calculate_radiation_patternコマンドで用いている計算式

(Formula used in calculate_radiation_pattern command)

1. 共通の記号 (Common notations)

calculate_radiation_patternコマンドでは点ソースを仮定し、ソース位置を\(\posxi=(\xi_1,\xi_2,\xi_3)^T\)とする。
A point source is assumed in calculate_radiation_pattern command, and the source location is expressed by \(\posxi=(\xi_1,\xi_2,\xi_3)^T\).
観測点位置(波形を計算する地点)を\(\posx=(x_1,x_2,x_3)^T\)とする。
The station location (i.e., the location where the waveforms are to be computed) is expressed by \(\posx=(x_1,x_2,x_3)^T\).
ソースから観測点までの距離を \[\begin{equation} r \equiv |\posx-\posxi| \label{eq.r} \end{equation}\] とする。
The distance from the source to the station is expressed by \(r\) (eq. \ref{eq.r}).
ソースから観測点に向かう単位方向ベクトルを \[\begin{equation} \myvector{\gamma} \equiv \frac{\posx-\posxi}{r} \label{eq.gamma} \end{equation}\] とする。
The unit directional vector from the source to the station is expressed by \(\myvector{\gamma}\) (eq. \ref{eq.gamma}).
計算する観測点での変位波形を \(\myvector{u}(t)=(u_1(t),u_2(t),u_3(t))\)とする。 \(t\)は時刻を表す。
The displacement waveform at the station to compute is expressed by \(\myvector{u}(t)=(u_1(t),u_2(t),u_3(t))\), where \(t\) is time.
変位の成分を\(n\)、地震波動ソースとして作用する力の成分を\(p\)、力の空間微分の成分を\(q\)で表す。
Indices \(n\), \(p\), and \(q\) represent a component of the displacement, a component of the force exerted as the source of the seismic wave, and a direction for the spatial derivative of the force, respectively.
calculate_radiation_patternコマンドでは均質媒質を仮定し、密度を\(\rho\)、P波速度を\(\alpha\)、S波速度を\(\beta\)とする。
A homogeneous medium is assumed in calculate_radiation_pattern command, and \(\rho\), \(\alpha\), and \(\beta\) represent the density, P-wave velocity, and S-wave velocity of the medium, respectively.
\(\pi\)は円周率を表す。
\(\pi\) is the circular constant.
\(\delta_{np}\)はクロネッカーのデルタを表す。
\(\delta_{np}\) is Cronecker delta.

2. シングルフォースが作る変位波形 (Amplitudes of displacement waveforms generated by a single force)

2-1. 一般式 (General formula)

Aki and Richards (2002)の(4.27)式を使用する。シングルフォースの時間関数を \(\myvector{F}(t)=(F_1(t),F_2(t),F_3(t))^T\) とおくと、変位波形は以下の式で与えられる。
Eq. (4.27) of Aki and Richards (2002) is used. Let \(\myvector{F}(t)=(F_1(t),F_2(t),F_3(t))^T\) be the time function of the single force. Then the displacement waveform is given by the formula below.

\[\begin{equation} u_n(t)=u_n^{N}(t)+u_n^{FP}(t)+u_n^{FS}(t) \hspace{1em} (n=1,2,3) \label{eq.F} \end{equation}\]

近地項 (Near-field term) : \[\begin{equation} u_n^{N}(t)= \sum_{p=1}^3 \frac{1}{4\pi\rho}(3\gamma_n\gamma_p-\delta_{np}) \frac{1}{r^3}\int_{r/\alpha}^{r/\beta}\tau F_p(t-\tau)d\tau \label{eq.F.N} \end{equation}\]
遠地P波項 (Far-field P-wave term) : \[\begin{equation} u_n^{FP}(t)= \sum_{p=1}^3\frac{1}{4\pi\rho\alpha^2}\gamma_n\gamma_p \frac{1}{r}F_p\left(t-\frac{r}{\alpha}\right) \label{eq.F.FP} \end{equation}\]
遠地S波項 (Far-field S-wave term) : \[\begin{equation} u_n^{FS}(t)= -\sum_{p=1}^3\frac{1}{4\pi\rho\beta^2}(\gamma_n\gamma_p-\delta_{np}) \frac{1}{r}F_p\left(t-\frac{r}{\beta}\right) \label{eq.F.FS} \end{equation}\]

