partial u_r}{\partial r} + \frac{\partial u_r}{\partial r} \Biggr)
+ \lambda \mathrm{\,div\,}\boldsymbol{u} \\
&=2\mu \frac{\partial u_r}{\partial r} + \lambda \mathrm{\,div\,}\boldsymbol{u} \\
&=2\rho_{1} (\omega / k_{2})^{2}
\biggl(\frac{\partial u_r}{\partial r} + \frac{\nu}{1 - 2\nu} \mathrm{\,div\,}\boldsymbol{u} \biggr) \\
&= \frac{2\rho_{1}\omega^{2} }{(k_{2}r)^{2}}
\sum_{n=0}^{\infty} P_{n}(\cos\theta) \\
&\qquad
\biggl\{
a_{n} (k_{1}r)^{2}
\biggl[\frac{\nu}{1 - 2\nu} j_{n}(k_{1} r)
- j_{n}^{\prime\prime} (k_{1} r) \biggr] \\
&\qquad\qquad
+ b_{n} n(n + 1) [j_{n}(k_{2} r) - k_{2} r j_{n}^{\prime}(k_{2} r)]
\biggr\}
\end{aligned}
$\sigma_{r\theta}$
\begin{aligned}
&\mu \frac{1}{r}
\Biggl[\frac{\partial u_r}{\partial \theta}
+ r^2 \frac{\partial }{\partial r} \biggl( \frac{u_\theta }{r} \biggr) \Biggr] \\
&=\mu \biggl(\frac{1}{r} \frac{\partial u_r}{\partial \theta}
+\frac{\partial u_\theta}{\partial r}
-\frac{u_\theta}{r}
\biggr) \\
&= \mu \frac{1}{r^{2}} \sum_{n=0}^{\infty}
\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta) \\
&\qquad
\biggl\{
-a_{n} k_{1}r j_{n}^{\prime} (k_{1} r)
-n(n + 1) b_{n} j_{n}(k_{2} r) \\
&\qquad
- a_{n} k_{1} r j_{n}^{\prime}(k_{1} r)
- b_{n} [2 k_{2} r j_{n}^{\prime}(k_{2} r)
+ (k_{2} r)^{2} j_{n}^{\prime\prime}(k_{2} r)] \\
&\qquad
+ 2 a_{n} j_{n}(k_{1} r)
+ 2 b_{n} [j_{n}(k_{2} r) + k_{2} r j_{n}^{\prime}(k_{2} r)]
\biggr\} \\
&= - \mu \frac{1}{r^{2}} \sum_{n=0}^{\infty}
\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta) \\
&\qquad
\biggl\{
2a_{n} [k_{1}r j_{n}^{\prime} (k_{1} r) - j_{n}(k_{1} r)] \\
&\qquad
+ b_{n} [(n^{2} + n - 2) j_{n}(k_{2} r)
+ (k_{2} r)^{2} j_{n}^{\prime\prime}(k_{2} r)]
\biggr\}
\end{aligned}
&\mu \frac{1}{r}
\Biggl[\frac{\partial u_r}{\partial \theta}
+ r^2 \frac{\partial }{\partial r} \biggl( \frac{u_\theta }{r} \biggr) \Biggr] \\
&=\mu \biggl(\frac{1}{r} \frac{\partial u_r}{\partial \theta}
+\frac{\partial u_\theta}{\partial r}
-\frac{u_\theta}{r}
\biggr) \\
&= \mu \frac{1}{r^{2}} \sum_{n=0}^{\infty}
\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta) \\
&\qquad
\biggl\{
-a_{n} k_{1}r j_{n}^{\prime} (k_{1} r)
-n(n + 1) b_{n} j_{n}(k_{2} r) \\
&\qquad
- a_{n} k_{1} r j_{n}^{\prime}(k_{1} r)
- b_{n} [2 k_{2} r j_{n}^{\prime}(k_{2} r)
+ (k_{2} r)^{2} j_{n}^{\prime\prime}(k_{2} r)] \\
&\qquad
+ 2 a_{n} j_{n}(k_{1} r)
+ 2 b_{n} [j_{n}(k_{2} r) + k_{2} r j_{n}^{\prime}(k_{2} r)]
\biggr\} \\
&= - \mu \frac{1}{r^{2}} \sum_{n=0}^{\infty}
\frac{\mathrm{d}}{\mathrm{d}\theta} P_{n}(\cos\theta) \\
&\qquad
\biggl\{
2a_{n} [k_{1}r j_{n}^{\prime} (k_{1} r) - j_{n}(k_{1} r)] \\
&\qquad
+ b_{n} [(n^{2} + n - 2) j_{n}(k_{2} r)
+ (k_{2} r)^{2} j_{n}^{\prime\prime}(k_{2} r)]
