From 820663ab684b81918d73fb915c856295b7e8991e Mon Sep 17 00:00:00 2001 From: "J.A. de Jong - ASCEE" Date: Wed, 24 Oct 2018 10:25:20 +0200 Subject: [PATCH] Some updates --- lrftubes.lyx | 442 ++++++++++++++++++++++++++++++++++++++++++++++++++- 1 file changed, 438 insertions(+), 4 deletions(-) diff --git a/lrftubes.lyx b/lrftubes.lyx index 3e2a229..537bae4 100644 --- a/lrftubes.lyx +++ b/lrftubes.lyx @@ -1,5 +1,5 @@ -#LyX 2.2 created this file. For more info see http://www.lyx.org/ -\lyxformat 508 +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 \begin_document \begin_header \save_transient_properties true @@ -25,6 +25,8 @@ \font_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures false \graphics default \default_output_format pdf2 \output_sync 1 @@ -65,6 +67,7 @@ \suppress_date false \justification true \use_refstyle 0 +\use_minted 0 \index Index \shortcut idx \color #008000 @@ -78,7 +81,10 @@ \tocdepth 3 \paragraph_separation indent \paragraph_indentation default -\quotes_language english +\is_math_indent 0 +\math_numbering_side default +\quotes_style english +\dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default @@ -462,6 +468,7 @@ LatexCommand nomenclature prefix "A" symbol "$\\mathbf{e}_x$" description "Unit vector in $x$-direction\\nomunit{-}" +literal "true" \end_inset @@ -474,6 +481,7 @@ LatexCommand nomenclature prefix "A" symbol "$H$" description "Total enthalpy per unit mass \\nomunit{\\si{\\joule\\per\\kilogram}}" +literal "true" \end_inset @@ -486,6 +494,7 @@ LatexCommand nomenclature prefix "A" symbol "$\\mathbf{I}$" description "Identity tensor\\nomunit{-}" +literal "true" \end_inset @@ -498,6 +507,7 @@ LatexCommand nomenclature prefix "A" symbol "$h_\\nu$" description "Viscothermal shape function for the velocity\\nomunit{-}" +literal "true" \end_inset @@ -510,6 +520,7 @@ LatexCommand nomenclature prefix "A" symbol "$h_\\kappa$" description "Viscothermal shape function for the temperature\\nomunit{-}" +literal "true" \end_inset @@ -522,6 +533,7 @@ LatexCommand nomenclature prefix "A" symbol "$E$" description "Total energy per unit mass \\nomunit{\\si{\\joule\\per\\kilogram}}" +literal "true" \end_inset @@ -536,6 +548,7 @@ LatexCommand nomenclature prefix "A" symbol "$c$" description "Speed of sound\\nomunit{\\si{\\metre\\per\\second}}" +literal "true" \end_inset @@ -545,6 +558,7 @@ LatexCommand nomenclature prefix "A" symbol "$c_p$" description "Specific heat at constant pressure \\nomunit{\\si{\\joule\\per\\kilogram\\kelvin}}" +literal "true" \end_inset @@ -554,6 +568,7 @@ LatexCommand nomenclature prefix "A" symbol "$c_s$" description "Specific heat of the solid\\nomunit{\\si{\\joule\\per\\kilogram\\kelvin}}" +literal "true" \end_inset @@ -563,6 +578,7 @@ LatexCommand nomenclature prefix "A" symbol "$c_v$" description "Specific heat at constant density \\nomunit{\\si{\\joule\\per\\kilogram\\kelvin}}" +literal "true" \end_inset @@ -572,6 +588,7 @@ LatexCommand nomenclature prefix "A" symbol "$D$" description "Diameter\\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -581,6 +598,7 @@ LatexCommand nomenclature prefix "A" symbol "$f_\\kappa$" description "Thermal Rott function \\nomunit{-}" +literal "true" \end_inset @@ -590,6 +608,7 @@ LatexCommand nomenclature prefix "A" symbol "$f_\\nu$" description "Viscous Rott function \\nomunit{-}" +literal "true" \end_inset @@ -599,6 +618,7 @@ LatexCommand nomenclature prefix "A" symbol "$f$" description "Frequency\\nomunit{\\si{\\hertz}}" +literal "true" \end_inset @@ -608,6 +628,7 @@ LatexCommand nomenclature prefix "A" symbol "$i$" description "Imaginary unit\\nomunit{-}" +literal "true" \end_inset @@ -617,6 +638,7 @@ LatexCommand nomenclature prefix "A" symbol "$J_\\alpha$" description "Bessel function of the first kind and order $\\alpha$\\nonomunit" +literal "true" \end_inset @@ -626,6 +648,7 @@ LatexCommand nomenclature prefix "A" symbol "$k$" description "Wave number\\nomunit{\\si{\\radian\\per\\metre}}" +literal "true" \end_inset @@ -635,6 +658,7 @@ LatexCommand nomenclature prefix "A" symbol "$L$" description "Length\\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -644,6 +668,7 @@ LatexCommand nomenclature prefix "A" symbol "$\\ell$" description "Characteristic length scale of a fluid space \\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -653,6 +678,7 @@ LatexCommand nomenclature prefix "A" symbol "$N$" description "Number of\\nomunit{-}" +literal "true" \end_inset @@ -662,6 +688,7 @@ LatexCommand nomenclature prefix "A" symbol "$\\mathbf{n}$" description "Normal vector pointing from the solid into the fluid\\nomunit{-}" +literal "true" \end_inset @@ -671,6 +698,7 @@ LatexCommand nomenclature prefix "A" symbol "$p$" description "Pressure, acoustic pressure \\nomunit{\\si{\\pascal}}" +literal "true" \end_inset @@ -680,6 +708,7 @@ LatexCommand nomenclature prefix "A" symbol "$r_h$" description "Hydraulic radius \\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -689,6 +718,7 @@ LatexCommand nomenclature prefix "A" symbol "$\\mathbf{r}$" description "Transverse position vector\\nomunit{-}" +literal "true" \end_inset @@ -698,6 +728,7 @@ LatexCommand nomenclature prefix "A" symbol "$S$" description "Cross-sectional area, surface area\\nomunit{\\si{\\square\\metre}}" +literal "true" \end_inset @@ -707,6 +738,7 @@ LatexCommand nomenclature prefix "A" symbol "$t$" description "Time \\nomunit{\\si{\\second}}" +literal "true" \end_inset @@ -716,6 +748,7 @@ LatexCommand nomenclature prefix "A" symbol "$T$" description "Temperature\\nomunit{\\si{\\kelvin}}" +literal "true" \end_inset @@ -725,6 +758,7 @@ LatexCommand nomenclature prefix "A" symbol "$\\mathbf{u}$" description "Velocity vector\\nomunit{\\si{\\metre\\per\\second}}" +literal "true" \end_inset @@ -734,6 +768,7 @@ LatexCommand nomenclature prefix "A" symbol "$u$" description "Velocity in wave propagation direction\\nomunit{\\si{\\metre\\per\\second}}" +literal "true" \end_inset @@ -743,6 +778,7 @@ LatexCommand nomenclature prefix "A" symbol "$U$" description "Volume flow\\nomunit{\\si{\\cubic\\metre\\per\\second}}" +literal "true" \end_inset @@ -752,6 +788,7 @@ LatexCommand nomenclature prefix "A" symbol "$V$" description "Volume \\nomunit{\\si{\\cubic\\metre}}" +literal "true" \end_inset @@ -761,6 +798,7 @@ LatexCommand nomenclature prefix "A" symbol "$\\mathbf{x}$" description "Position vector \\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -770,6 +808,7 @@ LatexCommand nomenclature prefix "A" symbol "$z$" description "Specific acoustic impedance\\nomunit{\\si{\\pascal\\second\\per\\metre}}" +literal "true" \end_inset @@ -779,6 +818,7 @@ LatexCommand nomenclature prefix "A" symbol "$Z$" description "Volume flow impedance\\nomunit{\\si{\\pascal\\second\\per\\cubic\\metre}}" +literal "true" \end_inset @@ -817,6 +857,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\Delta$" description "Difference\\nonomunit" +literal "true" \end_inset @@ -831,6 +872,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\gamma$" description "Ratio of specific heats\\nomunit{-}" +literal "true" \end_inset @@ -840,6 +882,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\Gamma$" description "Viscothermal wave number for a prismatic duct \\nomunit{\\si{\\radian\\per\\metre}}" +literal "true" \end_inset @@ -849,6 +892,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\delta_{\\kappa}$" description "Thermal penetration