2-2. 単振動ソースに対する変位波形 (Displacement waveforms for a monochromatic oscillation source)

震源時間関数が単振動 \[\begin{equation} F_p(t)=F_p^0e^{i\omega t} \label{eq.Fp.monochromatic} \end{equation}\] で与えられる場合、(\ref{eq.F.N})-(\ref{eq.F.FS})式は以下のように変形できる。
When the source time function is given by eq. (\ref{eq.Fp.monochromatic}), eqs. (\ref{eq.F.N})-(\ref{eq.F.FS}) can be arranged as below.

近地項 (Near-field term) : \[\begin{eqnarray} u_n^{N}(t) &=& \sum_{p=1}^3 \frac{1}{4\pi\rho}(3\gamma_n\gamma_p-\delta_{np}) \frac{1}{r^3}\int_{r/\alpha}^{r/\beta}\tau F_p^0 e^{i\omega(t-\tau)}d\tau \nonumber \\ &=& \frac{1}{4\pi\rho r^3} \sum_{p=1}^3 (3\gamma_n\gamma_p-\delta_{np}) F_p^0 \int_{r/\alpha}^{r/\beta}\tau e^{i\omega(t-\tau)}d\tau \nonumber \\ &=& \frac{1}{4\pi\rho r^3} \sum_{p=1}^3 (3\gamma_n\gamma_p-\delta_{np}) F_p^0 \nonumber \\ & & \left\{ \left[ \frac{\tau}{-i\omega}e^{i\omega(t-\tau)} \right]_{r/\alpha}^{r/\beta} -\int_{r/\alpha}^{r/\beta}\frac{1}{-i\omega}e^{i\omega(t-\tau)}d\tau \right\} \nonumber \\ &=& \frac{1}{4\pi\rho r^3} \sum_{p=1}^3 (3\gamma_n\gamma_p-\delta_{np}) F_p^0 \nonumber \\ & & \left\{ \left[ -\frac{\tau}{i\omega}e^{i\omega(t-\tau)} \right]_{r/\alpha}^{r/\beta} +\left[ \frac{1}{\omega^2}e^{i\omega(t-\tau)} \right]_{r/\alpha}^{r/\beta} \right\} \nonumber \\ &=& \frac{1}{4\pi\rho r^3} \sum_{p=1}^3 (3\gamma_n\gamma_p-\delta_{np}) F_p^0 \nonumber \\ & & \left[ \left(-\frac{\tau}{i\omega}+\frac{1}{\omega^2}\right) e^{i\omega(t-\tau)} \right]_{r/\alpha}^{r/\beta} \nonumber \\ &=& \frac{1}{4\pi\rho r^3} \sum_{p=1}^3 (3\gamma_n\gamma_p-\delta_{np}) F_p^0 \nonumber \\ & & \left[ \left(-\frac{r}{i\omega\beta}+\frac{1}{\omega^2}\right) e^{i\omega(t-r/\beta)} -\left(-\frac{r}{i\omega\alpha}+\frac{1}{\omega^2}\right) e^{i\omega(t-r/\alpha)} \right] \label{eq.F.N.monochromatic} \end{eqnarray}\]
遠地P波項 (Far-field P-wave term) : \[\begin{eqnarray} u_n^{FP}(t) &=& \sum_{p=1}^3\frac{1}{4\pi\rho\alpha^2}\gamma_n\gamma_p \frac{1}{r}F_p^0 e^{i\omega(t-r/\alpha)} \nonumber \\ &=& \frac{1}{4\pi\rho\alpha^2 r}\sum_{p=1}^3\gamma_n\gamma_p F_p^0 e^{i\omega(t-r/\alpha)} \label{eq.F.FP.monochromatic} \end{eqnarray}\]
遠地S波項 (Far-field S-wave term) : \[\begin{eqnarray} u_n^{FS}(t) &=& -\sum_{p=1}^3\frac{1}{4\pi\rho\beta^2}(\gamma_n\gamma_p-\delta_{np}) \frac{1}{r}F_p^0 e^{i\omega(t-r/\beta)} \nonumber \\ &=& -\frac{1}{4\pi\rho\beta^2 r}\sum_{p=1}^3(\gamma_n\gamma_p-\delta_{np}) F_p^0 e^{i\omega(t-r/\beta)} \label{eq.F.FS.monochromatic} \end{eqnarray}\]