\biggr\}
\end{aligned}
$\sigma_{r\varphi}$
\begin{aligned}
&\mu \frac{1}{r\sin\theta}
\Biggl[\frac{\partial u_r}{\partial \varphi}
+ r^2\sin^2\theta
\frac{\partial }{\partial r} \Biggl(\frac{u_\varphi}{r\sin\theta}\Biggr) \Biggr] \\
&=\mu \Biggl( \frac{1}{r\sin\theta}\frac{\partial u_r}{\partial \varphi}
+ \frac{\partial u_\varphi}{\partial r}
-\frac{u_\varphi}{r}
\Biggr) \\
&=0
\end{aligned}
&\mu \frac{1}{r\sin\theta}
\Biggl[\frac{\partial u_r}{\partial \varphi}
+ r^2\sin^2\theta
\frac{\partial }{\partial r} \Biggl(\frac{u_\varphi}{r\sin\theta}\Biggr) \Biggr] \\
&=\mu \Biggl( \frac{1}{r\sin\theta}\frac{\partial u_r}{\partial \varphi}
+ \frac{\partial u_\varphi}{\partial r}
-\frac{u_\varphi}{r}
\Biggr) \\
&=0
\end{aligned}
境界条件
$p_{\mathrm{i}} + p_{\mathrm{s}} = - \sigma_{rr}$
\begin{aligned}
& x_{1}^{2}\biggl[ \frac{\nu}{1-2\nu} j_{n}(x_{1}) - j_{n}^{\prime\prime}(x_{1}) \biggr] a_{n} \\
& + n(n + 1) [j_{n}(x_{2}) - x_{2} j_{n}^{\prime}(x_{2})] b_{n} \\
& + \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} [j_{n}(x_{3}) - i n_{n}(x_{3})] c_{n} \\
&= \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} [ -P_{0} (2n + 1) (-i)^{n} j_{n}(x_{3})]
\end{aligned}
& x_{1}^{2}\biggl[ \frac{\nu}{1-2\nu} j_{n}(x_{1}) - j_{n}^{\prime\prime}(x_{1}) \biggr] a_{n} \\
& + n(n + 1) [j_{n}(x_{2}) - x_{2} j_{n}^{\prime}(x_{2})] b_{n} \\
& + \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} [j_{n}(x_{3}) - i n_{n}(x_{3})] c_{n} \\
&= \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} [ -P_{0} (2n + 1) (-i)^{n} j_{n}(x_{3})]
\end{aligned}
$u_{\mathrm{i},r} + u_{\mathrm{s},r} = - u_{r}$
\begin{aligned}
& x_{1} j_{n}^{\prime}(x_{1}) a_{n} \\
& + n(n + 1) j_{n}(x_{2}) b_{n} \\
& + \frac{x_{3}}{\rho_{3} \omega^{2}} [j_{n}^{\prime}(x_{3}) - i n_{n}^{\prime}(x_{3})] c_{n} \\
&= \frac{x_{3}}{\rho_{3} \omega^{2}} [ -P_{0} (2n + 1) (-i)^{n} j_{n}^{\prime}(x_{3})]
\end{aligned}
& x_{1} j_{n}^{\prime}(x_{1}) a_{n} \\
& + n(n + 1) j_{n}(x_{2}) b_{n} \\
& + \frac{x_{3}}{\rho_{3} \omega^{2}} [j_{n}^{\prime}(x_{3}) - i n_{n}^{\prime}(x_{3})] c_{n} \\
&= \frac{x_{3}}{\rho_{3} \omega^{2}} [ -P_{0} (2n + 1) (-i)^{n} j_{n}^{\prime}(x_{3})]
\end{aligned}
$\sigma_{r\theta} = 0$
\begin{aligned}
& 2[x_{1} j_{n}^{\prime}(x_{1}) - j_{n}(x_{1})] a_{n} \\
& + [(n^{2} + n - 2) j_{n}(x_{2}) + x_{2}^{2} j_{n}^{\prime\prime}(x_{2})] b_{n} \\
&= 0
\end{aligned}
& 2[x_{1} j_{n}^{\prime}(x_{1}) - j_{n}(x_{1})] a_{n} \\
& + [(n^{2} + n - 2) j_{n}(x_{2}) + x_{2}^{2} j_{n}^{\prime\prime}(x_{2})] b_{n} \\
&= 0
\end{aligned}
散乱波の計算
\begin{aligned}
\frac{\xi}{\mathrm{Re\,} M_{13}}
= \frac{\eta}{\mathrm{Re\,} M_{23}}
= X
= - P_{0} (2n + 1) (- i)^{n}
\end{aligned}