depth\\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -858,6 +902,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\delta_{\\nu}$" description "Viscous penetration depth\\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -867,6 +912,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\epsilon_s$" description "Ideal stack correction factor \\nomunit{-}" +literal "true" \end_inset @@ -876,6 +922,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\lambda$" description "Wavelength \\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -885,6 +932,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\pi$" description "Ratio of the circumference to the diameter of a circle \\nomunit{-}" +literal "true" \end_inset @@ -894,6 +942,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\Pi$" description "Wetted perimeter (contact length between solid and fluid) \\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -932,6 +981,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\otimes$" description "Dyadic product\\nonomunit" +literal "true" \end_inset @@ -946,6 +996,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\Re$" description "Real part\\nonomunit" +literal "true" \end_inset @@ -955,6 +1006,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\Im$" description "Imaginary part\\nonomunit" +literal "true" \end_inset @@ -964,6 +1016,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\nabla$" description "Gradient \\nomunit{\\si{\\per\\metre}}" +literal "true" \end_inset @@ -973,6 +1026,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\nabla^2$" description "Laplacian\\nomunit{\\si{\\per\\square\\metre}}" +literal "true" \end_inset @@ -982,6 +1036,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\sim$" description "Same order of magnitude\\nonomunit" +literal "true" \end_inset @@ -991,6 +1046,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\left\\Vert \\bullet \\right\\Vert $" description "Eucledian norm\\nonomunit" +literal "true" \end_inset @@ -1000,6 +1056,7 @@ LatexCommand nomenclature prefix "M" symbol "d" description "Infinitesimal\\nonomunit" +literal "true" \end_inset @@ -1009,6 +1066,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\partial$" description "Infinitesimal\\nonomunit" +literal "true" \end_inset @@ -1018,6 +1076,7 @@ LatexCommand nomenclature prefix "M" symbol "$\\bullet$" description "Placeholder for an operand\\nonomunit" +literal "true" \end_inset @@ -1045,6 +1104,7 @@ LatexCommand nomenclature prefix "S" symbol "wall" description "At the wall" +literal "true" \end_inset @@ -1054,6 +1114,7 @@ LatexCommand nomenclature prefix "S" symbol "0" description "Evaluated at the reference condition" +literal "true" \end_inset @@ -1063,6 +1124,7 @@ LatexCommand nomenclature prefix "S" symbol "$f$" description "Fluid" +literal "true" \end_inset @@ -1072,6 +1134,7 @@ LatexCommand nomenclature prefix "S" symbol "$s$" description "Solid" +literal "true" \end_inset @@ -1081,6 +1144,7 @@ LatexCommand nomenclature prefix "S" symbol "$w$" description "Wall" +literal "true" \end_inset @@ -1090,6 +1154,7 @@ LatexCommand nomenclature prefix "S" symbol "$R$" description "Right side" +literal "true" \end_inset @@ -1099,6 +1164,7 @@ LatexCommand nomenclature prefix "S" symbol "$L$" description "Left side" +literal "true" \end_inset @@ -1108,6 +1174,7 @@ LatexCommand nomenclature prefix "S" symbol "$s$" description "Solid" +literal "true" \end_inset @@ -1117,6 +1184,7 @@ LatexCommand nomenclature prefix "S" symbol "$s$" description "Squeeze" +literal "true" \end_inset @@ -1126,6 +1194,7 @@ LatexCommand nomenclature prefix "S" symbol "$i$" description "Inner" +literal "true" \end_inset @@ -1135,6 +1204,7 @@ LatexCommand nomenclature