Some docs improvements. Added Zwikker Kosten explicit in bibtex
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lrftubes.bib
44
lrftubes.bib
@ -275,28 +275,6 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo
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file = {Karal - 1953 - The analogous acoustical impedance for discontinui.pdf:/home/anne/.literature/storage/ZSJSCHMS/Karal - 1953 - The analogous acoustical impedance for discontinui.pdf:application/pdf}
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}
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@article{keefe_acoustical_1984,
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title = {Acoustical wave propagation in cylindrical ducts: Transmission line parameter approximations for isothermal and nonisothermal boundary conditions},
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volume = {75},
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pages = {58--62},
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number = {1},
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journaltitle = {The Journal of the Acoustical Society of America},
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author = {Keefe, Douglas H},
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date = {1984},
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file = {Keefe - 1984 - Acoustical wave propagation in cylindrical ducts .pdf:/home/anne/.literature/storage/WPM2TBDL/Keefe - 1984 - Acoustical wave propagation in cylindrical ducts .pdf:application/pdf}
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}
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@article{thompson_analog_2014,
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title = {Analog model for thermoviscous propagation in a cylindrical tube},
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volume = {135},
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pages = {585--590},
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number = {2},
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journaltitle = {The Journal of the Acoustical Society of America},
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author = {Thompson, Stephen C and Gabrielson, Thomas B and Warren, Daniel M},
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date = {2014},
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file = {Thompson e.a. - 2014 - Analog model for thermoviscous propagation in a cy.pdf:/home/anne/.literature/storage/ZGSV8RWF/Thompson e.a. - 2014 - Analog model for thermoviscous propagation in a cy.pdf:application/pdf}
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}
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@article{benade_propagation_1968,
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title = {On the propagation of sound waves in a cylindrical conduit},
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volume = {44},
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@ -383,19 +361,11 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo
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file = {Kino et al. - 2009 - Investigation of non-acoustical parameters of comp.pdf:/home/anne/.literature/storage/I9P5SZAE/Kino et al. - 2009 - Investigation of non-acoustical parameters of comp.pdf:application/pdf}
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}
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@article{leniowska_plate_resonance_1999,
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title = {Vibrations of circular plate interacting with an ideal compressible fluid},
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volume = {24},
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url = {https://acoustics.ippt.pan.pl/index.php/aa/article/viewFile/1117/952},
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pages = {427--441},
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number = {4},
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journaltitle = {Archives of acoustics},
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author = {Leniowska, L.},
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date = {1999}
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}
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@misc{calcdevice,
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title = {Natural frequency calculators (web page)},
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url = {https://calcdevice.com/natural-frequency-of-circular-plate-id224.html},
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urldate = {2022-05-25}
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@book{zwikker_sound_1949,
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title = {Sound Absorbing Materials},
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url = {https://books.google.com/books?id=ezUOnQEACAAJ},
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publisher = {Elsevier Publishing Company},
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author = {Zwikker, C. and Kosten, C.W.},
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date = {1949},
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lccn = {50006127}
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}
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108
lrftubes.lyx
108
lrftubes.lyx
@ -3436,6 +3436,8 @@ status open
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\backslash
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lrftubes
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\backslash
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\end_layout
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\end_inset
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@ -4509,9 +4511,10 @@ C_{2} & =\frac{J_{0}\left(\alpha_{0}\right)-J_{0}\left(\alpha_{1}\right)}{J_{0}\
