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publisher = {Acoustical Society of America}, + author = {Swift, G. W.}, + date = {2003}, + file = {Swift - 2003 - Thermoacoustics A unifying perspective for some e.pdf:/home/anne/.literature/storage/TX4X2FEP/Swift - 2003 - Thermoacoustics A unifying perspective for some e.pdf:application/pdf;Swift en Garrett - 2003 - Thermoacoustics A unifying perspective for some e.pdf:/home/anne/.literature/storage/6RZNJADB/Swift en Garrett - 2003 - Thermoacoustics A unifying perspective for some e.pdf:application/pdf} +} + +@thesis{beltman_viscothermal_1998, + title = {Viscothermal Wave Propagation Including Acousto-Elastic Interaction}, + type = {phdthesis}, + author = {Beltman, W. M}, + date = {1998}, + keywords = {book available}, + file = {Beltman - 1998 - Viscothermal wave propagation including acousto-el.pdf:/home/anne/.literature/storage/GI8XS8GK/Beltman - 1998 - Viscothermal wave propagation including acousto-el.pdf:application/pdf} +} + +@article{beltman_viscothermal_1999, + title = {Viscothermal Wave Propagation Including Acousto-Elastic Interaction, Part I: Theory}, + volume = {227}, + issn = {0022-460X}, + doi = {06/jsvi.1999.2355}, + shorttitle = {{VISCOTHERMAL} {WAVE} {PROPAGATION} {INCLUDING} {ACOUSTO}-{ELASTIC} {INTERACTION}, {PART} I}, + abstract = {This research deals with pressure waves in a gas trapped in thin layers or narrow tubes. In these cases viscous and thermal effects can have a significant effect on the propagation of waves. This so-called viscothermal wave propagation is governed by a number of dimensionless parameters. The two most important parameters are the shear wave number and the reduced frequency. These parameters were used to put into perspective the models that were presented in the literature. The analysis shows that the complete parameter range is covered by three classes of models: the standard wave equation model, the low reduced frequency model and the full linearized Navier-Stokes model. For the majority of practical situations, the low reduced frequency model is sufficient and the most efficient to describe viscothermal wave propagation. The full linearized Navier-Stokes model should only be used under extreme conditions.}, + pages = {555--586}, + number = {3}, + journaltitle = {Journal of Sound and Vibration}, + author = {Beltman, W. M.}, + urldate = {2011-07-20}, + date = {1999-10-28}, + keywords = {Printed}, + file = {BELTMAN - 1999 - VISCOTHERMAL WAVE PROPAGATION INCLUDING ACOUSTO-EL.pdf:/home/anne/.literature/storage/VUEH48TH/BELTMAN - 1999 - VISCOTHERMAL WAVE PROPAGATION INCLUDING ACOUSTO-EL.pdf:application/pdf;ScienceDirect Snapshot:/home/anne/.literature/storage/7RZVXTAA/S0022460X99923556.html:text/html} +} + +@article{swift_thermoacoustic_1988, + title = {Thermoacoustic Engines}, + volume = {84}, + issn = {00014966}, + doi = {10.1121/1.396617}, + pages = {1145--1180}, + number = {4}, + journaltitle = {The Journal of the Acoustical Society of America}, + shortjournal = {J. Acoust. Soc. Am.}, + author = {Swift, G. W.}, + urldate = {2011-07-20}, + date = {1988}, + keywords = {Printed}, + file = {Swift - 1988 - Thermoacoustic engines.pdf:/home/anne/.literature/storage/NGUESBBB/Swift - 1988 - Thermoacoustic engines.pdf:application/pdf;Thermoacoustic engines | Browse - Journal of the Acoustical Society of America:/home/anne/.literature/storage/NSHR2HI3/p1145_s1.html:text/html} +} + +@article{rott_damped_1969, + title = {Damped and thermally driven acoustic oscillations in wide and narrow tubes}, + volume = {20}, + issn = {0044-2275}, + doi = {10.1007/BF01595562}, + pages = {230--243}, + number = {2}, + journaltitle = {Zeitschrift für angewandte Mathematik und Physik}, + shortjournal = {Journal of Applied Mathematics and Physics ({ZAMP})}, + author = {Rott, Nikolaus}, + date = {1969-03}, + file = {Rott - 1969 - Damped and thermally driven acoustic oscillations .pdf:/home/anne/.literature/storage/8FNJIHA2/Rott - 1969 - Damped and thermally driven acoustic oscillations .pdf:application/pdf;SpringerLink - Zeitschrift für Angewandte Mathematik und Physik (ZAMP), Volume 20, Number 2:/home/anne/.literature/storage/QNIQ6QWV/u22268322515314q.html:text/html} +} + +@article{kampinga_performance_2010, + title = {Performance of Several Viscothermal Acoustic Finite Elements}, + volume = {96}, + doi = {10.3813/AAA.918262}, + abstract = {Viscothermal acoustics can be described by the linearized Navier Stokes equations. Besides inertia and compressibility, these equations take the heat conductivity and the viscosity of the medium (air) into account. These 'viscothermal' effects are significant in, for example, miniature acoustic transducers and {MEMS} devices. A finite element for viscothermal acoustics, which can be used to model such devices, is presented. The particular set of equations used in the model of viscothermal acoustics leads to a complex symmetric finite element system matrix. Several different {FEM} discretizations are studied on a 2D thin gap problem. These discretizations are known, in the context of the Stokes equation, as the Taylor Hood quadrilateral and triangle elements, the Crouzeix Raviart element and the {MINI} element. All elements are implemented in the {FEM} software {COMSOL}. The elements with quadratic velocity and temperature shape functions show the best orders of convergence.}, + pages = {115--124}, + number = {1}, + journaltitle = {Acta Acustica united with Acustica}, + shortjournal = {Acta Acustica united with Acustica}, + author = {Kampinga, W.R. and Wijnant, Y.H. and de Boer, A.}, + date = {2010}, + file = {Kampinga et al. - 2010 - Performance of Several Viscothermal Acoustic Finit.pdf:/home/anne/.literature/storage/HJNPKD8V/Kampinga et al. - 2010 - Performance of Several Viscothermal Acoustic Finit.pdf:application/pdf} +} + +@article{tijdeman_propagation_1975, + title = {On the propagation of sound waves in cylindrical tubes}, + volume = {39}, + issn = {0022-460X}, + doi = {10.1016/S0022-460X(75)80206-9}, + abstract = {It is shown that the two main parameters governing the propagation of sound waves in gases contained in rigid cylindrical tubes, are the shear wave number, s = R ρ s ω / μ , and the reduced frequency, k=ωR/a0. It appears possible to rewrite the most significant analytical solutions for the propagation constant, Γ, as given in the literature, as simple expressions in terms of these two parameters. With the aid of these expressions the various solutions are put in perspective and their ranges of applicability are indicated. + +It is demonstrated that most of the analytical solutions are dependent only on the shear wave number, s, and that they are covered completely by the solution obtained for the first time by Zwikker and Kosten (1949). + +The full solution of the problem has been obtained by Kirchhoff (1868) in the form of a complicated, complex transcendental equation. In the present paper this equation is rewritten in terms of the mentioned basic parameters and brought in the attractive form F\<, s, k\>=0, which is solved numerically by using the Newton-Raphson procedure. As first estimate in this procedure the value ofaccording to the solution of Zwikker and Kosten is taken. Results are presented for a wide range of s and k values.}, + pages = {1--33}, + number = {1}, + journaltitle = {Journal of Sound and Vibration}, + author = {Tijdeman, H.}, + urldate = {2012-01-05}, + date = {1975-03-08}, + file = {ScienceDirect Snapshot:/home/anne/.literature/storage/9B2T7ZM8/S0022460X75802069.html:text/html;Tijdeman - 1975 - On the propagation of sound waves in cylindrical t.pdf:/home/anne/.literature/storage/368PZ6AN/Tijdeman - 1975 - On the propagation of sound waves in cylindrical t.pdf:application/pdf} +} + +@article{kampinga_efficient_2011, + title = {An Efficient Finite Element Model for Viscothermal Acoustics}, + volume = {97}, + doi = {10.3813/AAA.918442}, + abstract = {Standard isentropic acoustic models do not include the dissipative effects of viscous friction and heat conduction. These viscothermal effects can be important, for example in models of small acoustic transducers. Viscothermal acoustics can be modeled in arbitrary geometries with models that contain four or five coupled fields. Therefore, these fully coupled models are computationally costly. On the other hand, efficient approximate viscothermal acoustic models exist, but these are only applicable to certain simplified geometries. A new approximate model is presented which fills the gap between these two extremes. This new model can be used for arbitrary geometries and has a computational efficiency which is higher than the full model and lower than the models with geometrical constraints. The new model is derived and demonstrated on several problems, including acoustic-structure interaction problems.}, + pages = {618--631}, + number = {4}, + journaltitle = {Acta Acustica united with Acustica}, + author = {Kampinga, W.R. and Wijnant, Y.H. and de Boer, A.}, + date = {2011}, + file = {ingentaconnect An Efficient Finite Element Model for Viscothermal Acoustics:/home/anne/.literature/storage/68KJ2IVT/art00009.html:text/html;Kampinga et al. - 2011 - An Efficient Finite Element Model for Viscothermal.pdf:/home/anne/.literature/storage/QRGF9MR6/Kampinga et al. - 2011 - An Efficient Finite Element Model for Viscothermal.pdf:application/pdf} +} + +@article{bossart_hybrid_2003, + title = {Hybrid numerical and analytical solutions for acoustic boundary problems in thermo-viscous fluids}, + volume = {263}, + issn = {0022-460X}, + url = {http://www.sciencedirect.com/science/article/pii/S0022460X02010982}, + doi = {10.1016/S0022-460X(02)01098-2}, + abstract = {The present work aims at contributing to the investigation of methods to solve some classes of problems of acoustic propagation in thermo-viscous fluids, in unbounded or bounded media. The focus here is on thermal and vortical diffusion at the boundaries, which have to be considered for an accurate description of the acoustic field in small fluid-filled cavities and ducts. Existing boundary element or finite element acoustic software does not include these phenomena, as they are not compatible with the basic equations involved. A methodology is given to solve such problems when using this software, introducing a hybrid method which combines both numerical solutions and analytical solutions (for the fields inside the boundary layers). A detailed application is presented to validate the process using a boundary elements method.}, + pages = {69--84}, + number = {1}, + journaltitle = {Journal of Sound and Vibration}, + author = {Bossart, R. and Joly, N. and Bruneau, M.