From 2c1e9e6fd5f56843a07c181a553672458958cd75 Mon Sep 17 00:00:00 2001 From: "J.A. de Jong - Redu-Sone B.V., ASCEE V.O.F" Date: Tue, 10 Jan 2023 16:36:29 +0100 Subject: [PATCH] Updated lrftubes docs --- lrftubes.lyx | 2080 ++++++++++++++++++++++++++++---------------------- 1 file changed, 1186 insertions(+), 894 deletions(-) diff --git a/lrftubes.lyx b/lrftubes.lyx index 2f82a57..a30672f 100644 --- a/lrftubes.lyx +++ b/lrftubes.lyx @@ -10,18 +10,18 @@ \usepackage{ar} \end_preamble \options a4paper -\use_default_options true +\use_default_options false \maintain_unincluded_children false \language american \language_package babel \inputencoding utf8 \fontencoding global -\font_roman "default" "FreeSerif" +\font_roman "default" "Linux Libertine O" \font_sans "default" "Courier New" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default -\use_non_tex_fonts false +\use_non_tex_fonts true \font_sc false \font_osf false \font_sf_scale 100 100 @@ -29,7 +29,7 @@ \use_microtype false \use_dash_ligatures false \graphics default -\default_output_format pdf2 +\default_output_format default \output_sync 1 \output_sync_macro "\synctex=1" \bibtex_command biber @@ -107,6 +107,22 @@ status open \begin_layout Plain Layout +\backslash +pagestyle{fancy} +\end_layout + +\begin_layout Plain Layout + + +\backslash +setlength{ +\backslash +headheight}{2cm} +\end_layout + +\begin_layout Plain Layout + + \backslash thispagestyle{empty} \end_layout @@ -148,13 +164,6 @@ Ir. \end_inset -\end_layout - -\begin_layout Standard -\begin_inset VSpace bigskip -\end_inset - - \end_layout \begin_layout Standard @@ -181,11 +190,11 @@ ASCEE \begin_inset Newline newline \end_inset -Maximastraat 1 +Nikola Teslastraat 1-11 \begin_inset Newline newline \end_inset -7442 NW Nijverdal +7442 PC Nijverdal \begin_inset Newline newline \end_inset @@ -277,7 +286,7 @@ Document status: \begin_inset Text \begin_layout Plain Layout -Draft +Draft / Under constant improvement \end_layout \end_inset @@ -297,7 +306,7 @@ Document revision: \begin_inset Text \begin_layout Plain Layout -1 +2 \end_layout \end_inset @@ -317,6 +326,11 @@ Revision history: \begin_inset Text \begin_layout Plain Layout +2023-01-10: rev. + 2 +\begin_inset Newline newline +\end_inset + 2018-02-21: rev. 1 \end_layout @@ -371,22 +385,6 @@ year \end_inset -\end_layout - -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -pagestyle{fancy} -\end_layout - -\end_inset - - \end_layout \begin_layout Standard @@ -418,7 +416,7 @@ cleardoublepage \begin_layout Plain Layout - +% \backslash markboth{ \backslash @@ -467,6 +465,10 @@ thispagestyle{empty} \begin_inset Note Note status open +\begin_layout Plain Layout +\begin_inset Note Note +status collapsed + \begin_layout Plain Layout Unused: \end_layout @@ -834,7 +836,7 @@ literal "true" \end_layout -\begin_layout Standard +\begin_layout Plain Layout \begin_inset ERT status open @@ -852,9 +854,9 @@ status open \end_layout -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout Unused: @@ -958,7 +960,7 @@ literal "true" \end_layout -\begin_layout Standard +\begin_layout Plain Layout \begin_inset ERT status open @@ -976,9 +978,9 @@ status open \end_layout -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout Unused: @@ -1092,7 +1094,7 @@ literal "true" \end_layout -\begin_layout Standard +\begin_layout Plain Layout \begin_inset ERT status open @@ -1230,7 +1232,7 @@ literal "true" \end_layout -\begin_layout Standard +\begin_layout Plain Layout \begin_inset ERT status open @@ -1248,7 +1250,7 @@ status open \end_layout -\begin_layout Standard +\begin_layout Plain Layout \begin_inset CommandInset nomenclature LatexCommand nomenclature prefix "O" @@ -1279,6 +1281,11 @@ literal "true" \end_inset +\end_layout + +\end_inset + + \end_layout \begin_layout Standard @@ -1468,6 +1475,8 @@ status open \backslash lrftubes +\backslash + \end_layout \end_inset @@ -1789,6 +1798,82 @@ status collapsed \end_layout +\begin_layout Subsection +Mixing of mixtures +\end_layout + +\begin_layout Standard +Suppose we mix two mixtures of substances, mixture 1, and mixture 2. + We want to know the final concentrations / mass fraction in the mixed mixture. + Mix 1 comprises mass fractions +\begin_inset Formula $\omega_{1,i}$ +\end_inset + +, and mix 2 comprises mass fractions +\begin_inset Formula $\omega_{2,j}$ +\end_inset + +. + We assume that +\begin_inset Formula $i$ +\end_inset + + and +\begin_inset Formula $j$ +\end_inset + + can interfere. + For example, mixing air with Dutch natural gas, both contain nitrogen. + The first step is to determine the mass flow of the two, called +\begin_inset Formula $m_{1}$ +\end_inset + + and +\begin_inset Formula $m_{2}$ +\end_inset + +. + Then, assuming mass conservation under chemically inert conditions: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +m_{1}\omega_{1,i}+m_{2}\omega_{2,i}=m\omega_{i}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection* +Mixing air with natural gas +\end_layout + +\begin_layout Standard +The air factor +\begin_inset Formula $\lambda$ +\end_inset + + (not to be confused with wavelength in an acoustic context), is defined + as the ratio of air to the stoichiometric ratio. + The stoichiometric ratio can be determined by calculating the required + moles of oxygen such that all carbon atoms can become CO +\begin_inset Formula $_{2}$ +\end_inset + + , and +\emph on +half of +\emph default + all hydrogen atoms can become H +\begin_inset Formula $_{2}$ +\end_inset + +. +\end_layout + \begin_layout Subsection Ideal gas mixtures \end_layout @@ -2596,7 +2681,7 @@ y_{g,N_{2}} \begin_layout Standard Solving this results in: \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout \begin_inset Formula $x_{f,O}+2\times0.21y_{\mathrm{ox}}-y_{g,\mathrm{water}}-2y_{g,CO_{2}}=0$ @@ -3324,7 +3409,10 @@ second} . Eqs. - ( +\begin_inset space ~ +\end_inset + +( \begin_inset CommandInset ref LatexCommand ref reference "eq:contU" @@ -3369,7 +3457,10 @@ where \end_inset , we can solve for the acoustic pressure, upon using Eq. - +\begin_inset space ~ +\end_inset + + \begin_inset CommandInset ref LatexCommand ref reference "eq:momU" @@ -8672,7 +8763,11 @@ Radiation impedance of a baffled piston \begin_inset Formula $\pi a^{2}$ \end_inset - cross sectional area [m^2] + cross sectional area [m +\begin_inset Formula $^{2}$ +\end_inset + +] \end_layout \begin_layout Standard @@ -9023,7 +9118,7 @@ boundary condition: \begin_layout Standard \begin_inset Formula \begin{equation} -v=\frac{P_{i}-p_{t}(x=0)}{z_{rad}}\label{eq:bc-planewave-port-velocity} +v=\frac{P_{i}-p_{t}(x=0)}{z_{\mathrm{rad}}}\label{eq:bc-planewave-port-velocity} \end{equation} \end_inset @@ -9126,7 +9221,135 @@ name "fig:bc_planewave_port" \end_layout \begin_layout Chapter -Speaker +Thermoacoustic segments +\end_layout + +\begin_layout Standard +For relatively small temperature gradients, Swift's thermoacoustic equations + +\begin_inset CommandInset citation +LatexCommand cite +after "p. 91" +key "swift_thermoacoustics:_2003" +literal "false" + +\end_inset + +: +\begin_inset Formula +\begin{align} +\frac{\mathrm{d}p}{\mathrm{d}x} & =-\frac{-i\omega p_{m}}{R_{s}T_{m}S_{\mathrm{gas}}\left(1-f_{\nu}\right)}U,\\ +\frac{\mathrm{d}U}{\mathrm{d}x} & =\frac{-i\omega S_{\mathrm{gas}}}{\gamma p_{m}}\left[1+\left(\gamma-1\right)f_{\kappa}\right]p+\frac{f_{\kappa}-f_{\nu}}{\left(1-f_{\nu}\right)\left(1-\Pr\right)}\frac{1}{T_{m}}\frac{\mathrm{d}T_{m}}{\mathrm{d}x}U, +\end{align} + +\end_inset + +can be integrated. + Assuming +\begin_inset Formula $\frac{\mathrm{d}T_{m}}{\mathrm{d}x}L\ll T_{m}$ +\end_inset + +. + Then we find for the solution:: +\begin_inset Formula +\begin{equation} +p(x)=C_{1}\exp\left(\Gamma_{1}x\right)+C_{2}\exp\left(\Gamma_{2}x\right), +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +\Gamma_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a},\label{eq:Gammasol} +\end{equation} + +\end_inset + +wheren +\begin_inset Formula $1$ +\end_inset + + denotes to the +\begin_inset Formula $+$ +\end_inset + + and 2 to the +\begin_inset Formula $-$ +\end_inset + + sign. + In Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Gammasol" + +\end_inset + +, +\begin_inset Formula $a,$ +\end_inset + + +\begin_inset Formula $b,$ +\end_inset + + and +\begin_inset Formula $c$ +\end_inset + + are defined as: +\begin_inset Formula +\[ +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\left(\Gamma_{1}f_{\nu}p_{L}-\Gamma_{1}p_{L}+iU_{L}\omega\rho_{m}\right)e^{\Gamma_{2}L}+\left(-\Gamma_{2}f_{\nu}p_{L}+\Gamma_{2}p_{L}-iU_{L}\omega\rho_{m}\right)e^{\Gamma_{1}L}}{\Gamma_{1}f_{\nu}-\Gamma_{1}-\Gamma_{2}f_{\nu}+\Gamma_{2}}$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\begin{array}{c} +p_{R}\\ +U_{R} +\end{array}=\left[\begin{array}{cc} +\frac{\left(\Gamma_{1}f_{\nu}p_{L}-\Gamma_{1}p_{L}\right)e^{\Gamma_{2}L}+\left(-\Gamma_{2}f_{\nu}p_{L}+\Gamma_{2}p_{L}\right)e^{\Gamma_{1}L}}{\left(\Gamma_{2}-\Gamma_{1}\right)\left(1-f_{\nu}\right)} & i\frac{\left(iU_{L}\omega\rho_{m}\right)e^{\Gamma_{2}L}+\left(-iU_{L}\omega\rho_{m}\right)e^{\Gamma_{1}L}}{\left(\Gamma_{2}-\Gamma_{1}\right)\left(1-f_{\nu}\right)}\\ +\\ +\end{array}\right]\left\{ \begin{array}{c} +p_{L}\\ +U_{L} +\end{array}\right\} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Chapter +Speaker segment \end_layout \begin_layout Section @@ -9276,18 +9499,7 @@ where \end_inset is the equivalent impedance of the electrical circuit in -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -si{ -\backslash -ohm} -\end_layout - +\begin_inset Formula $\Omega$ \end_inset . @@ -9301,6 +9513,10 @@ z_{m}u=F_{\mathrm{emf}}+p_{l}S-p_{r}S, \end_inset + +\end_layout + +\begin_layout Standard where \begin_inset Formula $u$ \end_inset @@ -9644,7 +9860,7 @@ where \end_layout -\begin_layout Subsection +\begin_layout Standard Computing the voltage input for given velocity \end_layout @@ -9834,7 +10050,7 @@ The transfer matrix reads: \begin_layout Standard \begin_inset Note Note -status collapsed +status open \begin_layout Plain Layout Determinant: @@ -9848,29 +10064,14 @@ Determinant: \end_layout -\end_inset - +\begin_layout Plain Layout For a closed back-cavity volume, the back-cavity is: \end_layout -\begin_layout Standard -Then again: -\begin_inset Note Note -status open - \begin_layout Plain Layout - +Then again: \end_layout -\end_inset - - -\end_layout - -\begin_layout Standard -\begin_inset Note Note -status collapsed - \begin_layout Plain Layout Compute determinant: \end_layout @@ -9885,804 +10086,6 @@ Compute determinant: \end_inset -\end_layout - -\begin_layout Chapter -Optimized reactive silencers -\end_layout - -\begin_layout Section -Parallel Helmholtz resonator transfer function and transmission loss -\end_layout - -\begin_layout Standard -Equations for a side branch Helmholtz resonator: -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{align} -p_{R} & =p_{L},\\ -U_{R} & =U_{L}-p_{L}/Z_{h}, -\end{align} - -\end_inset - -where -\begin_inset Formula $Z_{h}$ -\end_inset - - is the side branch impedance of the Helmholtz resonator, defined as -\begin_inset Formula -\begin{equation} -Z_{h}=\left(\frac{\rho_{0}z_{0}}{i\omega V}+R_{v}+i\omega m_{\mathrm{neck}}\right), -\end{equation} - -\end_inset - -where -\begin_inset Formula -\begin{equation} -m_{\mathrm{neck}}=\frac{\rho_{0}\ell_{\mathrm{eff},\mathrm{neck}}}{S_{\mathrm{neck}}},\label{eq:acoustic_mass_neck} -\end{equation} - -\end_inset - -and for relatively large holes, air at STP, the resistance term can be estimated - as: -\begin_inset Formula -\begin{equation} -R_{v}\approx7.2\times10^{-3}z_{0}/S_{h}, -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -Now, the following substitutions are made: -\begin_inset Formula -\begin{align} -C & =\frac{V}{\rho_{0}z_{0}},\\ -m_{\mathrm{neck}} & =\frac{1}{\omega_{r}^{2}C}\\ -\zeta & =\frac{1}{2}\omega_{r}CR_{v}. -\end{align} - -\end_inset - - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -\begin_inset Formula $\frac{2\zeta}{\omega_{r}C}=R_{v}.$ -\end_inset - - -\end_layout - -\end_inset - -such that we can write: -\begin_inset Formula -\begin{equation} -Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right) -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -The quality factor of the resonator is the ratio of the resonance frequency - to its bandwidth measure. - If we take -\begin_inset Formula -\begin{equation} -Q\overset{\mathrm{def}}{=}\frac{f_{r}}{\Delta f}, -\end{equation} - -\end_inset - -where -\begin_inset Formula $\Delta f$ -\end_inset - - is the full width at half the maximum value, i.e. - the frequency distance between two points lying at -\begin_inset Formula $-3$ -\end_inset - - -\begin_inset space ~ -\end_inset - -dB w.r.t. - the maximum value. - The damping ratio -\begin_inset Formula $\zeta$ -\end_inset - - is related to -\begin_inset Formula $Q$ -\end_inset - - as: -\begin_inset Formula -\begin{equation} -\zeta=\frac{1}{2Q}=\frac{1}{2}\frac{\Delta f}{f_{r}} -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -Assembling the transfer matrix -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -\left\{ \begin{array}{c} -p\\ -U -\end{array}\right\} _{R}=\left[\begin{array}{cc} -T_{11} & T_{12}\\ -T_{21} & T_{22} -\end{array}\right]\left\{ \begin{array}{c} -p\\ -U -\end{array}\right\} _{L}, -\end{equation} - -\end_inset - -where -\begin_inset Formula -\begin{align} -T_{11} & =1\\ -T_{12} & =0\\ -T_{21} & =-Z_{h}^{-1}\\ -T_{22} & =1 -\end{align} - -\end_inset - - -\end_layout - -\begin_layout Subsection -Transmission loss -\end_layout - -\begin_layout Standard -The transmission coefficient can be computed as: -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -\tau=\frac{C}{A}=\frac{Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right)}{\frac{1}{2}\left(p_{L}+Z_{0}U_{L}\right)}, -\end{equation} - -\end_inset - -using -\begin_inset Formula -\begin{equation} -T_{11}p_{L}+T_{12}U_{L}=p_{R}=Z_{0}U_{R}=Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right), -\end{equation} - -\end_inset - -we get -\begin_inset Note Note -status collapsed - -\begin_layout Plain Layout -\begin_inset Formula $Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right)=T_{11}p_{L}+T_{12}U_{L}$ -\end_inset - - -\end_layout - -\begin_layout Plain Layout -– -\end_layout - -\begin_layout Plain Layout -\begin_inset Formula $U_{L}=\frac{\left(T_{11}-Z_{0}T_{21}\right)}{\left(Z_{0}T_{22}-T_{12}\right)}p_{L}$ -\end_inset - - -\end_layout - -\end_inset - - -\begin_inset Formula -\begin{equation} -U_{L}=\frac{\left(T_{11}-Z_{0}T_{21}\right)}{\left(Z_{0}T_{22}-T_{12}\right)}p_{L}, -\end{equation} - -\end_inset - -filling in: -\begin_inset Formula -\begin{equation} -\tau=\frac{2}{Z_{0}}\frac{T_{11}T_{22}-T_{12}T_{21}}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}}, -\end{equation} - -\end_inset - -assuming that the determinant of the transfer matrix be unity -\begin_inset Formula $(T_{11}T_{22}-T_{12}T_{21}\equiv1$ -\end_inset - -), this can be further simplified: -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -\tau=\frac{2}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}}, -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -For a Helmholtz resonator, this results in: -\begin_inset Note Note -status collapsed - -\begin_layout Plain Layout -\begin_inset Formula -\[ -\tau=\frac{2}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}}, -\] - -\end_inset - - -\end_layout - -\begin_layout Plain Layout -Filling in: -\begin_inset Formula $T_{11}=1$ -\end_inset - -, -\begin_inset Formula $T_{12}=0$ -\end_inset - -, -\begin_inset Formula $T_{21}=-1/Z_{h}$ -\end_inset - - -\begin_inset Formula $T_{22}=1$ -\end_inset - - -\end_layout - -\begin_layout Plain Layout -\begin_inset Formula -\[ -\tau=\frac{2Z_{h}}{2Z_{h}+Z_{0}}, -\] - -\end_inset - - -\end_layout - -\end_inset - - -\begin_inset Formula -\begin{equation} -\tau(\omega)=\frac{2Z_{h}(\omega)}{Z_{0}+2Z_{h}(\omega)}, -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -Filling in the Helmholtz resonator equation: -\begin_inset Formula -\begin{equation} -\tau(\omega)=\frac{2\left(1+2\frac{\omega}{\omega_{r}}\zeta-\left(\frac{\omega}{\omega_{r}}\right)^{2}\right)}{2\left(1+2\frac{\omega}{\omega_{r}}\zeta-\left(\frac{\omega}{\omega_{r}}\right)^{2}\right)+i\frac{\omega}{\omega_{r}}\left(\frac{Cz_{0}\omega_{r}}{S}\right)} -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -The peak height, filling in for -\begin_inset Formula $\omega/\omega_{r}=1$ -\end_inset - -: -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -\tau=\frac{4\zeta}{4\zeta+\beta}, -\end{equation} - -\end_inset - -where -\begin_inset Formula $\beta$ -\end_inset - - is defined as the resonator strength: -\begin_inset Formula -\begin{equation} -\beta=\frac{V\omega_{r}}{Sc_{0}} -\end{equation} - -\end_inset - -In terms of transmission loss: -\begin_inset Formula -\begin{equation} -\mathrm{TL}_{\omega=\omega_{r}}=20\log\left(\frac{\beta+4\zeta}{4\zeta}\right) -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -The peak half width is the distance over which the transmission loss has - dropped 3 -\begin_inset space ~ -\end_inset - -dB w.r.t. - the transmission loss at the resonance frequency. - This is an important design parameter. - We can compute it by setting -\begin_inset Formula -\begin{equation} -|\frac{\tau|_{\omega_{r}+\Delta\omega}}{\tau|_{\omega_{r}}}|=\sqrt{2}, -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -For light relative damping, and -\begin_inset Formula $\Delta\omega/\omega_{r}\approx1$ -\end_inset - -, -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -|\frac{\tau|_{\omega_{r}+\Delta\omega}}{\tau|_{\omega_{r}}}|\approx\frac{\alpha-1}{\zeta}, -\end{equation} - -\end_inset - -So given the -3 -\begin_inset space ~ -\end_inset - -dB point, and the maximum required transmission loss, we can compute -\begin_inset Formula $\zeta$ -\end_inset - - and -\begin_inset Formula $\beta$ -\end_inset - -: -\begin_inset Note Note -status collapsed - -\begin_layout Plain Layout -Eq 1: -\end_layout - -\begin_layout Plain Layout - -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none -\begin_inset Formula $\frac{\alpha_{-3\mathrm{dB}}-1}{\zeta}=\sqrt{2}\Rightarrow\zeta=\frac{\alpha_{-3\mathrm{dB}}-1}{\sqrt{2}}$ -\end_inset - - -\begin_inset Newline newline -\end_inset - -Eq 2: -\end_layout - -\begin_layout Plain Layout - -\family roman -\series medium -\shape up -\size normal -\emph off -\bar no -\strikeout off -\xout off -\uuline off -\uwave off -\noun off -\color none -\begin_inset Formula $\mathrm{TL}_{\mathrm{max}}=20\log\left(\frac{\beta+4\zeta}{4\zeta}\right)\Rightarrow\frac{\beta+4\zeta}{4\zeta}=10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}$ -\end_inset - - -\end_layout - -\begin_layout Plain Layout -\begin_inset Formula $\beta=4\zeta\left(10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}-1\right)$ -\end_inset - - -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Itemize -\begin_inset Formula $\zeta=\frac{\alpha_{-3\mathrm{dB}}-1}{\sqrt{2}}$ -\end_inset - - -\end_layout - -\begin_layout Itemize -\begin_inset Formula $\beta=4\zeta\left(10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}-1\right)$ -\end_inset - - -\end_layout - -\begin_layout Standard -Required volume in terms of resonator strength: -\begin_inset Formula -\begin{equation} -V=\frac{Sc_{0}\beta}{\omega_{r}} -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Subsection -Insertion loss -\end_layout - -\begin_layout Standard -For computation of the insertion loss, we require two more parameters: -\end_layout - -\begin_layout Itemize -The load impedance at the downstream end of the silencer -\end_layout - -\begin_layout Itemize -The output impedance of the source ( -\begin_inset Formula $Z_{\mathrm{rad}}$ -\end_inset - -) -\end_layout - -\begin_layout Standard -Suppose the source strength is defined by -\begin_inset Formula $\mathcal{S}$ -\end_inset - -. - Situation without silencer: -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{align} -U_{L} & =\mathcal{S}/\left(Z_{s}+Z_{l}\right),\\ -U_{R} & =U_{L},\\ -p_{R} & =Z_{\mathrm{rad}}U_{R}, -\end{align} - -\end_inset - -where -\begin_inset Formula $Z_{s}$ -\end_inset - - denotes the source output impedance, and -\begin_inset Formula $Z_{l}$ -\end_inset - - denotes the load impedance as felt by the source. -\end_layout - -\begin_layout Standard -For the reference case, the load impedance equals the radiation impedance, - and the radiated acoustic power is: -\begin_inset Note Note -status collapsed - -\begin_layout Plain Layout -\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[p_{R}U_{R}^{*}\right]$ -\end_inset - - -\end_layout - -\begin_layout Plain Layout -\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[Z_{\mathrm{rad}}\left(\mathcal{S}/Z_{s}\right)\left(\mathcal{S}/Z_{s}\right)^{*}\right]$ -\end_inset - - -\end_layout - -\begin_layout Plain Layout -\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}|\mathcal{S}/Z_{s}|^{2}\Re\left[Z_{\mathrm{rad}}\right]$ -\end_inset - - -\end_layout - -\end_inset - - -\begin_inset Formula -\begin{equation} -P_{\mathrm{ref}}=\frac{1}{2}\frac{|\mathcal{S}|^{2}}{|Z_{\mathrm{rad}}+Z_{s}|^{2}}\Re\left[Z_{\mathrm{rad}}\right] -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -Now, situation including silencer, with in general, transfer matrix -\begin_inset Formula $\boldsymbol{T}$ -\end_inset - -. -\begin_inset Note Note -status collapsed - -\begin_layout Plain Layout -\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[p_{R}U_{R}^{*}\right]$ -\end_inset - - -\end_layout - -\begin_layout Plain Layout -\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[Z_{\mathrm{rad}}U_{R}U_{R}^{*}\right]$ -\end_inset - - -\end_layout - -\begin_layout Plain Layout -Using: -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -P_{\mathrm{with\,silencer}}=\frac{1}{4}|\mathcal{S}|^{2}\frac{\Re\left[Z_{\mathrm{rad}}\right]}{|T_{22}Z_{\mathrm{rad}}-T_{12}+Z_{s}\left(T_{11}-T_{21}Z_{\mathrm{rad}}\right)|^{2}} -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -From that, computing the power ratio, that -\begin_inset Formula $\det\boldsymbol{T}\equiv1$ -\end_inset - - for a reciprocal system: -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -R_{P}=\frac{P_{\mathrm{with\,silencer}}}{P_{\mathrm{ref}}}=\frac{|Z_{\mathrm{rad}}+Z_{s}|^{2}}{|T_{22}Z_{\mathrm{rad}}-T_{12}+Z_{s}\left(T_{11}-T_{21}Z_{\mathrm{rad}}\right)|^{2}} -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Subsection -Insertion loss for a Helmholtz side branch resonator -\end_layout - -\begin_layout Standard -Filling in for a simple Helmholtz side branch resonator: -\begin_inset Formula -\begin{equation} -R_{P,\mathrm{Helmholtz}}=\frac{|Z_{\mathrm{rad}}+Z_{s}|^{2}}{|Z_{\mathrm{rad}}+Z_{s}\left(1+\frac{Z_{\mathrm{rad}}}{Z_{h}}\right)|^{2}}. -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -Comparing this to the transmission loss curve: -\begin_inset Formula -\begin{equation} -|\tau|_{\mathrm{Helmholtz}}^{2}=\frac{4|Z_{h}|^{2}}{|2Z_{h}+Z_{0}|^{2}} -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Subsubsection -High output impedance limit -\begin_inset Formula $(Z_{s}\gg Z_{\mathrm{rad}})$ -\end_inset - -, volume flow source -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+Z_{\mathrm{rad}}|^{2}}. -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Subsubsection -Low output impedance limit -\begin_inset Formula $(Z_{s}\ll Z_{\mathrm{rad}})$ -\end_inset - -, pressure source -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+Z_{s}|^{2}} -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Subsubsection -Special case: barrier in an infinite space -\begin_inset Formula $(Z_{s}=Z_{\mathrm{rad}})$ -\end_inset - - -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+\frac{1}{2}Z_{\mathrm{rad}}|^{2}}. -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -Comparing limits to power transmission ratio -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -|\tau|^{2}=\frac{|Z_{h}|^{2}}{|Z_{h}+\frac{1}{2}Z_{0}|^{2}}, -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Standard -So the transmission loss is the reduction in transmitted sound power for - the situation where the source output impedance equals the radiation impedance - on the other side of the silencer. -\end_layout - -\begin_layout Subsection -Multiple Helmholtz resonators at a single inlet -\end_layout - -\begin_layout Standard -In case multiple resonators are connected to the same inlet, the parallel - impedance can be computed by computing the equivalent parallel impedance: -\begin_inset Formula -\begin{equation} -\frac{1}{Z_{h,\mathrm{tot}}}=\frac{1}{Z_{h,1}}+\frac{1}{Z_{h,2}}+\dots -\end{equation} - -\end_inset - - -\end_layout - -\begin_layout Section -Transmission of the duct -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -\left\{ \begin{array}{c} -p_{R}\\ -U_{R} -\end{array}\right\} =\left[\begin{array}{cc} -\cos\left(kL\right) & -iZ_{0}\sin\left(kL\right)\\ --iZ_{0}^{-1}\sin\left(kL\right) & \cos\left(kL\right) -\end{array}\right]\left\{ \begin{array}{c} -p_{L}\\ -U_{L} -\end{array}\right\} -\end{equation} - -\end_inset - - \end_layout \begin_layout Chapter @@ -10906,7 +10309,6 @@ where due to the added mass effect, for the situation of negligible hole-hole interaction. [Paper: Tayong, 2013]. - \end_layout \begin_layout Standard @@ -10968,7 +10370,7 @@ Here we use a more advanced model, which includes the shear wave number. , the added mass end correction can be computed as: \begin_inset Note Note -status collapsed +status open \begin_layout Plain Layout Equation according to Temiz for added mass effect: @@ -11103,7 +10505,6 @@ In the large hole limit, without hole-hole interaction and \end_layout \begin_layout Standard -– \begin_inset Note Note status collapsed @@ -11215,7 +10616,7 @@ reference "eq:omgr_largeholes" yields \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout \begin_inset Formula $\phi\approx\frac{V\left(1.54D+t_{w}\right)\omega_{r,\mathrm{lh}}^{2}}{Sc_{0}^{2}}$ @@ -11644,7 +11045,6 @@ status open \begin_layout Plain Layout \align center -ss \begin_inset Graphics filename img/hexagonal_pattern.pdf width 50text% @@ -11671,10 +11071,6 @@ name "fig:hexagonal_pitch" \end_inset -\end_layout - -\begin_layout Plain Layout - \end_layout \end_inset @@ -11830,7 +11226,11 @@ Rectangular orifice Slit orifice \end_layout -\begin_layout Chapter +\begin_layout Standard +==================== +\end_layout + +\begin_layout Standard Lookup model \end_layout @@ -12925,6 +12325,873 @@ A compact square-shaped quadrupole with distances of \end_inset +\end_layout + +\begin_layout Chapter +Optimized reactive silencers +\end_layout + +\begin_layout Section +Parallel Helmholtz resonator transfer function and transmission loss +\end_layout + +\begin_layout Standard +Equations for a side branch Helmholtz resonator: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +p_{R} & =p_{L},\\ +U_{R} & =U_{L}-p_{L}/Z_{h}, +\end{align} + +\end_inset + +where +\begin_inset Formula $Z_{h}$ +\end_inset + + is the side branch impedance of the Helmholtz resonator, defined as +\begin_inset Formula +\begin{equation} +Z_{h}=\left(\frac{\rho_{0}z_{0}}{i\omega V}+R_{v}+i\omega m_{\mathrm{neck}}\right), +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +m_{\mathrm{neck}}=\frac{\rho_{0}\ell_{\mathrm{eff},\mathrm{neck}}}{S_{\mathrm{neck}}},\label{eq:acoustic_mass_neck} +\end{equation} + +\end_inset + +and for relatively large holes, air at STP, the resistance term can be estimated + as [SOURCE HERE!]: +\begin_inset Formula +\begin{equation} +R_{v}\approx7.2\times10^{-3}z_{0}/S_{h}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Now, the following substitutions are made: +\begin_inset Formula +\begin{align} +C & =\frac{V}{\rho_{0}z_{0}},\\ +m_{\mathrm{neck}} & =\frac{1}{\omega_{r}^{2}C}\\ +\zeta & =\frac{1}{2}\omega_{r}CR_{v}. +\end{align} + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\frac{2\zeta}{\omega_{r}C}=R_{v}.