3. モーメントテンソルが作る変位波形 (Displacement waveforms generated by a moment tensor)

3-1. 一般式 (General formula)

Aki and Richards (2002)の(4.29)式を使用する。モーメントテンソルの時間関数を \[\begin{equation} \myvector{M}(t)= \begin{pmatrix} M_{11}(t) & M_{12}(t) & M_{13}(t) \\ M_{21}(t) & M_{22}(t) & M_{23}(t) \\ M_{31}(t) & M_{32}(t) & M_{33}(t) \end{pmatrix} \label{eq.moment} \end{equation}\] とおくと、変位波形は以下の式で与えられる。
Eq. (4.29) of Aki and Richards (2002) is used. Let \(\myvector{M}(t)\) (eq. \ref{eq.moment}) be the time function of the moment tensor. Then the displacement waveform is given by the formula below.

\[\begin{equation} u_n(t)=u_n^{N}(t)+u_n^{IP}(t)+u_n^{IS}(t)+u_n^{FP}(t)+u_n^{FS}(t) \hspace{1em} (n=1,2,3) \label{eq.M} \end{equation}\]

近地項 (Near-field term) : \[\begin{eqnarray} u_n^{N}(t) &=& \sum_{p=1}^3\sum_{q=1}^3 \frac{15\gamma_n\gamma_p\gamma_q-3\gamma_n\delta_{pq} -3\gamma_p\delta_{nq}-3\gamma_q\delta_{np}}{4\pi\rho} \nonumber \\ & & \frac{1}{r^4}\int_{r/\alpha}^{r/\beta}\tau M_{pq}(t-\tau)d\tau \label{eq.M.N} \end{eqnarray}\]
中間P波項 (Intermediate-field P-wave term) : \[\begin{equation} u_n^{IP}(t)= \sum_{p=1}^3\sum_{q=1}^3 \frac{6\gamma_n\gamma_p\gamma_q-\gamma_n\delta_{pq} -\gamma_p\delta_{nq}-\gamma_q\delta_{np}}{4\pi\rho\alpha^2} \frac{1}{r^2}M_{pq}\left(t-\frac{r}{\alpha}\right) \label{eq.M.IP} \end{equation}\]
中間S波項 (Intermediate-field S-wave term) : \[\begin{equation} u_n^{IS}(t)= -\sum_{p=1}^3\sum_{q=1}^3 \frac{6\gamma_n\gamma_p\gamma_q-\gamma_n\delta_{pq} -\gamma_p\delta_{nq}-2\gamma_q\delta_{np}}{4\pi\rho\beta^2} \frac{1}{r^2}M_{pq}\left(t-\frac{r}{\beta}\right) \label{eq.M.IS} \end{equation}\]
遠地P波項 (Far-field P-wave term) : \[\begin{equation} u_n^{FP}(t)= \sum_{p=1}^3\sum_{q=1}^3 \frac{\gamma_n\gamma_p\gamma_q}{4\pi\rho\alpha^3} \frac{1}{r}\dot{M}_{pq}\left(t-\frac{r}{\alpha}\right) \label{eq.M.FP} \end{equation}\]
遠地S波項 (Far-field S-wave term) : \[\begin{equation} u_n^{FS}(t)= -\sum_{p=1}^3\sum_{q=1}^3 \frac{\gamma_n\gamma_p-\delta_{np}}{4\pi\rho\beta^3}\gamma_q \frac{1}{r}\dot{M}_{pq}\left(t-\frac{r}{\beta}\right) \label{eq.M.FS} \end{equation}\]

3-2. 単振動ソースに対する変位波形 (Displacement waveforms for a monochromatic oscillation source)

震源時間関数が単振動 \[\begin{equation} M_{pq}(t)=M_{pq}^0e^{i\omega t} \label{eq.Mpq.monochromatic} \end{equation}\] で与えられる場合、(\ref{eq.M.N})-(\ref{eq.M.FS})式は以下のように変形できる。
When the source time function is given by eq. (\ref{eq.Mpq.monochromatic}), eqs. (\ref{eq.M.N})-(\ref{eq.M.FS}) can be arranged as below.