}, + urldate = {2012-03-08}, + date = {2003-05-22}, + file = {Bossart et al. - 2003 - Hybrid numerical and analytical solutions for acou.pdf:/home/anne/.literature/storage/5HTQ7MAA/Bossart et al. - 2003 - Hybrid numerical and analytical solutions for acou.pdf:application/pdf;ScienceDirect Snapshot:/home/anne/.literature/storage/VFCQVD9W/Bossart et al. - 2003 - Hybrid numerical and analytical solutions for acou.html:text/html} +} + +@article{christensen_modeling_2011, + title = {Modeling the Effects of Viscosity and Thermal Conduction on Acoustic Propagation in Rigid Tubes with Various Cross-Sectional Shapes}, + volume = {97}, + doi = {10.3813/AAA.918398}, + abstract = {When modeling acoustics with viscothermal effects included, typically of importance for narrow tubes and slits, one can often use the so-called low reduced frequency model. With this model a characteristic length is assumed for which the sound pressure is constant. For example for a circular cylindrical tube the characteristic length is the radius. A triangular cross-section does not have a characteristic length, but as will be shown in this paper the model can in fact be used as long as 1) the cross-sectional pressure is constant and 2) a characteristic impedance and propagation wavenumber can be established for the geometry. These parameters can be found for a tube with a triangular cross-section and an implementation of the low reduced frequency which can handle tubes with both circular, rectangular triangular cross-sections has been made in {COMSOL} Multiphysics. For the circular and the rectangular tube results found using this implementation have been compared to results from an analytical model, a so-called full Navier-Stokes implementation in {COMSOL} Multiphysics and the commercial package {FFT} {ACTRAN} which also uses the low reduced frequency model. The triangular tube implementation has been compared to the analytical case as well as the full Navier-Stokes implementation.}, + pages = {193--201}, + number = {2}, + journaltitle = {Acta Acustica united with Acustica}, + author = {Christensen, René}, + date = {2011}, + file = {Christensen - 2011 - Modeling the Effects of Viscosity and Thermal Cond.pdf:/home/anne/.literature/storage/I7TPZTVK/Christensen - 2011 - Modeling the Effects of Viscosity and Thermal Cond.pdf:application/pdf;ingentaconnect Modeling the Effects of Viscosity and Thermal Conduction on Acous...:/home/anne/.literature/storage/5KPM7UNC/art00002.html:text/html} +} + +@thesis{nijhof_viscothermal_2010, + location = {E}, + title = {Viscothermal wave propagation}, + institution = {University of Twente}, + type = {phdthesis}, + author = {Nijhof, M. J. J.}, + date = {2010}, + file = {Nijhof - 2010 - Viscothermal wave propagation.pdf:/home/anne/.literature/storage/F4WV3C2J/Nijhof - 2010 - Viscothermal wave propagation.pdf:application/pdf} +} + +@book{blackstock_fundamentals_2000, + location = {Hoboken, {NJ}, {USA}}, + title = {Fundamentals of physical acoustics}, + pagetotal = {541}, + publisher = {John Wiley \& Sons}, + author = {Blackstock, D.T.}, + date = {2000} +} + +@thesis{kampinga_viscothermal_2010, + location = {Enschede, The Netherlands}, + title = {Viscothermal acoustics using finite elements: analysis tools for engineers}, + institution = {University of Twente}, + type = {phdthesis}, + author = {Kampinga, W.R.}, + date = {2010}, + file = {Kampinga - 2010 - Viscothermal acoustics using finite elements anal.pdf:/home/anne/.literature/storage/E596NW5B/Kampinga - 2010 - Viscothermal acoustics using finite elements anal.pdf:application/pdf} +} + +@misc{ward_deltaec_2017, + title = {{DeltaEC} Users Guide version 6.4b2.7}, + url = {www.lanl.gov/thermoacoustics}, + author = {Ward, W. C. and Clark, J. P. and Swift, G. W.}, + urldate = {2018-01-22}, + date = {2017-12-04}, + file = {Ward e.a. - 2017 - DeltaEC Users Guide version 6.4b2.7.pdf:/home/anne/.literature/storage/MQKGHJ9I/Ward e.a. - 2017 - DeltaEC Users Guide version 6.4b2.7.pdf:application/pdf} +} + +@thesis{van_der_eerden_noise_2000, + location = {Enschede, The Netherlands}, + title = {Noise reduction with coupled prismatic tubes}, + abstract = {The present investigation focuses on an accurate description of sound absorption. Within this research a new technique to create sound absorption for a predefined frequency band has been developed. Additionally, a simple and efficient numerical model for conventional sound absorbing materials, such as glass wool or foams, has been formulated. It is also demonstrated that the newly gained insights are useful in applications not directly related to sound absorption.}, + institution = {University of Twente}, + type = {phdthesis}, + author = {van der Eerden, F.J.M.}, + date = {2000-11}, + file = {Eerden - 2000 - Noise reduction with coupled prismatic tubes.pdf:/home/anne/.literature/storage/JTAD25WQ/Eerden - 2000 - Noise reduction with coupled prismatic tubes.pdf:application/pdf} +} + +@article{rienstra_introduction_2015, + title = {An introduction to acoustics}, + volume = {18}, + pages = {296}, + journaltitle = {Eindhoven University of Technology}, + author = {Rienstra, Sjoerd W and Hirschberg, Avraham}, + date = {2015}, + file = {Rienstra and Hirschberg - 2015 - An introduction to acoustics.pdf:/home/anne/.literature/storage/7E6VWDZ4/Rienstra and Hirschberg - 2015 - An introduction to acoustics.pdf:application/pdf} +} + +@thesis{de_jong_numerical_2015, + location = {Enschede}, + title = {Numerical modeling of thermoacoustic systems}, + rights = {All rights reserved}, + url = {http://doc.utwente.nl/96275/}, + abstract = {The subject of this thesis is a relatively new class of heat engines and refrigerators, called thermoacoustic ({TA}) systems. {TA} systems have gained commercial interest due to their low number of moving parts and potentially high efficiency. In the case of a {TA} engine, heat is converted to acoustic power. This power can subsequently be converted to electricity using a ?reversed? loudspeaker, called a linear alternator. In a {TA} refrigerator, a speaker or linear alternator is used to generate a strong acoustic wave, which is used to pump heat. To achieve competitive power densities, thermoacoustic systems are generally run at such high amplitudes, that performance deteriorating nonlinear effects can no longer be neglected. To accu- rately predict performance in the nonlinear regime, nonlinear models are required. This thesis describes two contributions to the field of thermoacoustic system modeling. Firstly, a one-dimensional heat transfer model has been developed. This model can be used to estimate the performance of often used parallel-plate heat exchangers for thermoacoustic systems. These heat exchangers are located close to the stack or regenerator of a {TA} system and are responsible for the heat in/output required to let the system execute its thermodynamic cycle. The results of the model show a good match with a different heat transfer model from the literature, and the model provides guidelines for future heat exchanger design. Secondly, a nonlinear frequency domain method is developed with which the initial transient start-up process can be skipped in the simulations. The method can be used to directly simulate a {TA} system in its periodic steady-state. This significantly reduces computational cost, since the initial transient regime often involves several hundred oscillation cycles. The method is applied to a one-dimensional nonlinear model of {TA} systems. The model is used to simulate an experimental standing wave thermoacoustic engine from the literature. The obtained results are in agreement with literature results.}, + institution = {Universiteit Twente}, + type = {phdthesis}, + author = {De Jong, J.A.}, + date = {2015}, + keywords = {my}, + file = {Jong - 2015 - Numerical modeling of thermoacoustic systems.pdf:/home/anne/.literature/storage/GQTWDG7B/Jong - 2015 - Numerical modeling of thermoacoustic systems.pdf:application/pdf} +} + +@article{aarts_approximation_2003, + title = {Approximation of the Struve function H1 occurring in impedance calculations}, + volume = {113}, + issn = {0001-4966}, + url = {http://asa.scitation.org/doi/10.1121/1.1564019}, + doi = {10.1121/1.1564019}, + pages = {2635--2637}, + number = {5}, + journaltitle = {The Journal of the Acoustical Society of America}, + author = {Aarts, Ronald M. and Janssen, Augustus J. E. M.}, + urldate = {2017-10-21}, + date = {2003-05}, + langid = {english}, + file = {Aarts and Janssen - 2003 - Approximation of the Struve function H1 occurring .pdf:/home/anne/.literature/storage/LUM2PYYD/Aarts and Janssen - 2003 - Approximation of the Struve function H1 occurring .pdf:application/pdf} +} + +@thesis{christensen_acoustic_2010, + title = {Acoustic Modeling of Hearing Aid Components}, + institution = {Syddansk Universitet}, + type = {phdthesis}, + author = {Christensen, René}, + date = {2010}, + file = {Christensen - 2010 - Acoustic Modeling of Hearing Aid Components.pdf:/home/anne/.literature/storage/ABISVJM8/Christensen - 2010 - Acoustic Modeling of Hearing Aid Components.pdf:application/pdf} +} + +@article{karal_analogous_1953, + title = {The analogous acoustical impedance for discontinuities and constrictions of circular cross section}, + volume = {25}, + pages = {327--334}, + number = {2}, + journaltitle = {The Journal of the Acoustical Society of America}, + author = {Karal, {FC}}, + date = {1953}, + 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}, + pages = {616--623}, + number = {2}, + journaltitle = {The Journal of the Acoustical Society of America}, + author = {Benade, Arthur H}, + date = {1968}, + file = {Benade - 1968 - On the propagation of sound waves in a cylindrical.pdf:/home/anne/.literature/storage/E4GR6AXF/Benade - 1968 - On the propagation of sound waves in a cylindrical.pdf:application/pdf} +} + +@book{morse_theoretical_1968, + title = {Theoretical acoustics}, + publisher = {{McGraw}-Hill International Edition}, + author = {Morse, Philip {McCord} and Ingard, K Uno}, + date = {1968}, + file = {Morse en Ingard - 1968 - Theoretical acoustics.pdf:/home/anne/.literature/storage/UD52GPB6/Morse en Ingard - 1968 - Theoretical acoustics.pdf:application/pdf} +} + +@article{tsilingiris_thermophysical_2008, + title = {Thermophysical and transport properties of humid air at temperature range between 0 and 100 C}, + volume = {49}, + pages = {1098--1110}, + number = {5}, + journaltitle = {Energy Conversion and Management}, + author = {Tsilingiris, {PT}}, + date = {2008}, + file = {Tsilingiris - 2008 - Thermophysical and transport properties of humid a.pdf:/home/anne/.literature/storage/65SE4BDX/Tsilingiris - 2008 - Thermophysical and transport properties of humid a.pdf:application/pdf} +} + +@article{cramer_variation_1993, + title = {The variation of the specific heat ratio and the speed of sound in air with temperature, pressure, humidity, and {CO}2 concentration}, + volume = {93}, + pages = {2510--2516}, + number = {5}, + journaltitle = {The Journal of the Acoustical Society of America}, + author = {Cramer, Owen}, + date = {1993}, + file = {Cramer - 1993 - The variation of the specific heat ratio and the s.pdf:/home/anne/.literature/storage/ZCGL7MPK/Cramer - 1993 - The variation of the specific heat ratio and the s.pdf:application/pdf} +} + +@book{young_roarks_2002, + title = {Roark's formulas for stress and strain}, + volume = {7}, + publisher = {{McGraw}-Hill New York}, + author = {Young, Warren Clarence and Budynas, Richard Gordon}, + date = {2002}, + file = {Young en Budynas - 2002 - Roark's formulas for stress and strain.pdf:/home/anne/.literature/storage/5VJDJLYP/Young en Budynas - 2002 - Roark's formulas for stress and strain.pdf:application/pdf} +} + +@inproceedings{kuipers_investigations_2017, + location = {Kiel, Germany}, + title = {Investigations on acoustic radiation by hearing aid tubes}, + eventtitle = {{DAGA} 2017}, + author = {Kuipers, Erwin Reinder and Westhausen, Nils}, + date = {2017} +} + +@article{welch_use_1967, + title = {The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms}, + volume = {15}, + pages = {70--73}, + number = {2}, + journaltitle = {{IEEE} Transactions on audio and electroacoustics}, + author = {Welch, Peter}, + date = {1967} +} \ No newline at end of file diff --git a/lrftubes.lyx b/lrftubes.lyx new file mode 100644 index 0000000..3e2a229 --- /dev/null +++ b/lrftubes.lyx @@ -0,0 +1,6915 @@ +#LyX 2.2 created this file. 