$ +\end_inset + + +\end_layout + +\end_inset + +such that we can write: +\begin_inset Formula +\begin{equation} +Z_{h}=\frac{1}{\omega_{r}C}\left(\frac{\omega_{r}}{i\omega}+2\zeta+\frac{i\omega}{\omega_{r}}\right) +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The quality factor of the resonator is the ratio of the resonance frequency + to its bandwidth measure. + If we take +\begin_inset Formula +\begin{equation} +Q\overset{\mathrm{def}}{=}\frac{f_{r}}{\Delta f}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\Delta f$ +\end_inset + + is the full width at half the maximum value, i.e. + the frequency distance between two points lying at +\begin_inset Formula $-3$ +\end_inset + + +\begin_inset space ~ +\end_inset + +dB w.r.t. + the maximum value. + The damping ratio +\begin_inset Formula $\zeta$ +\end_inset + + is related to +\begin_inset Formula $Q$ +\end_inset + + as: +\begin_inset Formula +\begin{equation} +\zeta=\frac{1}{2Q}=\frac{1}{2}\frac{\Delta f}{f_{r}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Assembling the transfer matrix +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p\\ +U +\end{array}\right\} _{R}=\left[\begin{array}{cc} +T_{11} & T_{12}\\ +T_{21} & T_{22} +\end{array}\right]\left\{ \begin{array}{c} +p\\ +U +\end{array}\right\} _{L}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{align} +T_{11} & =1\\ +T_{12} & =0\\ +T_{21} & =-Z_{h}^{-1}\\ +T_{22} & =1 +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Transmission loss +\end_layout + +\begin_layout Standard +The transmission coefficient can be computed as: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\tau=\frac{C}{A}=\frac{Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right)}{\frac{1}{2}\left(p_{L}+Z_{0}U_{L}\right)}, +\end{equation} + +\end_inset + +using +\begin_inset Formula +\begin{equation} +T_{11}p_{L}+T_{12}U_{L}=p_{R}=Z_{0}U_{R}=Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right), +\end{equation} + +\end_inset + +we get +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $Z_{0}\left(T_{21}p_{L}+T_{22}U_{L}\right)=T_{11}p_{L}+T_{12}U_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +– +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U_{L}=\frac{\left(T_{11}-Z_{0}T_{21}\right)}{\left(Z_{0}T_{22}-T_{12}\right)}p_{L}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +U_{L}=\frac{\left(T_{11}-Z_{0}T_{21}\right)}{\left(Z_{0}T_{22}-T_{12}\right)}p_{L}, +\end{equation} + +\end_inset + +filling in: +\begin_inset Formula +\begin{equation} +\tau=\frac{2}{Z_{0}}\frac{T_{11}T_{22}-T_{12}T_{21}}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}}, +\end{equation} + +\end_inset + +assuming that the determinant of the transfer matrix be unity +\begin_inset Formula $(T_{11}T_{22}-T_{12}T_{21}\equiv1$ +\end_inset + +) [THIS IS TRUE, BUT WHERE DOES THIS ASSUMPTION COME FROM??], this can be + further simplified: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\tau=\frac{2}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +For a Helmholtz resonator, this results in: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\tau=\frac{2}{T_{11}-T_{12}/Z_{0}-T_{21}Z_{0}+T_{22}}, +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Filling in: +\begin_inset Formula $T_{11}=1$ +\end_inset + +, +\begin_inset Formula $T_{12}=0$ +\end_inset + +, +\begin_inset Formula $T_{21}=-1/Z_{h}$ +\end_inset + + +\begin_inset Formula $T_{22}=1$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +\tau=\frac{2Z_{h}}{2Z_{h}+Z_{0}}, +\] + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\tau(\omega)=\frac{2Z_{h}(\omega)}{Z_{0}+2Z_{h}(\omega)}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Filling in the Helmholtz resonator equation: +\begin_inset Formula +\begin{equation} +\tau(\omega)=\frac{2\left(1+2\frac{\omega}{\omega_{r}}\zeta-\left(\frac{\omega}{\omega_{r}}\right)^{2}\right)}{2\left(1+2\frac{\omega}{\omega_{r}}\zeta-\left(\frac{\omega}{\omega_{r}}\right)^{2}\right)+i\frac{\omega}{\omega_{r}}\left(\frac{Cz_{0}\omega_{r}}{S}\right)}\label{eq:tau_hhres} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\left(\frac{Cz_{0}\omega_{r}}{S}\right)=\left(\frac{V\omega_{r}}{c_{0}S}\right)$ +\end_inset + + +\end_layout + +\end_inset + +The peak height, filling in for +\begin_inset Formula $\omega/\omega_{r}=1$ +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\tau=\frac{4\zeta}{4\zeta+\beta}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\beta$ +\end_inset + + is defined as the resonator strength: +\begin_inset Formula +\begin{equation} +\beta=\frac{V\omega_{r}}{Sc_{0}} +\end{equation} + +\end_inset + +In terms of transmission loss: +\begin_inset Formula +\begin{equation} +\mathrm{TL}_{\omega=\omega_{r}}=20\log\left(\frac{\beta+4\zeta}{4\zeta}\right) +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +In case of weak damping ( +\begin_inset Formula $\zeta\ll1$ +\end_inset + +), Eq. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:tau_hhres" + +\end_inset + + can be reduced to: +\begin_inset Formula +\begin{equation} +\tau(\omega)=\frac{1-\left(\frac{\omega}{\omega_{r}}\right)^{2}}{1-\left(\frac{\omega}{\omega_{r}}\right)^{2}+\frac{1}{2}i\frac{\omega}{\omega_{r}}\beta} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The width of the peak over which a certain transmission loss is higher than + a value of +\begin_inset Formula $\mathrm{TL_{\mathrm{min}}}$ +\end_inset + + +\begin_inset space ~ +\end_inset + +dB, can be computed as: +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $|\tau(\omega_{r}+\Delta\omega)|=|\frac{1-\left(\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\right)^{2}}{1-\left(\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\right)^{2}+\frac{1}{2}i\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\beta}|=10^{\frac{\mathrm{TL}_{\mathrm{min}}}{20}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $|\tau(\omega_{r}+\Delta\omega)|=|\frac{1-\left(\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\right)^{2}}{1-\left(\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\right)^{2}+\frac{1}{2}i\frac{\omega_{r}+\Delta\omega}{\omega_{r}}\beta}|=10^{\frac{\mathrm{TL}_{\mathrm{min}}}{20}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\beta=\frac{\Delta\omega}{\omega_{r}}4\sqrt{10^{^{\frac{\mathrm{TL_{\mathrm{min}}}}{10}}}-1} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +The peak half width is the distance over which the transmission loss has + dropped 3 +\begin_inset space ~ +\end_inset + +dB w.r.t. + the transmission loss at the resonance frequency. + This is an important design parameter. + We can compute it by setting: +\begin_inset Formula +\begin{equation} +|\frac{\tau|_{\omega_{r}+\Delta\omega}}{\tau|_{\omega_{r}}}|=\sqrt{2}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +So given the -3 +\begin_inset space ~ +\end_inset + +dB point, and the maximum required transmission loss, we can compute +\begin_inset Formula $\zeta$ +\end_inset + + and +\begin_inset Formula $\beta$ +\end_inset + +: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Eq 1: +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $\frac{\alpha_{-3\mathrm{dB}}-1}{\zeta}=\sqrt{2}\Rightarrow\zeta=\frac{\alpha_{-3\mathrm{dB}}-1}{\sqrt{2}}$ +\end_inset + + +\begin_inset Newline newline +\end_inset + +Eq 2: +\end_layout + +\begin_layout Plain Layout + +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none +\begin_inset Formula $\mathrm{TL}_{\mathrm{max}}=20\log\left(\frac{\beta+4\zeta}{4\zeta}\right)\Rightarrow\frac{\beta+4\zeta}{4\zeta}=10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\beta=4\zeta\left(10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}-1\right)$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\zeta=\frac{\alpha_{-3\mathrm{dB}}-1}{\sqrt{2}}$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\beta=4\zeta\left(10^{\frac{\mathrm{TL}_{\mathrm{max}}}{20}}-1\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +Required volume in terms of resonator strength: +\begin_inset Formula +\begin{equation} +V=\frac{Sc_{0}\beta}{\omega_{r}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Insertion loss +\end_layout + +\begin_layout Standard +For computation of the insertion loss, we require two more parameters: +\end_layout + +\begin_layout Itemize +The load impedance at the downstream end of the silencer +\end_layout + +\begin_layout Itemize +The output impedance of the source ( +\begin_inset Formula $Z_{\mathrm{rad}}$ +\end_inset + +) +\end_layout + +\begin_layout Standard +Suppose the source strength is defined by +\begin_inset Formula $\mathcal{S}$ +\end_inset + +. + Situation without silencer: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +U_{L} & =\mathcal{S}/\left(Z_{s}+Z_{l}\right),\\ +U_{R} & =U_{L},\\ +p_{R} & =Z_{\mathrm{rad}}U_{R}, +\end{align} + +\end_inset + +where +\begin_inset Formula $Z_{s}$ +\end_inset + + denotes the source output impedance, and +\begin_inset Formula $Z_{l}$ +\end_inset + + denotes the load impedance as felt by the source. +\end_layout + +\begin_layout Standard +For the reference case, the load impedance equals the radiation impedance, + and the radiated acoustic power is: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[p_{R}U_{R}^{*}\right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[Z_{\mathrm{rad}}\left(\mathcal{S}/Z_{s}\right)\left(\mathcal{S}/Z_{s}\right)^{*}\right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}|\mathcal{S}/Z_{s}|^{2}\Re\left[Z_{\mathrm{rad}}\right]$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +P_{\mathrm{ref}}=\frac{1}{2}\frac{|\mathcal{S}|^{2}}{|Z_{\mathrm{rad}}+Z_{s}|^{2}}\Re\left[Z_{\mathrm{rad}}\right] +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Now, situation including silencer, with in general, transfer matrix +\begin_inset Formula $\boldsymbol{T}$ +\end_inset + +. +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[p_{R}U_{R}^{*}\right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $P_{\mathrm{ref}}=\frac{1}{2}\Re\left[Z_{\mathrm{rad}}U_{R}U_{R}^{*}\right]$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Using: +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +P_{\mathrm{with\,silencer}}=\frac{1}{4}|\mathcal{S}|^{2}\frac{\Re\left[Z_{\mathrm{rad}}\right]}{|T_{22}Z_{\mathrm{rad}}-T_{12}+Z_{s}\left(T_{11}-T_{21}Z_{\mathrm{rad}}\right)|^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +From that, computing the power ratio, that +\begin_inset Formula $\det\boldsymbol{T}\equiv1$ +\end_inset + + for a reciprocal system: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R_{P}=\frac{P_{\mathrm{with\,silencer}}}{P_{\mathrm{ref}}}=\frac{|Z_{\mathrm{rad}}+Z_{s}|^{2}}{|T_{22}Z_{\mathrm{rad}}-T_{12}+Z_{s}\left(T_{11}-T_{21}Z_{\mathrm{rad}}\right)|^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsection +Insertion loss for a Helmholtz side branch resonator +\end_layout + +\begin_layout Standard +Filling in for a simple Helmholtz side branch resonator: +\begin_inset Formula +\begin{equation} +R_{P,\mathrm{Helmholtz}}=\frac{|Z_{\mathrm{rad}}+Z_{s}|^{2}}{|Z_{\mathrm{rad}}+Z_{s}\left(1+\frac{Z_{\mathrm{rad}}}{Z_{h}}\right)|^{2}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Comparing this to the transmission loss curve: +\begin_inset Formula +\begin{equation} +|\tau|_{\mathrm{Helmholtz}}^{2}=\frac{4|Z_{h}|^{2}}{|2Z_{h}+Z_{0}|^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +High output impedance limit +\begin_inset Formula $(Z_{s}\gg Z_{\mathrm{rad}})$ +\end_inset + +, volume flow source +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+Z_{\mathrm{rad}}|^{2}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Low output impedance limit +\begin_inset Formula $(Z_{s}\ll Z_{\mathrm{rad}})$ +\end_inset + +, pressure source +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+Z_{s}|^{2}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Special case: barrier in an infinite space +\begin_inset Formula $(Z_{s}=Z_{\mathrm{rad}})$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +R_{P,\mathrm{Helmholtz}}=\frac{|Z_{h}|^{2}}{|Z_{h}+\frac{1}{2}Z_{\mathrm{rad}}|^{2}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Comparing limits to power transmission ratio +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +|\tau|^{2}=\frac{|Z_{h}|^{2}}{|Z_{h}+\frac{1}{2}Z_{0}|^{2}}, +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +So the transmission loss is the reduction in transmitted sound power for + the situation where the source output impedance equals the radiation impedance + on the other side of the silencer. +\end_layout + +\begin_layout Subsection +Multiple Helmholtz resonators at a single inlet +\end_layout + +\begin_layout Standard +In case multiple resonators are connected to the same inlet, the parallel + impedance can be computed by computing the equivalent parallel impedance: +\begin_inset Formula +\begin{equation} +\frac{1}{Z_{h,\mathrm{tot}}}=\frac{1}{Z_{h,1}}+\frac{1}{Z_{h,2}}+\dots +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Transmission of the duct +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p_{R}\\ +U_{R} +\end{array}\right\} =\left[\begin{array}{cc} +\cos\left(kL\right) & -iZ_{0}\sin\left(kL\right)\\ +-iZ_{0}^{-1}\sin\left(kL\right) & \cos\left(kL\right) +\end{array}\right]\left\{ \begin{array}{c} +p_{L}\\ +U_{L} +\end{array}\right\} +\end{equation} + +\end_inset + + \end_layout \begin_layout Chapter @@ -13024,7 +13291,7 @@ Where \end_layout \begin_layout Plain Layout -Filling in the expression for eq of state, +Filling in the expression for eq of state, \end_layout \begin_layout Plain Layout @@ -13865,6 +14132,10 @@ where the solid, respectively. In frequency domain and using cylindrical coordinates, assuming axial symmetry, this can be written as +\begin_inset Note Note +status open + +\begin_layout Plain Layout \begin_inset CommandInset nomenclature LatexCommand nomenclature prefix "A" @@ -13875,6 +14146,11 @@ literal "true" \end_inset +\end_layout + +\end_inset + + \begin_inset Formula \begin{equation} \left(r^{2}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{\partial^{2}}{\partial x^{2}}\right)+r\frac{\partial}{\partial r}+\frac{2}{i\delta_{s}^{2}}r^{2}\right)T_{s}=0, @@ -14205,7 +14481,7 @@ Such that: \end_layout \begin_layout Plain Layout -Filling in +Filling in \end_layout \begin_layout Plain Layout @@ -14300,7 +14576,7 @@ status collapsed \end_layout \begin_layout Plain Layout -And for +And for \end_layout \begin_layout Plain Layout @@ -14311,7 +14587,7 @@ And for \end_layout \begin_layout Plain Layout -Filling this in into +Filling this in into \end_layout \begin_layout Plain Layout @@ -14328,6 +14604,10 @@ where \end_inset is the thermal effusivity +\begin_inset Note Note +status open + +\begin_layout Plain Layout \begin_inset CommandInset nomenclature LatexCommand nomenclature prefix "A" @@ -14335,6 +14615,11 @@ symbol "$e$" description "Thermal effusivity\\nomunit{\\si{\\joule\\per\\square\\metre\\kelvin\\second\\tothe{ \\frac{1}{2} } }}" literal "true" +\end_inset + + +\end_layout + \end_inset of the fluid, and @@ -14402,7 +14687,7 @@ name "chap:Derivation-of-Karal's" \begin_layout Standard \series bold -Note: this documentation is imcomplete. +Note: this documentation is incomplete. \end_layout \begin_layout Standard @@ -14440,6 +14725,10 @@ Schematic of a discontinuity at the interface between two tubes with different \end_inset +\end_layout + +\begin_layout Plain Layout +\align center \begin_inset CommandInset label LatexCommand label name "fig:karal-1" @@ -14629,7 +14918,10 @@ where is defined as the positive root of the r.h.s. of Eq. - +\begin_inset space ~ +\end_inset + + \begin_inset CommandInset ref LatexCommand ref reference "eq:beta_k" @@ -15324,7 +15616,7 @@ Setting \begin_inset Formula $p=n$ \end_inset - en + en \end_layout \begin_layout Plain Layout @@ -15746,7 +16038,7 @@ status collapsed \begin_layout Plain Layout \lang english -Filling in: +Filling in: \end_layout \begin_layout Plain Layout