近地項 (Near-field term) : \[\begin{eqnarray} u_n^{N}(t) &=& \sum_{p=1}^3\sum_{q=1}^3 \frac{15\gamma_n\gamma_p\gamma_q-3\gamma_n\delta_{pq} -3\gamma_p\delta_{nq}-3\gamma_q\delta_{np}}{4\pi\rho} \nonumber \\ & & \frac{1}{r^4}\int_{r/\alpha}^{r/\beta}\tau M_{pq}^0 e^{i\omega(t-\tau)}d\tau \nonumber \\ &=& \frac{1}{4\pi\rho r^4} \sum_{p=1}^3\sum_{q=1}^3 (15\gamma_n\gamma_p\gamma_q-3\gamma_n\delta_{pq} -3\gamma_p\delta_{nq}-3\gamma_q\delta_{np})M_{pq}^0 \nonumber \\ & & \int_{r/\alpha}^{r/\beta}\tau e^{i\omega(t-\tau)}d\tau \nonumber \\ &=& \frac{1}{4\pi\rho r^4} \sum_{p=1}^3\sum_{q=1}^3 (15\gamma_n\gamma_p\gamma_q-3\gamma_n\delta_{pq} -3\gamma_p\delta_{nq}-3\gamma_q\delta_{np})M_{pq}^0 \nonumber \\ & & \left\{ \left[ \frac{\tau}{-i\omega}e^{i\omega(t-\tau)}d\tau \right]_{r/\alpha}^{r/\beta} -\int_{r/\alpha}^{r/\beta} \frac{1}{-i\omega}e^{i\omega(t-\tau)}d\tau \right\} \nonumber \\ &=& \frac{1}{4\pi\rho r^4} \sum_{p=1}^3\sum_{q=1}^3 (15\gamma_n\gamma_p\gamma_q-3\gamma_n\delta_{pq} -3\gamma_p\delta_{nq}-3\gamma_q\delta_{np})M_{pq}^0 \nonumber \\ & & \left\{ \left[ -\frac{\tau}{i\omega}e^{i\omega(t-\tau)} \right]_{r/\alpha}^{r/\beta} +\left[ \frac{1}{\omega^2}e^{i\omega(t-\tau)} \right]_{r/\alpha}^{r/\beta} \right\} \nonumber \\ &=& \frac{1}{4\pi\rho r^4} \sum_{p=1}^3\sum_{q=1}^3 (15\gamma_n\gamma_p\gamma_q-3\gamma_n\delta_{pq} -3\gamma_p\delta_{nq}-3\gamma_q\delta_{np})M_{pq}^0 \nonumber \\ & & \left[ \left(-\frac{\tau}{i\omega}+\frac{1}{\omega^2}\right) e^{i\omega(t-\tau)} \right]_{r/\alpha}^{r/\beta} \nonumber \\ &=& \frac{1}{4\pi\rho r^4} \sum_{p=1}^3\sum_{q=1}^3 (15\gamma_n\gamma_p\gamma_q-3\gamma_n\delta_{pq} -3\gamma_p\delta_{nq}-3\gamma_q\delta_{np})M_{pq}^0 \nonumber \\ & & \left[ \left(-\frac{r}{i\omega\beta}+\frac{1}{\omega^2}\right) e^{i\omega(t-r/\beta)} -\left(-\frac{r}{i\omega\alpha}+\frac{1}{\omega^2}\right) e^{i\omega(t-r/\alpha)} \right] \label{eq.M.N.monochromatic} \end{eqnarray}\]