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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f_{i}=\delta_{i}\left(1+i\right)\frac{\left\{ H_{0}^{(1)}\left(\alpha_{0}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)\right\} \left[r_{0}H_{-1}^{(2)}\left(\alpha_{0}\right)-r_{1}H_{-1}^{(2)}\left(\alpha_{1}\right)\right]+\left\{ H_{0}^{(2)}\left(\alpha_{0}\right)-H_{0}^{(2)}\left(\alpha_{1}\right)\right\} \left[r_{1}H_{-1}^{(1)}\left(\alpha_{1}\right)-r_{0}H_{-1}^{(1)}\left(\alpha_{0}\right)\right]}{\left(r_{1}^{2}-r_{0}^{2}\right)\left[H_{0}^{(1)}\left(\alpha_{0}\right)H_{0}^{(2)}\left(\alpha_{1}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)H_{0}^{(2)}\left(\alpha_{0}\right)\right]}
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\end{equation}
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\begin{align}
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f_{i} & =\delta_{i}\left(1+i\right)\left[\frac{\left\{ H_{0}^{(1)}\left(\alpha_{0}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)\right\} \left[r_{0}H_{-1}^{(2)}\left(\alpha_{0}\right)-r_{1}H_{-1}^{(2)}\left(\alpha_{1}\right)\right]}{\left(r_{1}^{2}-r_{0}^{2}\right)\left[H_{0}^{(1)}\left(\alpha_{0}\right)H_{0}^{(2)}\left(\alpha_{1}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)H_{0}^{(2)}\left(\alpha_{0}\right)\right]}+\right.\\
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& \qquad\qquad\qquad\left.\frac{\left\{ H_{0}^{(2)}\left(\alpha_{0}\right)-H_{0}^{(2)}\left(\alpha_{1}\right)\right\} \left[r_{1}H_{-1}^{(1)}\left(\alpha_{1}\right)-r_{0}H_{-1}^{(1)}\left(\alpha_{0}\right)\right]}{\left(r_{1}^{2}-r_{0}^{2}\right)\left[H_{0}^{(1)}\left(\alpha_{0}\right)H_{0}^{(2)}\left(\alpha_{1}\right)-H_{0}^{(1)}\left(\alpha_{1}\right)H_{0}^{(2)}\left(\alpha_{0}\right)\right]}\right]
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\end{align}
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\end_inset
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@ -5198,7 +5201,7 @@ S_{f}=\exp\left(\alpha x\right)
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\begin_inset Note Note
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status open
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status collapsed
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\begin_layout Plain Layout
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\begin_inset Formula $\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\alpha\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0$
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@ -5559,7 +5562,7 @@ p=\frac{C_{1}\exp\left(-i\Gamma x\right)+C_{1}\exp\left(-i\Gamma x\right)}{r_{0}
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\begin_layout Standard
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\begin_inset Note Note
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status open
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status collapsed
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\begin_layout Plain Layout
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If we assume
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@ -7682,7 +7685,8 @@ Both tubes on either side of the discontinuity are cylindrical.
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\end_layout
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\begin_layout Itemize
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The wavelength is larger than transverse characteristic length scale.
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The wavelength is larger than transverse characteristic length scale (no
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propagating modes expect for the plane waves).
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\end_layout
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\begin_layout Itemize
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@ -8428,9 +8432,16 @@ noprefix "false"
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It would be beneficial for computing time to replace the outside world
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by a boundary condition on the port.
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Here it is approached as a scattering problem.
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More information is described in 'Sound absorbing materials' (1949) Zwikker
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et al., pp.
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132-134.
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More information is described in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "p. 132-134"
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key "zwikker_sound_1949"
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literal "false"
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\end_inset
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.
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The pressure field can be written as:
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\end_layout
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@ -8454,23 +8465,24 @@ in which
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\begin_inset Formula $p_{i}$
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\end_inset
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the incident pressure field and
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the incident pressure field (the field as if there were only an infinite
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wall) and
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\begin_inset Formula $p_{s}$
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\end_inset
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the scattered pressure field.
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All depend on both position and time.
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The combination of the incident and scattered field combined result in
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the total pressure field.
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All depend on both position and time (or frequency).