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Plain Layout + +% Roman (A) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Unused: +\end_layout + +\begin_layout Plain Layout +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$\\mathbf{e}_x$" +description "Unit vector in $x$-direction\\nomunit{-}" + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$H$" +description "Total enthalpy per unit mass \\nomunit{\\si{\\joule\\per\\kilogram}}" + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$\\mathbf{I}$" +description "Identity tensor\\nomunit{-}" + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$h_\\nu$" +description "Viscothermal shape function for the velocity\\nomunit{-}" + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$h_\\kappa$" +description "Viscothermal shape function for the temperature\\nomunit{-}" + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$E$" +description "Total energy per unit mass \\nomunit{\\si{\\joule\\per\\kilogram}}" + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$c$" +description "Speed of sound\\nomunit{\\si{\\metre\\per\\second}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$c_p$" +description "Specific heat at constant pressure \\nomunit{\\si{\\joule\\per\\kilogram\\kelvin}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$c_s$" +description "Specific heat of the solid\\nomunit{\\si{\\joule\\per\\kilogram\\kelvin}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$c_v$" +description "Specific heat at constant density \\nomunit{\\si{\\joule\\per\\kilogram\\kelvin}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$D$" +description "Diameter\\nomunit{\\si{\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$f_\\kappa$" +description "Thermal Rott function \\nomunit{-}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$f_\\nu$" +description "Viscous Rott function \\nomunit{-}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$f$" +description "Frequency\\nomunit{\\si{\\hertz}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$i$" +description "Imaginary unit\\nomunit{-}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$J_\\alpha$" +description "Bessel function of the first kind and order $\\alpha$\\nonomunit" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$k$" +description "Wave number\\nomunit{\\si{\\radian\\per\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$L$" +description "Length\\nomunit{\\si{\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$\\ell$" +description "Characteristic length scale of a fluid space \\nomunit{\\si{\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$N$" +description "Number of\\nomunit{-}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$\\mathbf{n}$" +description "Normal vector pointing from the solid into the fluid\\nomunit{-}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$p$" +description "Pressure, acoustic pressure \\nomunit{\\si{\\pascal}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$r_h$" +description "Hydraulic radius \\nomunit{\\si{\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$\\mathbf{r}$" +description "Transverse position vector\\nomunit{-}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$S$" +description "Cross-sectional area, surface area\\nomunit{\\si{\\square\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$t$" +description "Time \\nomunit{\\si{\\second}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$T$" +description "Temperature\\nomunit{\\si{\\kelvin}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$\\mathbf{u}$" +description "Velocity vector\\nomunit{\\si{\\metre\\per\\second}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$u$" +description "Velocity in wave propagation direction\\nomunit{\\si{\\metre\\per\\second}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$U$" +description "Volume flow\\nomunit{\\si{\\cubic\\metre\\per\\second}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$V$" +description "Volume \\nomunit{\\si{\\cubic\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$\\mathbf{x}$" +description "Position vector \\nomunit{\\si{\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$z$" +description "Specific acoustic impedance\\nomunit{\\si{\\pascal\\second\\per\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$Z$" +description "Volume flow impedance\\nomunit{\\si{\\pascal\\second\\per\\cubic\\metre}}" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + +% Greek (G) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Unused: +\end_layout + +\begin_layout Plain Layout +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\Delta$" +description "Difference\\nonomunit" + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "G" +symbol "$\\gamma$" +description "Ratio of specific heats\\nomunit{-}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "G" +symbol "$\\Gamma$" +description "Viscothermal wave number for a prismatic duct \\nomunit{\\si{\\radian\\per\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "G" +symbol "$\\delta_{\\kappa}$" +description "Thermal penetration depth\\nomunit{\\si{\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "G" +symbol "$\\delta_{\\nu}$" +description "Viscous penetration depth\\nomunit{\\si{\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "G" +symbol "$\\epsilon_s$" +description "Ideal stack correction factor \\nomunit{-}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "G" +symbol "$\\lambda$" +description "Wavelength \\nomunit{\\si{\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "G" +symbol "$\\pi$" +description "Ratio of the circumference to the diameter of a circle \\nomunit{-}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "G" +symbol "$\\Pi$" +description "Wetted perimeter (contact length between solid and fluid) \\nomunit{\\si{\\metre}}" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + +% Miscellaneous symbols and operators (M) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Unused: +\end_layout + +\begin_layout Plain Layout +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\otimes$" +description "Dyadic product\\nonomunit" + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\Re$" +description "Real part\\nonomunit" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\Im$" +description "Imaginary part\\nonomunit" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\nabla$" +description "Gradient \\nomunit{\\si{\\per\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\nabla^2$" +description "Laplacian\\nomunit{\\si{\\per\\square\\metre}}" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\sim$" +description "Same order of magnitude\\nonomunit" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\left\\Vert \\bullet \\right\\Vert $" +description "Eucledian norm\\nonomunit" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "d" +description "Infinitesimal\\nonomunit" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\partial$" +description "Infinitesimal\\nonomunit" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "M" +symbol "$\\bullet$" +description "Placeholder for an operand\\nonomunit" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + +% Subscripts (S) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "wall" +description "At the wall" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "0" +description "Evaluated at the reference condition" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$f$" +description "Fluid" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$s$" +description "Solid" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$w$" +description "Wall" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$R$" +description "Right side" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$L$" +description "Left side" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$s$" +description "Solid" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$s$" +description "Squeeze" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$i$" +description "Inner" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$o$" +description "Outer" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "S" +symbol "$t$" +description "Tube" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + +% Often used abbreviations (O) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "O" +symbol "Sec(s)." +description "Section(s)" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "O" +symbol "Eq(s)." +description "Equation(s)" + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "O" +symbol "LRF" +description "Low Reduced Frequency" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +printnomenclature[1.8cm] +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Chapter +Overview of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +lrftubes +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Introduction +\end_layout + +\begin_layout Standard +Welcome to the documentation of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +lrftubes +\end_layout + +\end_inset + +. + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +lrftubes +\backslash + +\end_layout + +\end_inset + + is a numerical code to solve one-dimensional acoustic duct systems using + the transfer matrix method. + Segments can be connected to generate simple one-dimensional acoustic systems + to model acoustic propagation problems in ducts in the frequency domain. + Viscothermal dissipation mechanisms are taken into account such that the + damping effects can be modeled accurately, below the cut-on frequency of + the duct. + For more information regarding the models and the theory behind the models, + the reader is referred to the work of +\begin_inset CommandInset citation +LatexCommand cite +key "van_der_eerden_noise_2000" + +\end_inset + +, +\begin_inset CommandInset citation +LatexCommand cite +key "kampinga_viscothermal_2010" + +\end_inset + + and +\begin_inset CommandInset citation +LatexCommand cite +key "ward_deltaec_2017" + +\end_inset + +. +\end_layout + +\begin_layout Standard +This documentation serves as a reference for the implemented models. + For examples on how to use the code, please take a look at the example + models as worked out in the IPython Notebooks. + For installation instructions, please refer the the +\begin_inset CommandInset href +LatexCommand href +name "README" +target "https://github.com/asceenl/lrftubes" + +\end_inset + + in the main repository. +\end_layout + +\begin_layout Standard +This document is very brief on the theory and it is assumed that the reader + has some knowledge on the basics of acoustics in general and viscothermal + acoustics as well. + If you are not falling in this category, I would please refer you first + to the book of Swift +\begin_inset CommandInset citation +LatexCommand cite +key "swift_thermoacoustics:_2003" + +\end_inset + +. + A more detailed introduction to the notation used in this documentation + can be found in the PhD thesis of de Jong +\begin_inset CommandInset citation +LatexCommand cite +key "de_jong_numerical_2015" + +\end_inset + +. +\end_layout + +\begin_layout Standard +Besides that, if you find the work interesting, but you are not sure how + to apply it, please contact ASCEE for more information. +\end_layout + +\begin_layout Section +License and disclaimer +\end_layout + +\begin_layout