中間P波項 (Intermediate-field P-wave term) : \[\begin{eqnarray} u_n^{IP}(t) &=& \sum_{p=1}^3\sum_{q=1}^3 \frac{6\gamma_n\gamma_p\gamma_q-\gamma_n\delta_{pq} -\gamma_p\delta_{nq}-\gamma_q\delta_{np}}{4\pi\rho\alpha^2} \frac{1}{r^2}M_{pq}^0 e^{i\omega(t-r/\alpha)} \nonumber \\ &=& \frac{1}{4\pi\rho\alpha^2 r^2} \sum_{p=1}^3\sum_{q=1}^3 (6\gamma_n\gamma_p\gamma_q-\gamma_n\delta_{pq} -\gamma_p\delta_{nq}-\gamma_q\delta_{np})M_{pq}^0 \nonumber \\ & & e^{i\omega(t-r/\alpha)} \label{eq.M.IP.monochromatic} \end{eqnarray}\]
中間S波項 (Intermediate-field S-wave term) : \[\begin{eqnarray} u_n^{IS}(t) &=& -\sum_{p=1}^3\sum_{q=1}^3 \frac{6\gamma_n\gamma_p\gamma_q-\gamma_n\delta_{pq} -\gamma_p\delta_{nq}-2\gamma_q\delta_{np}}{4\pi\rho\beta^2} \frac{1}{r^2}M_{pq}^0 e^{i\omega(t-r/\beta)} \nonumber \\ &=& -\frac{1}{4\pi\rho\beta^2 r^2} \sum_{p=1}^3\sum_{q=1}^3 (6\gamma_n\gamma_p\gamma_q-\gamma_n\delta_{pq} -\gamma_p\delta_{nq}-2\gamma_q\delta_{np})M_{pq}^0 \nonumber \\ & & e^{i\omega(t-r/\beta)} \label{eq.M.IS.monochromatic} \end{eqnarray}\]
遠地P波項 (Far-field P-wave term) : \[\begin{eqnarray} u_n^{FP}(t) &=& \sum_{p=1}^3\sum_{q=1}^3 \frac{\gamma_n\gamma_p\gamma_q}{4\pi\rho\alpha^3} \frac{1}{r}\frac{d}{dt}\left[M_{pq}^0 e^{i\omega(t-r/\alpha)}\right] \nonumber \\ &=& \frac{1}{4\pi\rho\alpha^3 r} \sum_{p=1}^3\sum_{q=1}^3 \gamma_n\gamma_p\gamma_q M_{pq}^0 i\omega e^{i\omega(t-r/\alpha)} \nonumber \\ &=& \frac{i\omega}{4\pi\rho\alpha^3 r} \sum_{p=1}^3\sum_{q=1}^3 \gamma_n\gamma_p\gamma_q M_{pq}^0 e^{i\omega(t-r/\alpha)} \label{eq.M.FP.monochromatic} \end{eqnarray}\]
遠地S波項 (Far-field S-wave term) : \[\begin{eqnarray} u_n^{FS}(t) &=& -\sum_{p=1}^3\sum_{q=1}^3 \frac{\gamma_n\gamma_p-\delta_{np}}{4\pi\rho\beta^3}\gamma_q \frac{1}{r}\frac{d}{dt}\left[M_{pq}^0 e^{i\omega(t-r/\beta)}\right] \nonumber \\ &=& -\frac{1}{4\pi\rho\beta^3 r} \sum_{p=1}^3\sum_{q=1}^3 (\gamma_n\gamma_p-\delta_{np})\gamma_q M_{pq}^0 i\omega e^{i\omega(t-r/\beta)} \nonumber \\ &=& -\frac{i\omega}{4\pi\rho\beta^3 r} \sum_{p=1}^3\sum_{q=1}^3 (\gamma_n\gamma_p-\delta_{np})\gamma_q M_{pq}^0 e^{i\omega(t-r/\beta)} \label{eq.M.FS.monochromatic} \end{eqnarray}\]