\frac{\xi}{\mathrm{Re\,} M_{13}}
= \frac{\eta}{\mathrm{Re\,} M_{23}}
= X
= - P_{0} (2n + 1) (- i)^{n}
\end{aligned}
\begin{aligned}
c_{n}
&= - P_{0} (2n + 1) (- i)^{n + 1} \sin\eta_{n} e^{i\eta_{n}} \\
\frac{B}{A}
&= \frac{M_{21} / M_{31} - M_{22} / M_{32}}
{M_{11} / M_{31} - M_{12} / M_{32}} \\
&= \scriptsize
\frac{x_{1} j_{n}^{\prime}(x_{1}) / (M_{31} / 2) - 2 \cdot n(n + 1) j_{n}(x_{2}) / M_{32}}
{x_{1}^{2}[\frac{\nu}{1-2\nu} j_{n}(x_{1}) - j_{n}^{\prime\prime}(x_{1})] / (M_{31} / 2) - 2 \cdot n(n + 1) [j_{n}(x_{2}) - x_{2} j_{n}^{\prime}(x_{2})] / M_{32}}
\end{aligned}
c_{n}
&= - P_{0} (2n + 1) (- i)^{n + 1} \sin\eta_{n} e^{i\eta_{n}} \\
\frac{B}{A}
&= \frac{M_{21} / M_{31} - M_{22} / M_{32}}
{M_{11} / M_{31} - M_{12} / M_{32}} \\
&= \scriptsize
\frac{x_{1} j_{n}^{\prime}(x_{1}) / (M_{31} / 2) - 2 \cdot n(n + 1) j_{n}(x_{2}) / M_{32}}
{x_{1}^{2}[\frac{\nu}{1-2\nu} j_{n}(x_{1}) - j_{n}^{\prime\prime}(x_{1})] / (M_{31} / 2) - 2 \cdot n(n + 1) [j_{n}(x_{2}) - x_{2} j_{n}^{\prime}(x_{2})] / M_{32}}
\end{aligned}
\begin{aligned}
\tan\eta_{n}
&= \frac{\mathrm{Re\,} M_{13}}{\mathrm{Im\,} M_{13}}
\frac{B/A - \mathrm{Re\,} M_{23} / \mathrm{Re\,} M_{13}}
{B/A - \mathrm{Im\,} M_{23} / \mathrm{Im\,} M_{13}} \\
&= - \frac{j_{n} (x_{3})}{n_{n} (x_{3})}
\frac{\dfrac{B}{A} - \frac{\frac{x_{3}}{\rho_{3} \omega^{2}} j_{n}^{\prime} (x_{3})}{\frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} j_{n} (x_{3})}}
{\dfrac{B}{A} - \frac{- \frac{x_{3}}{\rho_{3} \omega^{2}} n_{n}^{\prime} (x_{3})}{- \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} n_{n} (x_{3})}} \\
&= - \frac{j_{n} (x_{3})}{n_{n} (x_{3})}
\frac{\dfrac{\rho_{3}}{\rho_{1}} \dfrac{x_{2}^{2}}{2} \dfrac{B}{A} - x_{3} \dfrac{j_{n}^{\prime} (x_{3})}{j_{n} (x_{3})}}
{\dfrac{\rho_{3}}{\rho_{1}} \dfrac{x_{2}^{2}}{2} \dfrac{B}{A} - x_{3} \dfrac{n_{n}^{\prime} (x_{3})}{n_{n} (x_{3})}}
\end{aligned}
\tan\eta_{n}
&= \frac{\mathrm{Re\,} M_{13}}{\mathrm{Im\,} M_{13}}
\frac{B/A - \mathrm{Re\,} M_{23} / \mathrm{Re\,} M_{13}}
{B/A - \mathrm{Im\,} M_{23} / \mathrm{Im\,} M_{13}} \\
&= - \frac{j_{n} (x_{3})}{n_{n} (x_{3})}
\frac{\dfrac{B}{A} - \frac{\frac{x_{3}}{\rho_{3} \omega^{2}} j_{n}^{\prime} (x_{3})}{\frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} j_{n} (x_{3})}}
{\dfrac{B}{A} - \frac{- \frac{x_{3}}{\rho_{3} \omega^{2}} n_{n}^{\prime} (x_{3})}{- \frac{x_{2}^{2}}{2\rho_{1} \omega^{2}} n_{n} (x_{3})}} \\
&= - \frac{j_{n} (x_{3})}{n_{n} (x_{3})}
\frac{\dfrac{\rho_{3}}{\rho_{1}} \dfrac{x_{2}^{2}}{2} \dfrac{B}{A} - x_{3} \dfrac{j_{n}^{\prime} (x_{3})}{j_{n} (x_{3})}}
{\dfrac{\rho_{3}}{\rho_{1}} \dfrac{x_{2}^{2}}{2} \dfrac{B}{A} - x_{3} \dfrac{n_{n}^{\prime} (x_{3})}{n_{n} (x_{3})}}
\end{aligned}