Standard +Redistribution and use in source and binary forms are permitted provided + that the above copyright notice and this paragraph are duplicated in all + such forms and that any documentation, advertising materials, and other + materials related to such distribution and use acknowledge that the software + was developed by the ASCEE. + The name of the ASCEE may not be used to endorse or promote products derived + from this software without specific prior written permission. +\begin_inset Newline newline +\end_inset + + +\end_layout + +\begin_layout Standard +THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR IMPLIED WARRANTIE +S, INCLUDING, WITHOUT LIMITATION, THE IMPLIED WARRANTIES OF MERCHANTABILITY + AND FITNESS FOR A PARTICULAR PURPOSE. +\end_layout + +\begin_layout Section +Features +\end_layout + +\begin_layout Standard +Currently the +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +lrftubes +\end_layout + +\end_inset + + code provides acoustic models for the following physical entities: +\end_layout + +\begin_layout Itemize +Prismatic ducts with circular cross section, +\end_layout + +\begin_layout Itemize +Prismatic ducts with triangular cross section, +\end_layout + +\begin_layout Itemize +Prismatic ducts with parallel plate cross section, +\end_layout + +\begin_layout Itemize +Prismatic ducts with square cross section, +\end_layout + +\begin_layout Itemize +Acoustic compliance volumes +\end_layout + +\begin_layout Itemize +Discontinuity correction +\end_layout + +\begin_layout Itemize +End correction for a baffled piston +\end_layout + +\begin_layout Itemize +Lumped series impedance +\end_layout + +\begin_layout Standard +These segments can be connected to form one-dimensional acoustic systems + to model wave propagation below the cut-on frequency of higher order modes. + For a circular cross section, the cut-on frequency is +\begin_inset CommandInset citation +LatexCommand cite +key "van_der_eerden_noise_2000" + +\end_inset + +: +\begin_inset Formula +\begin{equation} +f_{c}\approx\frac{c_{0}}{3.4r}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $r$ +\end_inset + + is the tube radius and +\begin_inset Formula $c_{o}$ +\end_inset + + is the speed of sound. + Above the cut-on frequency, besides evanescent waves, there are also propagatin +g waves with a non-constant pressure distribution along the cross section + of the duct. +\end_layout + +\begin_layout Subsection +Limitations and future features +\end_layout + +\begin_layout Standard +The current version of has some limitations that will be resolved in a future + release. + These are: +\end_layout + +\begin_layout Subsubsection +Ducts with (turbulent) flow +\end_layout + +\begin_layout Standard +For thermoacoustic and HVAC (Heating, ventilation and Air Conditioning) + duct modeling it is imperative that mean flows can be taken into account. + An acoustic wave superimposed on a mean flow results in asymmetric wave + propagation. + More specifically, the phase velocity is higher in the direction of the + mean flow, and slower in the opposite direction. + In a future release, we will provide models for ducts including a mean + flow. + +\end_layout + +\begin_layout Subsubsection +Porous acoustic absorbers +\end_layout + +\begin_layout Standard +To model absorption of sound, a one-dimensional porous material model should + be implemented. + This work has been postponed to a later stage. +\end_layout + +\begin_layout Section +Overview of this documentation +\end_layout + +\begin_layout Standard +The next chapter of this documentation will describe the basic framework + of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +lrftubess +\end_layout + +\end_inset + + code: the transfer matrix method. + After that, in Chapter +\begin_inset CommandInset ref +LatexCommand ref +reference "chap:Provided-acoustic-models" + +\end_inset + +, an overview of the provided acoustic models is given, with which acoustic + networks can be built. + For each of the segments, the resulting transfer matrix model is derived. +\end_layout + +\begin_layout Chapter +The transfer matrix method +\end_layout + +\begin_layout Section +Introduction +\end_layout + +\begin_layout Standard +Each part of an acoustic system in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +lrftubess +\end_layout + +\end_inset + + is modeled using a so-called transfer matrix. + A transfer matrix maps the state quantities on one side of the segment + (node) to the other side of the segment (node). +\end_layout + +\begin_layout Standard +For one-dimensional wave propagation, analytical solutions for the velocity, + temperature and density field in the transverse direction can be found. + The state variables in frequency domain satisfy a system of first order + ordinary differential equations. + Once the solution is known on one end of a segment, the solution on the + other end can be deduced. + The transfer matrix couples the state variables +\begin_inset Formula $\boldsymbol{\phi}$ +\end_inset + + on one end of a segment to the other end, in frequency domain: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\boldsymbol{\phi}_{R}(\omega)=\boldsymbol{T}(\omega)\boldsymbol{\phi}_{L}(\omega)+\mathbf{s}(\omega), +\end{equation} + +\end_inset + +where +\begin_inset Formula $L$ +\end_inset + + and +\begin_inset Formula $R$ +\end_inset + + denote the left and right side, respectively, +\begin_inset Formula $\boldsymbol{T}$ +\end_inset + + denotes the transfer matrix and +\begin_inset Formula $\boldsymbol{s}$ +\end_inset + + is a source term. + In the code and in this documentation +\begin_inset Formula $e^{+i\omega t}$ +\end_inset + + convention is used. + A common choice of state variables is such that their product has the unit + of power. + For the acoustic systems in this work the state variables are acoustic + pressure +\begin_inset Formula $p\left(\omega\right)$ +\end_inset + + and volume flow +\begin_inset Formula $U\left(\omega\right)$ +\end_inset + +. + The acoustic power flow can then be computed as: +\begin_inset Formula +\begin{equation} +E=\frac{1}{2}\Re\left[pU^{*}\right], +\end{equation} + +\end_inset + +where +\begin_inset Formula $\Re[\bullet]$ +\end_inset + + denotes the real part of +\begin_inset Formula $\bullet$ +\end_inset + +, and * denotes the complex conjugation. +\end_layout + +\begin_layout Section +Example transfer matrix of an acoustic duct +\end_layout + +\begin_layout Standard +This section will provide the derivation of the transfer matrix of a simple + acoustic duct. + Starting with the isentropic acoustic continuity and momentum equation + : +\begin_inset Formula +\begin{align} +\frac{1}{c_{0}^{2}}\frac{\partial\hat{p}}{\partial\hat{t}}+\rho_{0}\nabla\cdot\hat{\boldsymbol{u}} & =0,\\ +\rho_{0}\frac{\partial\hat{\boldsymbol{u}}}{\partial t}+\nabla\hat{p} & =0. +\end{align} + +\end_inset + +The next step is to transform these equations to frequency domain and assuming + only wave propagation in the +\begin_inset Formula $x-$ +\end_inset + +direction, integrating over the cross section we find: +\begin_inset Formula +\begin{align} +\frac{i\omega}{c_{0}^{2}}p+\frac{\rho_{0}}{S_{f}}\frac{\mathrm{d}U}{\mathrm{d}x} & =0,\label{eq:contU}\\ +\rho_{0}i\omega U+S_{f}\frac{\mathrm{d}p}{\mathrm{d}x} & =0,\label{eq:momU} +\end{align} + +\end_inset + +where +\begin_inset Formula $U$ +\end_inset + + denotes the acoustic volume flow in +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +si{ +\backslash +cubic +\backslash +metre +\backslash +per +\backslash +second} +\end_layout + +\end_inset + +. + Eqs. + ( +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:contU" + +\end_inset + +- +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:momU" + +\end_inset + +) is a coupled set of ordinary differential equations, which can be solved + for the acoustic pressure to find +\begin_inset Formula +\begin{equation} +p(x)=A\exp\left(-ikx\right)+B\exp\left(ikx\right),\label{eq:HH_sol_prismaticinviscid} +\end{equation} + +\end_inset + +where +\begin_inset Formula $A$ +\end_inset + + and +\begin_inset Formula $B$ +\end_inset + + are constants, to be determined from the boundary conditions. + Setting +\begin_inset Formula $p=p_{L}$ +\end_inset + +, and +\begin_inset Formula $U=U_{L}$ +\end_inset + + at +\begin_inset Formula $x=0$ +\end_inset + +, we can solve for the acoustic pressure, upon using Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:momU" + +\end_inset + + as: +\begin_inset Formula +\begin{equation} +p(x)=p_{L}\cos\left(kx\right)-iZ_{0}\sin\left(kx\right)U_{L}, +\end{equation} + +\end_inset + +and for the acoustic volume flow we find: +\begin_inset Formula +\begin{equation} +U(x)=U_{L}\cos\left(kx\right)-\frac{i}{Z_{0}}\sin\left(kx\right)p_{L}. +\end{equation} + +\end_inset + +Now, we have all ingredients to derive the transfer matrix of an acoustic + duct. + Setting +\begin_inset Formula $p(x=L)=p_{R}$ +\end_inset + +, and +\begin_inset Formula $U(x=L)=U_{R}$ +\end_inset + +, we find the following two-port coupling between the pressure and the velocity + from the left side of the duct to the right side of the duct: +\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\} .\label{eq:transfer_inviscid} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Setting up the system of equations +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +lrftubes +\end_layout + +\end_inset + + has been set up to solve systems of acoustic segments such as this prismatic + duct. + The advantage of the transfer matrix method is the ease with which mixed + (impedance/pressure/velocity) boundary conditions can be implemented. +\end_layout + +\begin_layout Standard +In this section, the assembly of the global system of equations is explained. + The state variables of each segment are stacked in a column vector +\series bold + +\begin_inset Formula $\boldsymbol{\phi}_{\mbox{sys}}$ +\end_inset + + +\series default +, which has the size of +\begin_inset Formula $4N_{\mbox{segs}}$ +\end_inset + +, where +\begin_inset Formula $N_{\mbox{segs}}$ +\end_inset + + denotes the number of segments in the system. + The coupling equations between the nodes of each segment, are the transfer + matrices. + Since the transfer matrices are +\begin_inset Formula $2\times2$ +\end_inset + +, this fills only half of the required amount of equations. + The other half is filled with boundary conditions. + Each segments transfer matrix can be regarded as the element matrix, which + all have a form like: +\begin_inset Formula +\begin{equation} +\boldsymbol{\phi}_{R}=\boldsymbol{T}\cdot\boldsymbol{\phi}_{L}+\boldsymbol{s}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\boldsymbol{\phi}_{L},\boldsymbol{\phi}_{R}$ +\end_inset + + are the state vectors on the left and right sides of the segment, respectively, + +\begin_inset Formula $\boldsymbol{T}$ +\end_inset + + is the transfer matrix, and +\begin_inset Formula $\boldsymbol{s}$ +\end_inset + + is a source term. +\end_layout + +\begin_layout Standard +There are two kind of boundary conditions, called external and internal + boundary conditions. + External boundary conditions apply where a prescribed condition is given, + such as a prescribed pressure, voltage, volume flow, current or acoustic/electr +ic impedance. + Internal boundary conditions are used to couple different segments at a + connection point, which is recognized by a shared node number. + At a connection point, the effort variable is shared, which means that + the pressure at