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If only the infinite wall is taken into account and the port and system
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behind it are ignored, the amplitude of the incident plane wave and its
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reflection can be described as:
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behind it are ignored, the amplitude of the incident plane wave is:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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p_{i}(x,t)=\begin{cases}
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P_{i}\cdot\cos(kx) & x<0\\
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undefined & x=0\\
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p_{i}(x,\omega)=\begin{cases}
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P_{i}\cdot\cos(kx) & x<=0\\
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0 & x>0
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\end{cases}
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\end{equation}
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@ -8485,8 +8497,8 @@ in which
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\begin_inset Formula $P_{i}$
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\end_inset
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is half the amplitude of the incident plane wave (resulting in sound pressure
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is the amplitude of the incident plane wave at the wall (resulting in sound
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pressure
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\family roman
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\series medium
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\shape up
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@ -8528,7 +8540,7 @@ in which
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There is no scattered pressure field, so this is the total pressure field
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right away.
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When the port and system behind it are added, the total pressure field
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no longer is equal to the incident pressure field: a correction must be
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is no longer equal to the incident pressure field: a correction must be
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added, which is captured in
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\begin_inset Formula $p_{s}$
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\end_inset
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@ -8539,7 +8551,7 @@ in which
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\begin_inset Formula $x<0$
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\end_inset
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, this has the same effect als a baffled piston.
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, this has the same effect as a baffled piston.
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On the condition that the wavelength is much larger than the port size,
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the scattered field near the boundary (but still outside of the port) is
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given by:
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@ -8548,7 +8560,7 @@ in which
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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p_{s}(x=0^{-})=-Z_{rad}U
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p_{s}(x=0^{-})=-Z_{\mathrm{rad}}U
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\end{equation}
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\end_inset
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@ -8558,7 +8570,7 @@ p_{s}(x=0^{-})=-Z_{rad}U
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\begin_layout Standard
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in which
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\begin_inset Formula $Z_{rad}$
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\begin_inset Formula $Z_{\mathrm{rad}}$
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\end_inset
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is the radiation impedance of a baffled piston and
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@ -8642,7 +8654,7 @@ velocities
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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p_{t}(x=0)=P_{i}-z_{rad}v\label{eq:bc-planewave-port-pressure}
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p_{t}(x=0)=P_{i}-z_{\mathrm{rad}}v\label{eq:bc-planewave-port-pressure}
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\end{equation}
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\end_inset
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@ -8652,7 +8664,7 @@ p_{t}(x=0)=P_{i}-z_{rad}v\label{eq:bc-planewave-port-pressure}
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\begin_layout Standard
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in which
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\begin_inset Formula $z_{rad}$
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\begin_inset Formula $z_{\mathrm{rad}}$
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\end_inset
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is the specific radiation impedance of a baffled piston and
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@ -8669,8 +8681,16 @@ boundary condition in COMSOL.
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\begin_inset Formula $v$
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\end_inset
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can be 'measured' by averaging the normal component of the velocity and
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adding a minus sign to make it inwards.
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can be
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\begin_inset Quotes eld
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\end_inset
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measured
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\begin_inset Quotes erd
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\end_inset
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by averaging the normal component of the velocity and adding a minus sign
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to make it inwards.
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Alternatively, the equation can be solved for
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\begin_inset Formula $v$
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\end_inset
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@ -8698,7 +8718,39 @@ in which
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\begin_inset Formula $p_{t}(x=0)$
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\end_inset
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can be 'measured' by averaging it over the port's boundary.
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can be
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\begin_inset Quotes eld
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\end_inset
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measured
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\begin_inset Quotes erd
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\end_inset
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by averaging it over the port's boundary.
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The LRFTubes implementation of this
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\emph on
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mixed
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\emph default
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boundary condition is for a left wall:
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\begin_inset Formula
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\begin{equation}
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p_{L}+Z_{\mathrm{rad}}U_{L}=P_{i},
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\end{equation}
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\end_inset
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and the same on a right wall:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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p_{R}-Z_{\mathrm{rad}}U_{R}=P_{i}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -9294,7 +9346,7 @@ reference "eq:U_vs_V"
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a bit:
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\begin_inset Note Note
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status open
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status collapsed
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\begin_layout Plain Layout
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\begin_inset Formula $\frac{1}{S_{l}}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)U_{l}=p_{l}S_{l}-p_{r}S_{r}+\frac{B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}}$
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