diff --git a/lrftubes.bib b/lrftubes.bib index 946a177..ee10616 100644 --- a/lrftubes.bib +++ b/lrftubes.bib @@ -275,28 +275,6 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo 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} } -@article{keefe_acoustical_1984, - title = {Acoustical wave propagation in cylindrical ducts: Transmission line parameter approximations for isothermal and nonisothermal boundary conditions}, - volume = {75}, - pages = {58--62}, - number = {1}, - journaltitle = {The Journal of the Acoustical Society of America}, - author = {Keefe, Douglas H}, - date = {1984}, - 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} -} - -@article{thompson_analog_2014, - title = {Analog model for thermoviscous propagation in a cylindrical tube}, - volume = {135}, - pages = {585--590}, - number = {2}, - journaltitle = {The Journal of the Acoustical Society of America}, - author = {Thompson, Stephen C and Gabrielson, Thomas B and Warren, Daniel M}, - date = {2014}, - 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} -} - @article{benade_propagation_1968, title = {On the propagation of sound waves in a cylindrical conduit}, volume = {44}, @@ -383,19 +361,11 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo 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} } -@article{leniowska_plate_resonance_1999, - title = {Vibrations of circular plate interacting with an ideal compressible fluid}, - volume = {24}, - url = {https://acoustics.ippt.pan.pl/index.php/aa/article/viewFile/1117/952}, - pages = {427--441}, - number = {4}, - journaltitle = {Archives of acoustics}, - author = {Leniowska, L.}, - date = {1999} -} - -@misc{calcdevice, - title = {Natural frequency calculators (web page)}, - url = {https://calcdevice.com/natural-frequency-of-circular-plate-id224.html}, - urldate = {2022-05-25} -} +@book{zwikker_sound_1949, + title = {Sound Absorbing Materials}, + url = {https://books.google.com/books?id=ezUOnQEACAAJ}, + publisher = {Elsevier Publishing Company}, + author = {Zwikker, C. and Kosten, C.W.}, + date = {1949}, + lccn = {50006127} +} \ No newline at end of file diff --git a/lrftubes.lyx b/lrftubes.lyx index d9bfc6d..1291133 100644 --- a/lrftubes.lyx +++ b/lrftubes.lyx @@ -3435,7 +3435,9 @@ status open \backslash -lrftubes +lrftubes +\backslash + \end_layout \end_inset @@ -4509,9 +4511,10 @@ C_{2} & =\frac{J_{0}\left(\alpha_{0}\right)-J_{0}\left(\alpha_{1}\right)}{J_{0}\ \begin_layout Standard \begin_inset Formula -\begin{equation} -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]} -\end{equation} +\begin{align} +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.\\ + & \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] +\end{align} \end_inset @@ -5198,7 +5201,7 @@ S_{f}=\exp\left(\alpha x\right) \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout \begin_inset Formula $\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}+\alpha\frac{\mathrm{d}p}{\mathrm{d}x}+\Gamma^{2}p=0$ @@ -5559,7 +5562,7 @@ p=\frac{C_{1}\exp\left(-i\Gamma x\right)+C_{1}\exp\left(-i\Gamma x\right)}{r_{0} \begin_layout Standard \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout If we assume @@ -7682,7 +7685,8 @@ Both tubes on either side of the discontinuity are cylindrical. \end_layout \begin_layout Itemize -The wavelength is larger than transverse characteristic length scale. +The wavelength is larger than transverse characteristic length scale (no + propagating modes expect for the plane waves). \end_layout \begin_layout Itemize @@ -8428,9 +8432,16 @@ noprefix "false" It would be beneficial for computing time to replace the outside world by a boundary condition on the port. Here it is approached as a scattering problem. - More information is described in 'Sound absorbing materials' (1949) Zwikker - et al., pp. - 132-134. + More information is described in +\begin_inset CommandInset citation +LatexCommand cite +after "p. 132-134" +key "zwikker_sound_1949" +literal "false" + +\end_inset + +. The pressure field can be written as: \end_layout @@ -8454,23 +8465,24 @@ in which \begin_inset Formula $p_{i}$ \end_inset - the incident pressure field and + the incident pressure field (the field as if there were only an infinite + wall) and \begin_inset Formula $p_{s}$ \end_inset the scattered pressure field. - All depend on both position and time. + The