Added documentation for exponential horn implementation

lrftubes.lyx

@@ -7922,6 +7922,239 @@ Z_{cpm}\left(f\right)=\frac{1}{i2\pi fC_{dyn}\left(f\right)}
 \end_inset
 
 
+\end_layout
+
+\begin_layout Section
+Exponential horn
+\end_layout
+
+\begin_layout Standard
+hon.py/class ExpHorn(Duct)
+\end_layout
+
+\begin_layout Standard
+An exponential horn can be modelled using ExpHorn.
+ The transfer matrix is derived from Webster's horn equation.
+ The solution for an exponential horn is
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+p\left(x\right)=C_{1}e^{E_{1}x}+C_{2}e^{E_{2}x},\qquad U\left(x\right)=i\frac{S\left(x\right)}{\rho\omega}\left(C_{1}E_{1}e^{E_{1}x}+C_{2}E_{2}e^{E_{2}x}\right)
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which 
+\begin_inset Formula $S(x)$
+\end_inset
+
+ is the local cross-sectional area, 
+\begin_inset Formula $\rho$
+\end_inset
+
+ is the air density, 
+\begin_inset Formula $\omega$
+\end_inset
+
+ is the angular frequency.
+ The exponent terms 
+\begin_inset Formula $E_{1}$
+\end_inset
+
+ and 
+\begin_inset Formula $E_{2}$
+\end_inset
+
+ are given by
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+E_{1}=\frac{-m+\sqrt{-4k^{2}+m^{2}}}{2},\qquad E_{2}=-\frac{m+\sqrt{-4k^{2}+m^{2}}}{2}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which 
+\begin_inset Formula $k$
+\end_inset
+
+ is the wave number and 
+\begin_inset Formula $m$
+\end_inset
+
+ is the flare rate, which is defined as
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+m=\frac{1}{L}\ln\left(\frac{S_{R}}{S_{L}}\right)
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+with 
+\begin_inset Formula $L$
+\end_inset
+
+ being the horn length and 
+\begin_inset Formula $S_{R}$
+\end_inset
+
+ and 
+\begin_inset Formula $S_{L}$
+\end_inset
+
+ being the cross-sectional areas on the right-hand and left-hand side of
+ the horn respectively.
+ 
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+By setting the solutions on 
+\begin_inset Formula $x=0$
+\end_inset
+
+ to 
+\begin_inset Formula $p_{L}$
+\end_inset
+
+ and 
+\begin_inset Formula $U_{L}$
+\end_inset
+
+ and those on 
+\begin_inset Formula $x=L$
+\end_inset
+
+ to 
+\begin_inset Formula $p_{R}$
+\end_inset
+
+ and 
+\begin_inset Formula $U_{R}$
+\end_inset
+
+, the following transfer matrix can be derived:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+T_{ExpHorn}=\frac{1}{E_{2}-E_{1}}\left[\begin{array}{cc}
+E_{2}e^{E_{1}L}-E_{1}e^{E_{2}L} & i\frac{\rho\omega}{S_{L}}\left(e^{E_{1}L}-e^{E_{2}L}\right)\\
+i\frac{S_{R}}{\rho\omega}E_{1}E_{2}\left(e^{E_{1}L}-e^{E_{2}L}\right) & -\frac{S_{R}}{S_{L}}\left(E_{1}e^{E_{1}L}-E_{2}e^{E_{2}L}\right)
+\end{array}\right]
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $p_{L}=p\left(0\right)=C_{1}+C_{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $U_{L}=U\left(0\right)=i\frac{S_{L}}{\rho\omega}\left(C_{1}E_{1}+C_{2}E_{2}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $p_{R}=p\left(L\right)=C_{1}e^{E_{1}L}+C_{2}e^{E_{2}L}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $U_{R}=U\left(L\right)=i\frac{S_{L}}{\rho\omega}\left(C_{1}E_{1}e^{E_{1}L}+C_{2}E_{2}e^{E_{2}L}\right)$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+Solve for constants
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $C_{1}=P_{L}-C_{2}\rightarrow U_{L}=i\frac{S_{L}}{\rho\omega}\left[E_{1}p_{L}+\left(E_{2}-E_{1}\right)C_{2}\right]$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $C_{1}=\frac{1}{E_{2}-E_{1}}\left(i\frac{\rho\omega}{S_{L}}U_{L}+E_{2}P_{L}\right),\qquad C_{2}=-\frac{1}{E_{2}-E_{1}}\left(i\frac{\rho\omega}{S_{L}}U_{L}+E_{1}P_{L}\right)$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+Fill in in solution:
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $p_{R}=\frac{1}{E_{2}E_{1}}\left[\left(E_{2}e^{E_{1}L}-E_{1}e^{E_{2}L}\right)P_{L}+i\frac{\rho\omega}{S_{L}}\left(e^{E_{1}L}-e^{E_{2}L}\right)U_{L}\right]$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $U_{R}=\frac{1}{E_{2}E_{1}}\left[i\frac{S_{R}}{\rho\omega}E_{1}E_{2}\left(e^{E_{1}L}-e^{E_{2}L}\right)P_{L}-\frac{S_{R}}{S_{L}}\left(E_{1}e^{E_{1}L}-E_{2}e^{E_{2}L}\right)U_{L}\right]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Section