4. 統一された表記 (Unified expressions)

シングルフォースに対する式 (\ref{eq.F.N.monochromatic})-(\ref{eq.F.FS.monochromatic}) とモーメントテンソルに対する式 (\ref{eq.M.N.monochromatic})-(\ref{eq.M.FS.monochromatic}) は係数の違いを除いて同じ形をしており、以下の形に統一できる。
The formulas for single force (\ref{eq.F.N.monochromatic})-(\ref{eq.F.FS.monochromatic}) and moment tensor (\ref{eq.M.N.monochromatic})-(\ref{eq.M.FS.monochromatic}) have the same shapes except for the differences in coefficients, and can be unified into the expressions below.

近地項 (Near-field term) : \[\begin{equation} u_n^{N}(t)= A_n^N \left[ \left(-\frac{r}{i\omega\beta}+\frac{1}{\omega^2}\right) e^{i\omega(t-r/\beta)} -\left(-\frac{r}{i\omega\alpha}+\frac{1}{\omega^2}\right) e^{i\omega(t-r/\alpha)} \right] \label{eq.N.monochromatic.unified} \end{equation}\]
P波項 (P-wave terms) : \[\begin{equation} u_n^{IP}(t)+u_n^{FP}(t)= \left(A_n^{PR}+A_n^{PI}i\right)e^{i\omega(t-r/\alpha)} \label{eq.P.monochromatic.unified} \end{equation}\]
S波項 (S-wave terms) : \[\begin{equation} u_n^{IS}(t)+u_n^{FS}(t)= -\left(A_n^{SR}+A_n^{SI}i\right)e^{i\omega(t-r/\beta)} \label{eq.S.monochromatic.unified} \end{equation}\]

ここで係数は以下のように与えられる。
Here the coefficients are given as below.

係数 Coefficient	シングルフォース Single force	モーメントテンソル Moment tensor
\(A_n^N\)	\(\frac{1}{4\pi\rho r^3} \sum_{p=1}^3 (3\gamma_n\gamma_p-\delta_{np}) F_p^0\)	\(\frac{1}{4\pi\rho r^4} \sum_{p=1}^3\sum_{q=1}^3\) \((15\gamma_n\gamma_p\gamma_q-3\gamma_n\delta_{pq} -3\gamma_p\delta_{nq}-3\gamma_q\delta_{np}) M_{pq}^0\)
\(A_n^{PR}\)	\(\frac{1}{4\pi\rho\alpha^2 r}\sum_{p=1}^3 \gamma_n\gamma_p F_p^0\)	\(\frac{1}{4\pi\rho\alpha^2 r^2} \sum_{p=1}^3\sum_{q=1}^3\) \((6\gamma_n\gamma_p\gamma_q-\gamma_n\delta_{pq} -\gamma_p\delta_{nq}-\gamma_q\delta_{np}) M_{pq}^0\)
\(A_n^{PI}\)	\(0\)	\(\frac{\omega}{4\pi\rho\alpha^3 r} \sum_{p=1}^3\sum_{q=1}^3\) \(\gamma_n\gamma_p\gamma_q M_{pq}^0\)
\(A_n^{SR}\)	\(\frac{1}{4\pi\rho\beta^2 r}\sum_{p=1}^3 (\gamma_n\gamma_p-\delta_{np})F_p^0\)	\(\frac{1}{4\pi\rho\beta^2 r^2} \sum_{p=1}^3\sum_{q=1}^3\) \((6\gamma_n\gamma_p\gamma_q-\gamma_n\delta_{pq} -\gamma_p\delta_{nq}-2\gamma_q\delta_{np}) M_{pq}^0\)
\(A_n^{SI}\)	\(0\)	\(\frac{\omega}{4\pi\rho\beta^3 r} \sum_{p=1}^3\sum_{q=1}^3\) \((\gamma_n\gamma_p-\delta_{np})\gamma_q M_{pq}^0\)

(\ref{eq.N.monochromatic.unified})-(\ref{eq.S.monochromatic.unified}) 式を用いると、全ての項を足した変位波形は以下の形に書ける。
Using eqs. (\ref{eq.N.monochromatic.unified})-(\ref{eq.S.monochromatic.unified}), the displacement waveform including contributions from all terms is written as below.