prefix "S" symbol "$o$" description "Outer" +literal "true" \end_inset @@ -1144,6 +1214,7 @@ LatexCommand nomenclature prefix "S" symbol "$t$" description "Tube" +literal "true" \end_inset @@ -1174,6 +1245,7 @@ LatexCommand nomenclature prefix "O" symbol "Sec(s)." description "Section(s)" +literal "true" \end_inset @@ -1183,6 +1255,7 @@ LatexCommand nomenclature prefix "O" symbol "Eq(s)." description "Equation(s)" +literal "true" \end_inset @@ -1192,6 +1265,7 @@ LatexCommand nomenclature prefix "O" symbol "LRF" description "Low Reduced Frequency" +literal "true" \end_inset @@ -1277,6 +1351,7 @@ lrftubes \begin_inset CommandInset citation LatexCommand cite key "van_der_eerden_noise_2000" +literal "true" \end_inset @@ -1284,6 +1359,7 @@ key "van_der_eerden_noise_2000" \begin_inset CommandInset citation LatexCommand cite key "kampinga_viscothermal_2010" +literal "true" \end_inset @@ -1291,6 +1367,7 @@ key "kampinga_viscothermal_2010" \begin_inset CommandInset citation LatexCommand cite key "ward_deltaec_2017" +literal "true" \end_inset @@ -1306,6 +1383,7 @@ This documentation serves as a reference for the implemented models. LatexCommand href name "README" target "https://github.com/asceenl/lrftubes" +literal "false" \end_inset @@ -1321,6 +1399,7 @@ This document is very brief on the theory and it is assumed that the reader \begin_inset CommandInset citation LatexCommand cite key "swift_thermoacoustics:_2003" +literal "true" \end_inset @@ -1330,6 +1409,7 @@ key "swift_thermoacoustics:_2003" \begin_inset CommandInset citation LatexCommand cite key "de_jong_numerical_2015" +literal "true" \end_inset @@ -1425,6 +1505,7 @@ These segments can be connected to form one-dimensional acoustic systems \begin_inset CommandInset citation LatexCommand cite key "van_der_eerden_noise_2000" +literal "true" \end_inset @@ -2194,6 +2275,7 @@ For a formal derivation of the model for prismatic cylindrical tubes, the \begin_inset CommandInset citation LatexCommand cite key "tijdeman_propagation_1975" +literal "true" \end_inset @@ -2201,6 +2283,7 @@ key "tijdeman_propagation_1975" \begin_inset CommandInset citation LatexCommand cite key "nijhof_viscothermal_2010" +literal "true" \end_inset @@ -2210,6 +2293,7 @@ key "nijhof_viscothermal_2010" \begin_inset CommandInset citation LatexCommand cite key "swift_thermoacoustics:_2003,swift_thermoacoustic_1988" +literal "true" \end_inset @@ -2217,6 +2301,7 @@ key "swift_thermoacoustics:_2003,swift_thermoacoustic_1988" \begin_inset CommandInset citation LatexCommand cite key "rott_damped_1969" +literal "true" \end_inset @@ -2261,6 +2346,7 @@ where \begin_inset CommandInset citation LatexCommand cite key "rott_damped_1969" +literal "true" \end_inset @@ -2275,6 +2361,7 @@ key "rott_damped_1969" LatexCommand cite after "p. 88" key "swift_thermoacoustics:_2003" +literal "true" \end_inset @@ -2295,6 +2382,7 @@ LatexCommand nomenclature prefix "A" symbol "$j$" description "Index, subscript placeholder\\nomunit{-}" +literal "true" \end_inset @@ -3026,7 +3114,9 @@ name "fig:im_gamma" \begin_inset Note Note status open -\begin_layout Section +\begin_layout Plain Layout + +\series bold Duct with conical cross-sectional area \end_layout @@ -3040,6 +3130,7 @@ For conical ducts, i.e. LatexCommand cite after "p. 181" key "rienstra_introduction_2015" +literal "true" \end_inset @@ -3975,6 +4066,337 @@ Z_{c,0}=\frac{kz_{0}}{\left(1-f_{\nu}\right)\pi r_{0}^{2}\Gamma_{0}} \end_inset +\end_layout + +\begin_layout Section +Prismatic lined circular duct +\end_layout + +\begin_layout Standard +The Fourier transformed wave equation in axisymmetric cylindrical coordinates + can be written as: +\begin_inset Formula +\begin{equation} +\frac{\partial^{2}p}{\partial