combination of the incident and scattered field combined result in + the total pressure field. + All depend on both position and time (or frequency). If only the infinite wall is taken into account and the port and system - behind it are ignored, the amplitude of the incident plane wave and its - reflection can be described as: + behind it are ignored, the amplitude of the incident plane wave is: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} -p_{i}(x,t)=\begin{cases} -P_{i}\cdot\cos(kx) & x<0\\ -undefined & x=0\\ +p_{i}(x,\omega)=\begin{cases} +P_{i}\cdot\cos(kx) & x<=0\\ 0 & x>0 \end{cases} \end{equation} @@ -8485,8 +8497,8 @@ in which \begin_inset Formula $P_{i}$ \end_inset - is half the amplitude of the incident plane wave (resulting in sound pressure - + is the amplitude of the incident plane wave at the wall (resulting in sound + pressure \family roman \series medium \shape up @@ -8520,7 +8532,7 @@ in which \begin_inset Formula $k$ \end_inset - is the wavenumber and + is the wave number and \begin_inset Formula $x$ \end_inset @@ -8528,7 +8540,7 @@ in which There is no scattered pressure field, so this is the total pressure field right away. When the port and system behind it are added, the total pressure field - no longer is equal to the incident pressure field: a correction must be + is no longer equal to the incident pressure field: a correction must be added, which is captured in \begin_inset Formula $p_{s}$ \end_inset @@ -8539,7 +8551,7 @@ in which \begin_inset Formula $x<0$ \end_inset -, this has the same effect als a baffled piston. +, this has the same effect as a baffled piston. On the condition that the wavelength is much larger than the port size, the scattered field near the boundary (but still outside of the port) is given by: @@ -8548,7 +8560,7 @@ in which \begin_layout Standard \begin_inset Formula \begin{equation} -p_{s}(x=0^{-})=-Z_{rad}U +p_{s}(x=0^{-})=-Z_{\mathrm{rad}}U \end{equation} \end_inset @@ -8558,10 +8570,10 @@ p_{s}(x=0^{-})=-Z_{rad}U \begin_layout Standard in which -\begin_inset Formula $Z_{rad}$ +\begin_inset Formula $Z_{\mathrm{rad}}$ \end_inset -is the radiation impedance of a baffled piston and + is the radiation impedance of a baffled piston and \begin_inset Formula $U$ \end_inset @@ -8642,7 +8654,7 @@ velocities \begin_layout Standard \begin_inset Formula \begin{equation} -p_{t}(x=0)=P_{i}-z_{rad}v\label{eq:bc-planewave-port-pressure} +p_{t}(x=0)=P_{i}-z_{\mathrm{rad}}v\label{eq:bc-planewave-port-pressure} \end{equation} \end_inset @@ -8652,10 +8664,10 @@ p_{t}(x=0)=P_{i}-z_{rad}v\label{eq:bc-planewave-port-pressure} \begin_layout Standard in which -\begin_inset Formula $z_{rad}$ +\begin_inset Formula $z_{\mathrm{rad}}$ \end_inset -is the specific radiation impedance of a baffled piston and + is the specific radiation impedance of a baffled piston and \begin_inset Formula $v$ \end_inset @@ -8669,8 +8681,16 @@ boundary condition in COMSOL. \begin_inset Formula $v$ \end_inset - can be 'measured' by averaging the normal component of the velocity and - adding a minus sign to make it inwards. + can be +\begin_inset Quotes eld +\end_inset + +measured +\begin_inset Quotes erd +\end_inset + + by averaging the normal component of the velocity and adding a minus sign + to make it inwards. Alternatively, the equation can be solved for \begin_inset Formula $v$ \end_inset @@ -8698,7 +8718,39 @@ in which \begin_inset Formula $p_{t}(x=0)$ \end_inset - can be 'measured' by averaging it over the port's boundary. + can be +\begin_inset Quotes eld +\end_inset + +measured +\begin_inset Quotes erd +\end_inset + + by averaging it over the port's boundary. + The LRFTubes implementation of this +\emph on +mixed +\emph default + boundary condition is for a left wall: +\begin_inset Formula +\begin{equation} +p_{L}+Z_{\mathrm{rad}}U_{L}=P_{i}, +\end{equation} + +\end_inset + +and the same on a right wall: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +p_{R}-Z_{\mathrm{rad}}U_{R}=P_{i} +\end{equation} + +\end_inset + + \end_layout \begin_layout Standard @@ -9294,7 +9346,7 @@ reference "eq:U_vs_V" a bit: \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout \begin_inset Formula $\frac{1}{S_{l}}\left(z_{m}+\frac{\left(B\ell\right)^{2}}{Z_{\mathrm{el}}}\right)U_{l}=p_{l}S_{l}-p_{r}S_{r}+\frac{B\ell}{Z_{\mathrm{el}}}V_{\mathrm{in}}$