\[\begin{eqnarray} u_n(t) &=& A_n^N \left[ \left(-\frac{r}{i\omega\beta}+\frac{1}{\omega^2}\right) e^{i\omega(t-r/\beta)} -\left(-\frac{r}{i\omega\alpha}+\frac{1}{\omega^2}\right) e^{i\omega(t-r/\alpha)} \right] \nonumber \\ & & +\left(A_n^{PR}+A_n^{PI}i\right)e^{i\omega(t-r/\alpha)} \nonumber \\ & & -\left(A_n^{SR}+A_n^{SI}i\right)e^{i\omega(t-r/\beta)} \nonumber \\ &=& \left[ A_n^{PR}+A_n^{PI}i -A_n^N\left(-\frac{r}{i\omega\alpha}+\frac{1}{\omega^2}\right) \right]e^{i\omega(t-r/\alpha)} \nonumber \\ & & -\left[ A_n^{SR}+A_n^{SI}i -A_n^N\left(-\frac{r}{i\omega\beta}+\frac{1}{\omega^2}\right) \right]e^{i\omega(t-r/\beta)} \nonumber \\ &=& \left[ A_n^{PR}+A_n^{PI}i -A_n^N\left(\frac{r}{\omega\alpha}i+\frac{1}{\omega^2}\right) \right]e^{i\omega(t-r/\alpha)} \nonumber \\ & & -\left[ A_n^{SR}+A_n^{SI}i -A_n^N\left(\frac{r}{\omega\beta}i+\frac{1}{\omega^2}\right) \right]e^{i\omega(t-r/\beta)} \nonumber \\ &=& \left[ \left(A_n^{PR}-\frac{A_n^N}{\omega^2}\right) +\left(A_n^{PI}-\frac{rA_n^N}{\omega\alpha}\right)i \right]e^{i\omega(t-r/\alpha)} \nonumber \\ & & -\left[ \left(A_n^{SR}-\frac{A_n^N}{\omega^2}\right) +\left(A_n^{SI}-\frac{rA_n^N}{\omega\beta}\right)i \right]e^{i\omega(t-r/\beta)} \label{eq.un} \end{eqnarray}\]

5. 変位振幅の式 (Formula for displacement amplitude)

変位振幅は \[\begin{equation} |u_n(t)| =\sqrt{u_n(t)^{∗}u_n(t)} \label{eq.abs_un} \end{equation}\] で計算できる。ここで\(^{∗}\)は複素共役を表す。
The displacement amplitude is computed by eq. (\ref{eq.abs_un}), where \(^{∗}\) represents a complex conjugate.