r^{2}}+\frac{1}{r}\frac{\partial p}{\partial r}+\frac{\partial^{2}p}{\partial x^{2}}+k^{2}p=0, +\end{equation} + +\end_inset + +Using separation of variables: +\begin_inset Formula +\begin{equation} +p=\rho(r)\xi(x), +\end{equation} + +\end_inset + +this can be written as: +\begin_inset Formula +\begin{equation} +\frac{\rho^{''}}{\rho}+\frac{1}{r}\frac{\rho'}{\rho}+\frac{\xi^{''}}{\xi}+k^{2}=0 +\end{equation} + +\end_inset + +Solutions: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1}{r}\frac{\rho'}{\rho}+\frac{\rho^{''}}{\rho}=-k^{2}-\frac{\xi^{''}}{\xi}=-\epsilon^{2}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Try: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\xi=A\exp\left(\alpha x\right)$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $-k^{2}-\alpha^{2}=-\epsilon^{2}$ +\end_inset + + Or: +\begin_inset Formula $\alpha^{2}=\epsilon^{2}-k^{2}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +And +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1}{r}\frac{\rho'}{\rho}+\frac{\rho^{''}}{\rho}=-\epsilon^{2}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Means: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $r\frac{\rho'}{\rho}+r^{2}\frac{\rho^{''}}{\rho}+r^{2}\epsilon^{2}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Which has solution: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\rho=J_{0}\left(\epsilon r\right)$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align} +\xi & =\exp\left(-i\alpha x\right),\\ +\rho & =J_{0}\left(\epsilon r\right), +\end{align} + +\end_inset + +such that the solution for the pressure is: +\begin_inset Formula +\begin{equation} +p=J_{0}\left(\epsilon r\right)\exp\left(\alpha x\right) +\end{equation} + +\end_inset + +under the condition: +\begin_inset Formula +\begin{equation} +\alpha^{2}=k^{2}-\epsilon^{2}. +\end{equation} + +\end_inset + +At +\begin_inset Formula $r=R$ +\end_inset + + we have the boundary condition that +\begin_inset Formula $Z_{0}\zeta_{R}u=p$ +\end_inset + +. + After filling in and using the rule +\begin_inset Formula $J_{0}'(x)=J_{-1}(x)$ +\end_inset + +: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $\frac{i\zeta_{R}}{k}\frac{\partial p}{\partial x}|_{r=R}=p|r=R$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{i\zeta_{R}}{k}\epsilon J'_{0}\left(\epsilon r\right)=J_{0}\left(\epsilon r\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Or: +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\epsilon R\frac{J_{-1}\left(\epsilon R\right)}{J_{0}\left(\epsilon R\right)}=-i\upsilon, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\upsilon=\frac{kR}{\zeta_{R}}$ +\end_inset + +. + This is the characteristic eqation for +\begin_inset Formula $\epsilon R$ +\end_inset + +. + Solutions for +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $\Im\left[\epsilon R\right]<3$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +And +\begin_inset Formula $\Re\left[2\right]<2$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Using +\begin_inset Formula $\upsilon=\frac{kR}{\zeta_{R}}$ +\end_inset + +. +\end_layout + +\begin_layout Plain Layout +Solution: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(\epsilon R\right)^{2}\approx\frac{96+36i\upsilon\pm\sqrt{9216+2304i\upsilon-912\upsilon^{2}}}{12+i\upsilon}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Filling in for +\begin_inset Formula $U$ +\end_inset + +: +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\epsilon\approx+\frac{1}{R}\sqrt{\frac{96+36i\upsilon\pm\sqrt{9216+2304i\upsilon-912\upsilon^{2}}}{12+i\upsilon}} +\end{equation} + +\end_inset + +where +\begin_inset Formula $0\leq\Re[\epsilon R]\leq2$ +\end_inset + + and +\begin_inset Formula $0\leq\Im\left[\epsilon R\right]\leq3$ +\end_inset + + should be satisfied in order to guarantee precision, see Mechel, p. + 630. +\end_layout + +\begin_layout Subsection +Cremers impedance +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\frac{kR}{\zeta}=2.9803824+1.2796025i +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Or: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\zeta=y_{cr}\pi\frac{kR}{y_{cr}\pi}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\zeta=kR\left(0.28-0.12i\right) +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Attenuation reached when the liner impedance equals Cremer's impedance is + around 15 dB per unit of radius maximum. + It decreases with increasing frequency, when +\begin_inset Formula $fR\approx100$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Locally reacting lining with back-volume +\end_layout + +\begin_layout Standard +Impedance of concentric liner, outer radius is +\begin_inset Formula $R_{o}$ +\end_inset + +, inner radius is +\begin_inset Formula $R_{i}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\zeta_{\mathrm{back}}=i\frac{H_{0}^{(1)}\left(kR_{i}\right)-\frac{H_{1}^{(1)}\left(kR_{o}\right)}{H_{1}^{(2)}\left(kR_{o}\right)}H_{0}^{(2)}\left(kR_{i}\right)}{H_{1}^{(1)}\left(kR_{i}\right)-\frac{H_{1}^{(1)}\left(kR_{o}\right)}{H_{1}^{(2)}\left(kR_{o}\right)}H_{1}^{(2)}\left(kR_{i}\right)} +\end{equation} + +\end_inset + +Such that the total impedance is +\begin_inset Formula +\begin{equation} +\zeta=\zeta_{\mathrm{back}}+\zeta_{\mathrm{MPP}} +\end{equation} + +\end_inset + + \end_layout \begin_layout Section @@ -4084,6 +4506,7 @@ The lower the frequency, the more the second assumption is violated, while LatexCommand cite after "p. 156" key "ward_deltaec_2017" +literal "true" \end_inset @@ -4163,6 +4586,7 @@ LatexCommand nomenclature prefix "A" symbol "$C_c$" description "Acoustic capacitance of a compliance volume\\nomunit{\\si{\\cubic\\metre\\per\\pascal}}" +literal "true" \end_inset @@ -4267,6 +4691,7 @@ For discontinuities in the cross section of a waveguide, and the case of \begin_inset CommandInset citation LatexCommand cite key "karal_analogous_1953" +literal "true" \end_inset @@ -4332,6 +4757,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\alpha$" description "Ratio of tube radii\\nomunit{-}" +literal "true" \end_inset @@ -4379,6 +4805,7 @@ LatexCommand nomenclature prefix "A" symbol "$M_A$" description "Acoustic mass\\nomunit{\\si{\\kg\\per\\metre\\tothe{4}}}" +literal "true" \end_inset @@ -4388,6 +4815,7 @@ LatexCommand nomenclature prefix "A" symbol "$a$" description "Tube radius\\nomunit{\\si{\\metre}}" +literal "true" \end_inset @@ -4409,6 +4837,7 @@ LatexCommand nomenclature prefix "G" symbol "$\\chi$" description "Karal's discontinuity factor\\nomunit{-}" +literal "true" \end_inset @@ -4525,6 +4954,7 @@ LatexCommand nomenclature prefix "A" symbol "$N$" description "Number\\nomunit{-}" +literal "true" \end_inset @@ -4665,6 +5095,7 @@ We can derive the following impedance boundary condition LatexCommand cite after "p. 157" key "ward_deltaec_2017" +literal "true" \end_inset @@ -4922,6 +5353,7 @@ LatexCommand nomenclature prefix "A" symbol "$r$" description "Radial position in cylindrical coordinates\\nomunit{\\si{\\m}}" +literal "true" \end_inset @@ -5384,6 +5816,7 @@ LatexCommand nomenclature prefix "A" symbol "$e$" description "Thermal effusivity\\nomunit{\\si{\\joule\\per\\square\\metre\\kelvin\\second\\tothe{ \\frac{1}{2} } }}" +literal "true" \end_inset @@ -5538,6 +5971,7 @@ In cylindrical coordinates, the solution of the Helmholtz equation can be \begin_inset CommandInset citation LatexCommand cite key "blackstock_fundamentals_2000" +literal "true" \end_inset