the node is equal for each connected segment sharing the + node. + The flow variable is conserved, so the sum of the volume flow out of all + segments connected at the node is 0. +\end_layout + +\begin_layout Subsection* +Example: two ducts +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename img/tfm_expl.pdf + width 80text% + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Example of two simple duct segments connected together. +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:coupling_example" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +This procedure of creating a system matrix is explained by an example where + only two ducts are coupled. + A schematic of the situation is depicted in Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:coupling_example" + +\end_inset + +. + For the example situation, at the left node of segment (1), an impedance + boundary +\begin_inset Formula $Z_{L}$ +\end_inset + + is prescribed. + The right node of segment (1) is connected to the left node of segment + (2), and at the right side of segment (2), a volume flow boundary condition + is prescribed of +\begin_inset Formula $U_{R}$ +\end_inset + +. + The corresponding system of equations for this case is +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\left[\begin{array}{cccc} +\mathbf{T}_{1} & -\mathbf{I} & \mathbf{0} & \mathbf{0}\\ +\mathbf{0} & \mathbf{0} & \mathbf{T}_{2} & -\mathbf{I}\\ +\mathbf{0} & \left[\begin{array}{cc} +1 & 0\\ +0 & 1 +\end{array}\right] & \left[\begin{array}{cc} +-1 & 0\\ +0 & -1 +\end{array}\right] & \mathbf{0}\\ +\left[\begin{array}{cc} +1 & Z_{L}\\ +0 & 0 +\end{array}\right] & \mathbf{0} & \mathbf{0} & \left[\begin{array}{cc} +0 & 0\\ +0 & 1 +\end{array}\right] +\end{array}\right]\left\{ \begin{array}{c} +p_{1L}\\ +U_{1L}\\ +p_{1R}\\ +U_{1R}\\ +p_{2L}\\ +U_{2L}\\ +p_{2R}\\ +U_{2R} +\end{array}\right\} =\left\{ \begin{array}{c} +0\\ +0\\ +0\\ +0\\ +0\\ +0\\ +0\\ +U_{R} +\end{array}\right\} , +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +In this system matrix, +\begin_inset Formula $\mathbf{0}$ +\end_inset + + denotes a +\begin_inset Formula $2\times2$ +\end_inset + + sub matrix of zeros and +\begin_inset Formula $\mathbf{I}$ +\end_inset + + denotes a +\begin_inset Formula $2\times2$ +\end_inset + + identity sub matrix. + +\begin_inset Formula $\mathbf{T}_{i}$ +\end_inset + + is the transfer matrix of the +\begin_inset Formula $i$ +\end_inset + +-th segment. + The solution can be obtained by Gaussian elimination, for which in +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +lrftubess +\end_layout + +\end_inset + + the +\family typewriter +numpy.linalg.solve() +\family default + solver is used. + Once the solution on the nodes is known, the solution in each segment can + be computed as a post processing step. + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +lrftubess +\end_layout + +\end_inset + + provides some post processing routines to aid in visualization of the acoustic + field inside a non-lumped segment, such as an acoustic duct. +\end_layout + +\begin_layout Chapter +Provided acoustic models +\begin_inset CommandInset label +LatexCommand label +name "chap:Provided-acoustic-models" + +\end_inset + + +\end_layout + +\begin_layout Section +Introduction +\end_layout + +\begin_layout Standard +This chapter provides a concise overview of the provided acoustic models + implemented in +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +lrftubes +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Section +Prismatic duct +\begin_inset CommandInset label +LatexCommand label +name "subsec:Prismatic-duct" + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename img/prsduct.pdf + width 80text% + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Geometry of the prismatic duct +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:prsduct" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +A prismatic duct is used to model one-dimensional acoustic wave propagation. + The prismatic duct is implemented in +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +lrftubess +\end_layout + +\end_inset + + in the +\family typewriter +PrsDuct +\family default + class. + Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:prsduct" + +\end_inset + + shows this segment schematically. + In the thermal boundary layer, heat and momentum diffuse to the wall. + The thermal boundary layer can be a small layer w.r.t. + to the transverse characteristic length scale of the tube, or can fully + occupy the tube. + In the latter case, the solution converges to the classic laminar Poisseuille + flow solution. + The basic assumptions behind this model are +\end_layout + +\begin_layout Itemize +Prismatic cross sectional area. +\end_layout + +\begin_layout Itemize +\begin_inset Formula $L\gg r_{h}$ +\end_inset + +, (tube is long compared to its transverse length scale). +\end_layout + +\begin_layout Itemize +Radius is much smaller than the wave length. +\end_layout + +\begin_layout Itemize +Wave length is much larger than viscous penetration depth. +\end_layout + +\begin_layout Itemize +End effects and entrance effects are negligible. +\end_layout + +\begin_layout Standard +For a formal derivation of the model for prismatic cylindrical tubes, the + reader is referred to the work of Tijdeman +\begin_inset CommandInset citation +LatexCommand cite +key "tijdeman_propagation_1975" + +\end_inset + + and Nijhof +\begin_inset CommandInset citation +LatexCommand cite +key "nijhof_viscothermal_2010" + +\end_inset + +. + For a somewhat more pragmatic derivation, we would like to refer to the + work of Swift +\begin_inset CommandInset citation +LatexCommand cite +key "swift_thermoacoustics:_2003,swift_thermoacoustic_1988" + +\end_inset + + and Rott +\begin_inset CommandInset citation +LatexCommand cite +key "rott_damped_1969" + +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +\frac{\mathrm{d}p}{\mathrm{d}x} & =\frac{\omega\rho_{0}}{i\left(1-f_{\nu}\right)S_{f}}U,\label{eq:momentum_LRF}\\ +\frac{\mathrm{d}U}{\mathrm{d}x} & =\frac{k}{iZ_{0}}\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)p,\label{eq:continuity_LRF} +\end{align} + +\end_inset + +where +\begin_inset Formula $S_{f}$ +\end_inset + + is the cross-sectional area filled with fluid, +\begin_inset Formula $k$ +\end_inset + + is the inviscid wave number, and +\begin_inset Formula $Z_{0}$ +\end_inset + + the inviscid characteristic impedance of a tube ( +\begin_inset Formula $Z_{0}=z_{0}/S_{f}$ +\end_inset + +). + +\begin_inset Formula $f_{\nu}$ +\end_inset + + and +\begin_inset Formula $f_{\kappa}$ +\end_inset + + are the viscous and thermal Rott functions, respectively +\begin_inset CommandInset citation +LatexCommand cite +key "rott_damped_1969" + +\end_inset + +. + They model the viscous and thermal effects with the wall. + For circular tubes, the +\begin_inset Formula $f$ +\end_inset + +'s are defined as +\begin_inset CommandInset citation +LatexCommand cite +after "p. 88" +key "swift_thermoacoustics:_2003" + +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +f_{j,\mathrm{circ}}=\frac{J_{1}\left[\left(i-1\right)\frac{2r_{h}}{\delta_{j}}\right]}{\left(i-1\right)\frac{r_{h}}{\delta}J_{0}\left[\left(i-1\right)\frac{2r_{h}}{\delta_{j}}\right]},\label{eq:f_cylindrical} +\end{equation} + +\end_inset + + +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$j$" +description "Index, subscript placeholder\\nomunit{-}" + +\end_inset + +where +\begin_inset Formula $\delta_{j}=\delta_{\nu}$ +\end_inset + + for +\begin_inset Formula $f_{\nu,\mathrm{circ}}$ +\end_inset + + and +\begin_inset Formula $\delta_{j}=\delta_{\kappa}$ +\end_inset + + for +\begin_inset Formula $f_{\kappa,\mathrm{circ}}$ +\end_inset + +. + +\begin_inset Formula $J_{\alpha}$ +\end_inset + + denotes the cylindrical Bessel function of the first kind and order +\begin_inset Formula $\alpha$ +\end_inset + +. + +\begin_inset Formula $r_{h}$ +\end_inset + + is the hydraulic radius, defined as the ratio of the cross sectional area + to the +\begin_inset Quotes eld +\end_inset + +wetted perimeter +\begin_inset Quotes erd +\end_inset + +: +\begin_inset Formula +\begin{equation} +r_{h}=S_{f}/\Pi. +\end{equation} + +\end_inset + +Note that for a circular tube with diameter +\begin_inset Formula $D$ +\end_inset + +, +\begin_inset Formula $r_{h}=\nicefrac{D}{4}$ +\end_inset + +. + The parameter +\begin_inset Formula $\epsilon_{s}$ +\end_inset + + in Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:continuity_LRF" + +\end_inset + + is the ideal solid correction factor, which corrects for solids that have + a finite heat capacity. + This parameter is dependent on the thermal properties and the geometry + of the solid. + An example of +\begin_inset Formula $\epsilon_{s}$ +\end_inset + + is derived in Section +\begin_inset CommandInset ref +LatexCommand ref +reference "subsec:Thermal-relaxation-effect" + +\end_inset + +. + For the case of an thermally ideal solid, +\begin_inset Formula $\epsilon_{s}$ +\end_inset + + can be set to 0. +\end_layout + +\begin_layout Standard +Upon solving for Eqs. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:momentum_LRF" + +\end_inset + +- +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:continuity_LRF" + +\end_inset + +, a transfer matrix can be derived which couples the pressure and volume + flow on the left side to the right side as: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula +\begin{align*} +\frac{\mathrm{d}p}{\mathrm{d}x} & =\frac{\omega\rho_{0}}{i\left(1-f_{\nu}\right)S_{f}}U,\\ +\frac{\mathrm{d}U}{\mathrm{d}x} & =\frac{k}{iZ_{0}}\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)p, +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +We know the solution for +\begin_inset Formula $p$ +\end_inset + + is +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p=A\exp\left(-i\Gamma x\right)+B\exp\left(i\Gamma x\right)$ +\end_inset + + where +\begin_inset Formula $\Gamma^{2}=k^{2}\frac{\left(1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\varepsilon_{s}}\right)}{1-f_{\nu}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Then +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U=\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\frac{\mathrm{d}p}{\mathrm{d}x}=\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\Gamma i\left(-A\exp\left(-i\Gamma x\right)+B\exp\left(i\Gamma x\right)\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U=-\frac{\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\Gamma\left(B\exp\left(i\Gamma x\right)-A\exp\left(-i\Gamma x\right)\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Now: +\begin_inset Formula $p(x=0)=p_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +And: +\begin_inset Formula $U(x=0)=U_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Then: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U_{L}=\frac{\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\Gamma\left(A-B\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p_{L}=A+B\Rightarrow B=p_{L}-A$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Hence: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U_{L}=\frac{\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\Gamma\left(2A-p_{L}\right)$ +\end_inset + + or +\begin_inset Formula $A=\frac{1}{2}p_{L}+\frac{1}{2}\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f}\Gamma}U_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +And: +\begin_inset Formula $B=p_{L}-A=\frac{1}{2}p_{L}-\frac{1}{2}\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f}\Gamma}U_{L}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +So, finally for +\begin_inset Formula $p$ +\end_inset + + we find: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p=\left(\frac{1}{2}p_{L}+\frac{1}{2}\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f}\Gamma}U_{L}\right)\exp\left(-i\Gamma x\right)+\left(\frac{1}{2}p_{L}-\frac{1}{2}\frac{\omega\rho_{0}}{\left(1-f_{\nu}\right)S_{f}\Gamma}U_{L}\right)\exp\left(i\Gamma x\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p=\left(\frac{1}{2}p_{L}+\frac{1}{2}Z_{c}U_{L}\right)\exp\left(-i\Gamma x\right)+\left(\frac{1}{2}p_{L}-\frac{1}{2}Z_{c}U_{L}\right)\exp\left(i\Gamma x\right)$ +\end_inset + + where +\begin_inset Formula $Z_{c}=\frac{kZ_{0}}{\left(1-f_{\nu}\right)\Gamma}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Or, working to transfer matrices +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p=\frac{1}{2}p_{L}\exp\left(-i\Gamma x\right)+\frac{1}{2}Z_{c}U_{L}\exp\left(-i\Gamma x\right)+\frac{1}{2}p_{L}\exp\left(i\Gamma x\right)-Z_{c}U_{L}\exp\left(i\Gamma x\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p=p_{L}\cos\left(\Gamma x\right)+\frac{1}{2}Z_{c}U_{L}\exp\left(-i\Gamma x\right)-Z_{c}U_{L}\exp\left(i\Gamma x\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Using the rule: +\begin_inset Formula $\sin\left(x\right)=\frac{1}{2i}\left(e^{ix}-e^{-ix}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $p=p_{L}\cos\left(\Gamma x\right)-iZ_{c}U_{L}\sin\left(\Gamma x\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Using +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $U=\frac{i\left(1-f_{\nu}\right)S_{f}}{\omega\rho_{0}}\frac{\mathrm{d}p}{\mathrm{d}x}=\frac{i}{Z_{c}}\left[-p_{L}\sin\left(\Gamma x\right)-iZ_{c}U_{L}\cos\left(\Gamma x\right)\right]=\left[-\frac{i}{Z_{c}}p_{L}\sin\left(\Gamma x\right)+U_{L}\cos\left(\Gamma x\right)\right]$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\left\{ \begin{array}{c} +p_{R}\\ +U_{R} +\end{array}\right\} =\left[\begin{array}{cc} +\cos\left(\Gamma L\right) & -iZ_{c}\sin\left(\Gamma L\right)\\ +-iZ_{c}^{-1}\sin\left(\Gamma L\right) & \cos\left(\Gamma L\right) +\end{array}\right]\left\{ \begin{array}{c} +p_{L}\\ +U_{L} +\end{array}\right\} ,\label{eq:transfer_matrix_prismatic_duct} +\end{equation} + +\end_inset + +where +\begin_inset Formula $Z_{c}$ +\end_inset + + is the characteristic impedance of the duct, i.e. + the impedance +\begin_inset Formula $p/U$ +\end_inset + + of a plane (although damped) propagating wave: +\begin_inset Formula +\begin{equation} +Z_{c}=\frac{kZ_{0}}{\left(1-f_{\nu}\right)\Gamma}.\label{eq:Z_c_prismduct} +\end{equation} + +\end_inset + +The parameter +\begin_inset Formula $\Gamma$ +\end_inset + + in Eqs. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:transfer_matrix_prismatic_duct" + +\end_inset + + and +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Z_c_prismduct" + +\end_inset + + is the viscothermal wave number, i.e. + the wave number corrected for viscothermal losses: +\begin_inset Formula +\begin{equation} +\Gamma=k\sqrt{\frac{1+\tfrac{\left(\gamma-1\right)f_{\kappa}}{1+\epsilon_{s}}}{1-f_{\nu}}}.\label{eq:Gamma} +\end{equation} + +\end_inset + +Due to the numerical implementation of the Bessel functions in many libraries, + the +\begin_inset Formula $f_{j}$ +\end_inset + + function for cylindrical ducts (Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:f_cylindrical" + +\end_inset + +) cannot be computed for high +\begin_inset Formula $r_{h}/\delta$ +\end_inset + + by computing this ratio +\begin_inset Formula $J_{1}/J_{0}$ +\end_inset + +. + The numerical result starts to break down at +\begin_inset Formula $r_{h}/\delta\sim100$ +\end_inset + +. + To resolve this problem, the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +lrftubess +\end_layout + +\end_inset + + code applies a smooth transition from the Bessel function ratio to the + boundary layer limit solution for +\begin_inset Formula $f$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +f_{j,\mathrm{bl}}=\frac{\left(1-i\right)\delta_{j}}{2r_{h}} +\end{equation} + +\end_inset + +in the range of +\begin_inset Formula $1000.07$ +\end_inset + +. + To limit possible faulty results, the +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +lrftubess +\end_layout + +\end_inset + + code gives a warning when the tube ratio is chosen such that an invalid + +\begin_inset Formula $\chi$ +\end_inset + + is computed. + When an +\begin_inset Formula $\alpha<0.07$ +\end_inset + + is desired, the user should choose a higher value of +\begin_inset Formula $N$ +\end_inset + +. + +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename img/chi_vs_alpha.pdf + width 90text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset Formula $\chi$ +\end_inset + + vs +\begin_inset Formula $\alpha$ +\end_inset + + for different truncations +\begin_inset Formula $\left(N\right)$ +\end_inset + + of the infinite system of equations. +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:chi_vs_alpha" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Hard wall +\end_layout + +\begin_layout Standard +A hard wall is the wall perpendicular to the wave propagation direction. + Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:hardwall" + +\end_inset + + shows the schematic configuration for this segment. + Due to thermal relaxation a hard wall consumes acoustic energy is consumed. + The hard wall segment models this thermal relaxation loss. + The assumptions behind the model are: +\end_layout + +\begin_layout Itemize +Normal incident waves. +\end_layout + +\begin_layout Itemize +Uniform normal velocity. +\end_layout + +\begin_layout Itemize +The wavelength is much larger than the thermal penetration depth ( +\begin_inset Formula $\lambda\gg\delta_{\kappa}$ +\end_inset + +). +\end_layout + +\begin_layout Standard +We can derive the following impedance boundary condition +\begin_inset CommandInset citation +LatexCommand cite +after "p. 157" +key "ward_deltaec_2017" + +\end_inset + +: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +Delta EC User guide: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +U_{R}=U_{L}-\frac{\omega p}{\rho_{0}c_{0}^{2}}\frac{\gamma-1}{1+\epsilon_{s}}S\frac{\delta_{\kappa}}{2} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Or: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\[ +U_{L}=\frac{k}{z_{0}}\frac{\gamma-1}{1+\epsilon_{s}}S\frac{\delta_{\kappa}}{2}p +\] + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +U=k\delta_{\kappa}\frac{S}{z_{0}}\frac{\left(\gamma-1\right)\left(1+i\right)}{2\left(1+\epsilon_{s}\right)}p. +\end{equation} + +\end_inset + +Hence the impedance of a hard wall scales with +\begin_inset Formula $Z\sim Z_{0}\frac{\lambda}{\delta_{\kappa}}$ +\end_inset + +. + For 1 kHz, this results in +\begin_inset Formula $\sim4100Z_{0}$ +\end_inset + +, which is practically already close to +\begin_inset Formula $\infty$ +\end_inset + +. + Except for really high frequencies this segment can often be replaced with + a boundary condition of +\begin_inset Formula $U=0$ +\end_inset + +. + An important point to make here is that this boundary condition is inconsistent + with the LRF solution for 1D wave propagation in ducts, as the velocity + profile in a duct is not uniform. + This is especially true for the case of small ducts where +\begin_inset Formula $r_{h}\sim\delta$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename img/hardwall.pdf + width 50text% + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Schematic of a hard acoustic wall where the thermal boundary layer dissipates + a bit of the acoustic energy ( +\begin_inset Formula $Z\neq\infty$ +\end_inset + +). +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:hardwall" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset CommandInset bibtex +LatexCommand bibtex +bibfiles "lrftubes" +options "plain" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +printbibliography +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Chapter +\start_of_appendix +Thermal relaxation in thick tubes +\end_layout + +\begin_layout Section +\begin_inset CommandInset label +LatexCommand label +name "subsec:Thermal-relaxation-effect" + +\end_inset + +Thermal relaxation effect in thick tubes +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename img/prsduct_thermal_relax.pdf + width 80text% + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Schematic situation of a tube surrounded by a thick solid. + Note that the transverse acoustic temperature is drawn to be not zero at + the wall. + This happens in case of thermal interaction with a solid with finite thermal + effusivity. +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:prsduct-heatwave-solid" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +In this section, a formulation for +\begin_inset Formula $\epsilon_{s}$ +\end_inset + + is given for tubes where the temperature wave in the solid is present. + Figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:prsduct-heatwave-solid" + +\end_inset + + shows a schematic overview of the situation. + As shown in the figure, the temperature wave accompanied with an acoustic + wave results in heat conduction to/from the wall of the tube. + To solve this interaction mathematically, the heat equation in the solid + has to be solved. + For constant thermal conductivity, density and heat capacity the heat equation + of the solid is +\begin_inset Formula +\begin{equation} +\rho_{s}c_{s}\frac{\partial\tilde{T}_{s}}{\partial t}=\kappa_{s}\nabla^{2}\tilde{T}_{s}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\rho_{s},c_{s},\tilde{T}_{s}$ +\end_inset + + and +\begin_inset Formula $\kappa_{s}$ +\end_inset + + are the density, specific heat, temperature and thermal conductivity of + the solid, respectively. + In frequency domain and using cylindrical coordinates, assuming axial symmetry, + this can be written as +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$r$" +description "Radial position in cylindrical coordinates\\nomunit{\\si{\\m}}" + +\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, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\delta_{s}$ +\end_inset + + is +\begin_inset Formula +\begin{equation} +\delta_{s}=\sqrt{\frac{2\kappa_{s}}{\rho_{s}c_{s}\omega}}. +\end{equation} + +\end_inset + +Now, since +\begin_inset Formula $\partial T_{s}/\partial x\sim\frac{\delta_{s}}{\lambda}\frac{\partial T_{s}}{\partial r}$ +\end_inset + +, the second order derivative of the temperature in the axial direction + can be neglected. + In that case, the differential equation to solve for is +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $\rho_{s}c_{s}i\omega T_{s}=\kappa_{s}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\right)T_{s}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $-\kappa_{s}\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\right)T_{s}+\rho_{s}c_{s}i\omega T_{s}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\right)T_{s}+2\frac{\rho_{s}c_{s}\omega}{2\kappa_{s}i}T_{s}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\delta_{s}^{2}=\frac{2\kappa_{s}}{\rho_{s}c_{s}\omega}$ +\end_inset + +<<< subst +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\right)T_{s}+\frac{2}{i\delta_{s}^{2}}T_{s}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Multiply with +\begin_inset Formula $r^{2}$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(r^{2}\frac{\partial^{2}}{\partial