簡単のため \[\begin{equation} B^{PR}\equiv A_n^{PR}-\frac{A_n^N}{\omega^2} \label{eq.B_PR} \end{equation}\] \[\begin{equation} B^{PI}\equiv A_n^{PI}-\frac{rA_n^N}{\omega\alpha} \label{eq.B_PI} \end{equation}\] \[\begin{equation} B^{SR}\equiv A_n^{SR}-\frac{A_n^N}{\omega^2} \label{eq.B_SR} \end{equation}\] \[\begin{equation} B^{SI}\equiv A_n^{SI}-\frac{rA_n^N}{\omega\beta} \label{eq.B_SI} \end{equation}\] とおくと(\ref{eq.un})式は \[\begin{equation} u_n(t)= \left(B^{PR}+B^{PI}i\right)e^{i\omega(t-r/\alpha)} -\left(B^{SR}+B^{SI}i\right)e^{i\omega(t-r/\beta)} \label{eq.un.use_B} \end{equation}\] その複素共役は \[\begin{equation} u_n(t)^{∗}= \left(B^{PR}-B^{PI}i\right)e^{-i\omega(t-r/\alpha)} -\left(B^{SR}-B^{SI}i\right)e^{-i\omega(t-r/\beta)} \label{eq.un_conjugate.use_B} \end{equation}\] であるので \[\begin{eqnarray} u_n(t)^{∗}u_n(t) &=& \left[ \left(B^{PR}-B^{PI}i\right)e^{-i\omega(t-r/\alpha)} -\left(B^{SR}-B^{SI}i\right)e^{-i\omega(t-r/\beta)} \right] \nonumber \\ & & \left[ \left(B^{PR}+B^{PI}i\right)e^{i\omega(t-r/\alpha)} -\left(B^{SR}+B^{SI}i\right)e^{i\omega(t-r/\beta)} \right] \nonumber \\ &=& \left(B^{PR}-B^{PI}i\right)\left(B^{PR}+B^{PI}i\right) \nonumber \\ & & -\left(B^{PR}-B^{PI}i\right)\left(B^{SR}+B^{SI}i\right) e^{-i\omega(t-r/\alpha)}e^{i\omega(t-r/\beta)} \nonumber \\ & & -\left(B^{SR}-B^{SI}i\right)\left(B^{PR}+B^{PI}i\right) e^{-i\omega(t-r/\beta)}e^{i\omega(t-r/\alpha)} \nonumber \\ & & +\left(B^{SR}-B^{SI}i\right)\left(B^{SR}+B^{SI}i\right) \nonumber \\ &=& \left[\left(B^{PR}\right)^2+\left(B^{PI}\right)^2\right] \nonumber \\ & & -\left[B^{PR}B^{SR}+B^{PR}B^{SI}i-B^{PI}B^{SR}i+B^{PI}B^{SI}\right] e^{i\omega(r/\alpha-r/\beta)} \nonumber \\ & & -\left[B^{SR}B^{PR}+B^{SR}B^{PI}i-B^{SI}B^{PR}i+B^{SI}B^{PI}\right] e^{i\omega(r/\beta-r/\alpha)} \nonumber \\ & & \left[\left(B^{SR}\right)^2+\left(B^{SI}\right)^2\right] \nonumber \\ &=& \left[\left(B^{PR}\right)^2+\left(B^{PI}\right)^2 +\left(B^{SR}\right)^2+\left(B^{SI}\right)^2\right] \nonumber \\ & & -\left[B^{PR}B^{SR}+B^{PR}B^{SI}i-B^{PI}B^{SR}i+B^{PI}B^{SI}\right] e^{-i\omega(r/\beta-r/\alpha)} \nonumber \\ & & -\left[B^{PR}B^{SR}-B^{PR}B^{SI}i+B^{PI}B^{SR}i+B^{PI}B^{SI}\right] e^{i\omega(r/\beta-r/\alpha)} \nonumber \\ &=& \left[\left(B^{PR}\right)^2+\left(B^{PI}\right)^2 +\left(B^{SR}\right)^2+\left(B^{SI}\right)^2\right] \nonumber \\ & & -\left[B^{PR}B^{SR}+B^{PI}B^{SI}\right] \left[e^{i\omega(r/\beta-r/\alpha)}+e^{-i\omega(r/\beta-r/\alpha)}\right] \nonumber \\ & & -\left[-B^{PR}B^{SI}i+B^{PI}B^{SR}i\right] \left[e^{i\omega(r/\beta-r/\alpha)}-e^{-i\omega(r/\beta-r/\alpha)}\right] \nonumber \\ &=& \left[\left(B^{PR}\right)^2+\left(B^{PI}\right)^2 +\left(B^{SR}\right)^2+\left(B^{SI}\right)^2\right] \nonumber \\ & & -\left[B^{PR}B^{SR}+B^{PI}B^{SI}\right]\cdot 2\cos\left(\frac{\omega r}{\beta}-\frac{\omega r}{\alpha}\right) \nonumber \\ & & +\left[B^{PR}B^{SI}-B^{PI}B^{SR}\right]i\cdot 2\sin\left(\frac{\omega r}{\beta}-\frac{\omega r}{\alpha}\right)i \nonumber \\ &=& \left[\left(B^{PR}\right)^2+\left(B^{PI}\right)^2 +\left(B^{SR}\right)^2+\left(B^{SI}\right)^2\right] \nonumber \\ & & -2\left[B^{PR}B^{SR}+B^{PI}B^{SI}\right] \cos\left(\frac{\omega r}{\beta}-\frac{\omega r}{\alpha}\right) \nonumber \\ & & -2\left[B^{PR}B^{SI}-B^{PI}B^{SR}\right] \sin\left(\frac{\omega r}{\beta}-\frac{\omega r}{\alpha}\right) \label{eq.abs_u2.use_B} \end{eqnarray}\] となる。 calculate_radiaiton_patternコマンドでは (\ref{eq.abs_un})(\ref{eq.abs_u2.use_B})式を用いて計算を行う。
For simplicity, \(B^{PR}\), \(B^{PI}\), \(B^{SR}\), and \(B^{SI}\) are defined by eqs. (\ref{eq.B_PR})-(\ref{eq.B_SI}). Then eq. (\ref{eq.un}) can be rewritten as (\ref{eq.un.use_B}), and the complex conjugate of it can be written as (\ref{eq.un_conjugate.use_B}). Therefore eq. (\ref{eq.abs_u2.use_B}) is derived. The calculate_radiation_pattern command uses eqs. (\ref{eq.abs_un}) and (\ref{eq.abs_u2.use_B}) for the computation.