r^{2}}+r\frac{\partial}{\partial r}+\frac{2}{i\delta_{s}^{2}}r^{2}\right)T_{s}=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Say: +\begin_inset Formula $\xi^{2}=\frac{2}{i\delta_{s}^{2}}r^{2}\Rightarrow$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Then: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\partial^{2}}{\partial r^{2}}=$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\left(r^{2}\frac{\partial^{2}}{\partial r^{2}}+r\frac{\partial}{\partial r}+\frac{2}{i\delta_{s}^{2}}r^{2}\right)T_{s}=0, +\end{equation} + +\end_inset + +which is a Bessel differential equation of the zero'th order in +\begin_inset Formula $T_{s}$ +\end_inset + +. + The solutions is sought in terms of traveling cylindrical waves: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $\sqrt{\frac{2}{i}}=\sqrt{-2i}=\pm\left(i-1\right)$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +T_{s}=C_{1}H_{0}^{(1)}\left(\left(i-1\right)\frac{r}{\delta_{s}}\right)+C_{2}H_{0}^{(2)}\left(\left(i-1\right)\frac{r}{\delta_{s}}\right), +\end{equation} + +\end_inset + +where +\begin_inset Formula $C_{1}$ +\end_inset + + and +\begin_inset Formula $C_{2}$ +\end_inset + + constants to be determined from the boundary conditions, and +\begin_inset Formula $H_{\alpha}^{(i)}$ +\end_inset + + is the cylindrical Hankel function of the +\begin_inset Formula $(i)^{\mathrm{th}}$ +\end_inset + + kind and order +\begin_inset Formula $\alpha$ +\end_inset + +. + If we require +\begin_inset Formula $T_{s}\to0$ +\end_inset + + as +\begin_inset Formula $r\to\infty$ +\end_inset + +, the constant +\begin_inset Formula $C_{2}$ +\end_inset + + is required to be +\begin_inset Formula $0$ +\end_inset + +. + From the acoustic energy equation, a similar differential equation can + be found for the acoustic temperature in the fluid: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $\rho_{0}c_{p}i\omega T=i\omega\alpha_{P}T_{0}p+\kappa\nabla^{2}T$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(\nabla^{2}-2\frac{\omega\rho_{0}c_{p}}{2\kappa}i\right)T=-\frac{1}{\kappa}i\omega\alpha_{P}T_{0}p$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\left(\nabla^{2}+\frac{2}{i\delta_{\kappa}^{2}}\right)T=\frac{2}{i\delta_{s}^{2}}\frac{\alpha_{P}T_{0}}{\rho_{0}c_{p}}p$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\[ +\left(r^{2}\frac{\partial^{2}}{\partial r^{2}}+r\frac{\partial}{\partial r}+\frac{2}{i\delta_{s}^{2}}r^{2}\right)T=\frac{2}{i\delta_{s}^{2}}\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p, +\] + +\end_inset + +for which the (partial) solution is +\begin_inset Formula +\begin{equation} +T=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\left(1-C_{3}J_{0}\left(\left(i-1\right)\frac{r}{\delta_{\kappa}}\right)\right).\label{eq:temp_partial_sol} +\end{equation} + +\end_inset + +To attain at Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:temp_partial_sol" + +\end_inset + +, use has been made of the fact that the temperature should be finite at + +\begin_inset Formula $r=0$ +\end_inset + +. + +\begin_inset Formula $C_{3}$ +\end_inset + + is a constant that is to be determined from the boundary conditions at + the solid-fluid interface. + These boundary conditions are: +\begin_inset Formula +\begin{align} +T_{s}|_{r=a} & =T|_{r=a},\\ +-\kappa_{s}\frac{\partial T_{s}}{\partial r}|_{r=a} & =-\kappa\frac{\partial T}{\partial r}|_{r=a}, +\end{align} + +\end_inset + +i.e. + continuity of the temperature and the heat flux at the interface. + This yields two equations for two unknowns ( +\begin_inset Formula $C_{1}$ +\end_inset + + and +\begin_inset Formula $C_{3}$ +\end_inset + +, +\begin_inset Formula $C_{2}$ +\end_inset + + is already argued to be +\begin_inset Formula $0$ +\end_inset + +). + Solving for the acoustic temperature yields: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +\begin_inset Formula $T|_{r=a}=T_{s}|_{r=a}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +– +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $C_{1}H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\left(1-C_{3}J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right)\Rightarrow C_{1}=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(1-C_{3}J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right)}{H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}$ +\end_inset + + (1) +\end_layout + +\begin_layout Plain Layout +Derivative b.c. +\end_layout + +\begin_layout Plain Layout +– +\begin_inset Formula $-\frac{\partial T}{\partial r}|_{r=a}=-\frac{\kappa_{s}}{\kappa}\frac{\partial T_{s}}{\partial r}|_{r=a}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +where +\begin_inset Formula $-\frac{\partial T}{\partial r}|_{r=a}=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(i-1\right)}{\delta_{\kappa}}C_{3}J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +using +\begin_inset Formula $\frac{\partial H_{0}^{(1)}(z)}{\partial z}=-H_{1}^{(1)}(z)$ +\end_inset + + ==> +\begin_inset Formula $-\frac{\kappa}{\kappa_{s}}\frac{\partial T_{s}}{\partial r}|_{r=a}=\frac{\kappa}{\kappa_{s}}C_{1}\frac{\left(i-1\right)}{\delta_{s}}H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Such that: +\begin_inset Formula $\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(i-1\right)}{\delta_{\kappa}}C_{3}J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)=\frac{\kappa_{s}}{\kappa}C_{1}\frac{\left(i-1\right)}{\delta_{s}}H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Filling in +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(i-1\right)}{\delta_{\kappa}}C_{3}J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)=\frac{\kappa_{s}}{\kappa}\left(\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}p\frac{\left(1-C_{3}J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right)}{H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}\right)\frac{\left(i-1\right)}{\delta_{s}}H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Solving for +\begin_inset Formula $C_{3}$ +\end_inset + + gives: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $C_{3}=\frac{1}{\left[\frac{J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}{\frac{\kappa_{s}}{\kappa}\frac{\delta_{\kappa}}{\delta_{s}}H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}+J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right]}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +or: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $C_{3}=\frac{1}{\left[\left(1+\epsilon_{s}\right)J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)\right]}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\epsilon_{s}=\frac{\kappa\delta_{s}}{\delta_{\kappa}\kappa_{s}}\frac{H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}{H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\kappa\delta_{s}}{\delta_{\kappa}\kappa_{s}}=\sqrt{\frac{\kappa^{2}\delta_{s}^{2}}{\kappa_{s}^{2}\delta_{\kappa}^{2}}}=\sqrt{\frac{\kappa\rho_{0}c_{p}}{\kappa\rho_{s}c_{s}}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\[ +T=\frac{\alpha_{p}T_{0}}{\rho_{0}c_{p}}\left(1-\frac{1}{\left(1+\epsilon_{s}\right)}\frac{J_{0}\left(\left(i-1\right)\frac{r}{\delta_{\kappa}}\right)}{J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}\right)p, +\] + +\end_inset + +where +\begin_inset Formula +\begin{equation} +\epsilon_{s}=\frac{e_{f}}{e_{s}}\frac{J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}{J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}, +\end{equation} + +\end_inset + + +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout +- +\end_layout + +\begin_layout Plain Layout +-Asymptotic form of the Hankel function for large argument, and +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $-\pi<\arg(z)<2\pi$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $H_{\alpha}^{(1)}(z)\sim\sqrt{\frac{2}{\pi z}}e^{i\left(z-\pi\frac{1+2\alpha}{4}\right)}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +And for +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $J_{\alpha}(z)\sim\sqrt{\frac{2}{\pi z}}\cos\left(z-\pi\frac{1+2\alpha}{4}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +Filling this in into +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula $\frac{e_{f}}{e_{s}}\cdot-ii=\frac{e_{f}}{e_{s}}$ +\end_inset + + +\end_layout + +\end_inset + +where +\begin_inset Formula $e_{f}$ +\end_inset + + is the thermal effusivity +\begin_inset CommandInset nomenclature +LatexCommand nomenclature +prefix "A" +symbol "$e$" +description "Thermal effusivity\\nomunit{\\si{\\joule\\per\\square\\metre\\kelvin\\second\\tothe{ \\frac{1}{2} } }}" + +\end_inset + + of the fluid, and +\begin_inset Formula $e_{s}$ +\end_inset + + the thermal effusivity of the solid, such that the ratio is +\begin_inset Formula +\begin{equation} +\frac{e_{f}}{e_{s}}=\sqrt{\frac{\kappa\rho_{0}c_{p}}{\kappa_{s}\rho_{s}c_{s}}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Note that for large +\begin_inset Formula $a/\delta_{\kappa}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +\frac{J_{1}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}{J_{0}\left(\left(i-1\right)\frac{a}{\delta_{\kappa}}\right)}\to i, +\end{equation} + +\end_inset + +and for large +\begin_inset Formula $a/\delta_{s}$ +\end_inset + + +\begin_inset Formula +\begin{equation} +\frac{H_{0}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}{H_{1}^{(1)}\left(\left(i-1\right)\frac{a}{\delta_{s}}\right)}\to-i, +\end{equation} + +\end_inset + +such that for both numbers large +\begin_inset Formula +\begin{equation} +\epsilon_{s}\to\frac{e_{f}}{e_{s}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Chapter +Derivation of Karal's discontinuity factor +\begin_inset CommandInset label +LatexCommand label +name "chap:Derivation-of-Karal's" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Note: this documentation is imcomplete. +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename img/discontinuity_appendix.pdf + width 60text% + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Schematic of a discontinuity at the interface between two tubes with different + radius. + Domain B is the smaller tube and domain C is the larger tube. + The radius of the tube in domain B is +\begin_inset Formula $b$ +\end_inset + +, and the radius of the tube in domain C is +\begin_inset Formula $c$ +\end_inset + +. +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:karal-1" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +This appendix describes the derivation of Karal's discontinuity factor. + The following assumptions underlie the model: +\end_layout + +\begin_layout Itemize +\begin_inset Formula $z=0$ +\end_inset + + : position of the discontinuity +\end_layout + +\begin_layout Itemize +Assume +\begin_inset Formula $f\ll f_{c}$ +\end_inset + +, such that far away from the discontinuity, only propagating modes exist. +\end_layout + +\begin_layout Itemize +Assume axial symmetry, so dependence of +\begin_inset Formula $\theta$ +\end_inset + + is dropped +\end_layout + +\begin_layout Standard +In cylindrical coordinates, the solution of the Helmholtz equation can be + written in terms of cylindrical harmonics +\begin_inset CommandInset citation +LatexCommand cite +key "blackstock_fundamentals_2000" + +\end_inset + +. + Assuming axial symmetrySuch that the acoustic pressure in for example tube + +\begin_inset Formula $B$ +\end_inset + + can be written as: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +p_{B}=\left\{ \begin{array}{c} +J_{m}\left(k_{r}r\right)\\ +N_{m}\left(k_{r}r\right) +\end{array}\right\} \left\{ \begin{array}{c} +e^{im\phi}\\ +e^{-im\phi} +\end{array}\right\} \left\{ \begin{array}{c} +e^{\beta z}\\ +e^{-\beta z} +\end{array}\right\} +\end{equation} + +\end_inset + +where +\begin_inset Formula $J_{m}$ +\end_inset + + is the cylindrical Bessel function of order +\begin_inset Formula +\begin{equation} +k_{r}^{2}-\beta^{2}=k^{2}. +\end{equation} + +\end_inset + +Using the boundary condition that +\begin_inset Formula +\begin{equation} +\frac{\partial p_{B}}{\partial r}|_{r=b}=0, +\end{equation} + +\end_inset + +and assuming axial symmetry (only the +\begin_inset Formula $m=0$ +\end_inset + + modes) it follows that +\begin_inset Formula +\begin{equation} +\frac{\partial J_{0}}{\partial r}\left(k_{r}b\right)|_{r=b}=0. +\end{equation} + +\end_inset + +Assuming that +\begin_inset Formula $\alpha_{k}$ +\end_inset + + is the +\begin_inset Formula $k^{\mathrm{th}}$ +\end_inset + + zero of +\begin_inset Formula $J_{0}^{'}(x)$ +\end_inset + +, we can write for +\begin_inset Formula $k_{r,k}$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +k_{r,k}=\frac{\alpha_{k}}{b}. +\end{equation} + +\end_inset + +Hence we find the following reduced expression for the pressure in tube + +\begin_inset Formula $B$ +\end_inset + +: +\begin_inset Formula +\begin{equation} +p_{B}=B_{0}^{0}\exp\left(ikz\right)+B_{0}^{1}\exp\left(-ikz\right)+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)\left\{ \begin{array}{c} +e^{\beta_{n}z}\\ +e^{-\beta_{n}z} +\end{array}\right\} , +\end{equation} + +\end_inset + +where accordingly, +\begin_inset Formula +\begin{equation} +\beta_{k}^{2}=\left(\frac{\alpha_{k}}{b}\right)^{2}-k^{2}\label{eq:beta_k} +\end{equation} + +\end_inset + +For +\begin_inset Formula $k^{2}<\left(\alpha_{k}/b\right)^{2}$ +\end_inset + +, +\begin_inset Formula $\beta_{k}^{2}>0$ +\end_inset + +, the modes are evanescent. + And since we only allow finite solutions for +\begin_inset Formula $z\leq0$ +\end_inset + +, the final results for +\begin_inset Formula $p_{B}$ +\end_inset + + is +\begin_inset Formula +\begin{equation} +p_{B}=B_{0}^{0}\exp\left(ikz\right)+B_{0}^{1}\exp\left(-ikz\right)+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z}, +\end{equation} + +\end_inset + +where +\begin_inset Formula $\beta_{n}$ +\end_inset + + is defined as the positive root of the r.h.s. + of Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:beta_k" + +\end_inset + +. + We simplify this relation to: +\begin_inset Formula +\begin{equation} +p_{B}(z)=p_{B}^{0}(z)+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z}. +\end{equation} + +\end_inset + +For the velocity we find +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $u=\frac{i}{\omega\rho_{0}}\frac{\partial p_{B}}{\partial z}=u_{B}^{0}(z)+\sum_{n=1}^{\infty}\frac{i\beta_{n}}{\omega\rho_{0}}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +u_{B}(z)=u_{B}^{0}(z)+\sum_{n=1}^{\infty}Y_{B,n}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +Y_{B,n}=\frac{i\beta_{n}}{\omega\rho_{0}}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Similarly, for the positive +\begin_inset Formula $z$ +\end_inset + + we find +\begin_inset Formula +\begin{equation} +p_{C}(z)=P_{C}^{0}(z)+\sum_{m=1}^{\infty}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)e^{-\gamma_{m}z}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +\gamma_{m}=\sqrt{\left(\frac{\alpha_{m}}{c}\right)^{2}-k^{2}}. +\end{equation} + +\end_inset + +and +\begin_inset Formula +\begin{equation} +u_{C}(z)=u_{C}^{0}(z)+\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)e^{-\gamma_{m}z}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +Y_{C,m}=-\frac{i\gamma_{m}}{\omega\rho_{0}} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Boundary conditions +\end_layout + +\begin_layout Standard +At the interface ( +\begin_inset Formula $z=0$ +\end_inset + +), the following boundary conditions are valid: +\begin_inset Formula +\begin{align} +u_{B}|_{z=0} & =u_{C}|_{z=0} & 0\leq r\leq b\label{eq:derivative1bc}\\ +u_{C}|_{z=0} & =0 & b\leq r\leq c\label{eq:derivative2bc}\\ +p_{B} & =p_{C} & 0\leq r\leq b\label{eq:continuitybc} +\end{align} + +\end_inset + +Taking Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:derivative1bc" + +\end_inset + +, multiply by +\begin_inset Formula $r$ +\end_inset + + and integrating from +\begin_inset Formula $0$ +\end_inset + + to +\begin_inset Formula $c$ +\end_inset + + , taking into account Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:derivative2bc" + +\end_inset + + yields: +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $u_{B}(z)=u_{B}^{0}(z)+\sum_{n=1}^{\infty}\zeta_{B,n}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)e^{\beta_{n}z}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Integrating from 0 to +\begin_inset Formula $b$ +\end_inset + + for +\begin_inset Formula $u_{B}$ +\end_inset + + and integrating from 0 to +\begin_inset Formula $c$ +\end_inset + + for +\begin_inset Formula $u_{C}$ +\end_inset + + cancels out the Bessel functions, as the primitive of +\begin_inset Formula $J_{0}(x)x$ +\end_inset + + is +\begin_inset Formula $J_{1}(x)x$ +\end_inset + +, for which due to the no-slip b.c. + the resulting integral is zero, and at +\begin_inset Formula $r=0$ +\end_inset + +, the integral is zero as well. + Hence we obtain only the propagating mode contribution to the volume flow. +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +b^{2}u_{B}^{0}=c^{2}u_{C}^{0} +\end{equation} + +\end_inset + +We require one more equation at the interface, which is found from the continuit +y boundary conditions as well. + Multiplying Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:derivative1bc" + +\end_inset + + with +\begin_inset Formula $J_{0}(\alpha_{q}\frac{r}{c})r$ +\end_inset + + and integrating setting +\begin_inset Formula $q=m$ +\end_inset + + and dividing by +\begin_inset Formula $bc$ +\end_inset + + yields: +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $u_{B}=u_{B}^{0}+\sum_{n=1}^{\infty}\zeta_{B,n}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $u_{C}=u_{C}^{0}+\sum_{m=1}^{\infty}\zeta_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +– +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +– Work out stuff, first line: +\end_layout + +\begin_layout Plain Layout + +\lang english +- Using the rule: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +\int J_{0}(C_{1}x)J_{0}(C_{2}x)x\mathrm{d}x=x\frac{C_{1}J_{1}(C_{1}x)J_{0}(C_{2}x)-C_{2}J_{0}\left(C_{1}x\right)J_{1}(C_{2}x)}{C_{1}^{2}-C_{2}^{2}} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +–> +\begin_inset Formula $C_{1}=\frac{\alpha_{q}}{c}$ +\end_inset + +; +\begin_inset Formula $C_{2}=\frac{\alpha_{n}}{b}$ +\end_inset + + +\begin_inset Formula $x=b$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=u_{B}^{0}J_{1}(\alpha_{q}\frac{b}{c})\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}b\frac{\frac{\alpha_{q}}{c}J_{1}(\frac{\alpha_{q}}{c}b)J_{0}(\frac{\alpha_{n}}{b}b)-\frac{\alpha_{n}}{b}J_{0}\left(\frac{\alpha_{q}}{c}b\right)J_{1}(\frac{\alpha_{n}}{b}b)}{\left(\frac{\alpha_{q}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}=$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Using: +\begin_inset Formula $J_{1}\left(\alpha_{i}\right)=0$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=u_{B}^{0}J_{1}(\alpha_{q}\frac{b}{c})\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{b}{\left(\frac{\alpha_{q}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}\frac{\alpha_{q}}{c}J_{1}(\frac{\alpha_{q}}{c}b)J_{0}(\frac{\alpha_{n}}{b}b)=$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Using: +\begin_inset Formula $\rho=\frac{b}{c}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=u_{B}^{0}J_{1}(\alpha_{q}\frac{b}{c})\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{q}\rho}{\left(\frac{\alpha_{q}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\frac{\alpha_{q}}{c}b)J_{0}(\frac{\alpha_{n}}{b}b)=$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Setting: +\begin_inset Formula $q=m$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{b}u_{B}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{m}\rho}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +——————————————————————— +\end_layout + +\begin_layout Plain Layout + +\lang english +And the rhs: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=\int\limits _{0}^{c}\left[u_{C}^{0}J_{0}(\alpha_{q}\frac{r}{c})r+\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)J_{0}(\alpha_{q}\frac{r}{c})r\right]\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=\int\limits _{0}^{c}\left[\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)J_{0}(\alpha_{q}\frac{r}{c})r\right]\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Setting: +\begin_inset Formula $q=m$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=\int\limits _{0}^{c}\left[\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)J_{0}(\alpha_{m}\frac{r}{c})r\right]\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Using the rule: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +\int J_{0}(C_{1}x)^{2}x\mathrm{d}x=\frac{1}{2}x^{2}\left(J_{0}(C_{1}x)^{2}+J_{1}(C_{1}x)^{2}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $C_{1}=\alpha_{m}\frac{r}{c}$ +\end_inset + +, +\begin_inset Formula $x=c$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=Y_{C,m}C_{m}\frac{1}{2}c^{2}\left(J_{0}(\alpha_{m}\frac{c}{c})^{2}+J_{1}(\alpha_{m}\frac{c}{c})^{2}\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int\limits _{0}^{c}u_{C}J_{0}(\alpha_{q}\frac{r}{c})r\mathrm{d}r=Y_{C,m}C_{m}\frac{1}{2}c^{2}J_{0}(\alpha_{m})^{2}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +— OR: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{bc}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{m}\rho}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})=Y_{C,m}C_{m}\frac{1}{2}c^{2}J_{0}(\alpha_{m})^{2} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +– Divide by bc: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{1}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{m}\rho}{\left[\rho\alpha_{m}^{2}-\rho^{-1}\alpha_{n}^{2}\right]}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})=Y_{C,m}C_{m}\frac{1}{2}\rho^{-1}J_{0}(\alpha_{m})^{2} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +- Deel teller en noemer in breuk door +\begin_inset Formula $\rho$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{1}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}B_{n}\frac{\alpha_{m}}{\alpha_{m}^{2}-\frac{\alpha_{n}^{2}}{\rho^{2}}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})=Y_{C,m}C_{m}\frac{1}{2}\rho^{-1}J_{0}(\alpha_{m})^{2} +\] + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{1}{\alpha_{q}}+\sum_{n=1}^{\infty}Y_{B,n}T_{mn}B_{n}=Y_{C,m}\frac{1}{2}\rho^{-1}J_{0}(\alpha_{m})^{2}C_{m}, +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +T_{mn}=\frac{\alpha_{m}}{\alpha_{m}^{2}-\frac{\alpha_{n}^{2}}{\rho^{2}}}J_{0}\left(\alpha_{n}\right)J_{1}\left(\alpha_{m}\rho\right). +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Setting +\begin_inset Formula $p_{B}=p_{C}$ +\end_inset + + +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)r\mathrm{d}r=\int_{0}^{b}\left[p_{B}^{0}+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)\right]r\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)r\mathrm{d}r=\frac{b^{2}}{2}p_{B}^{0}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +———————————————– +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)r\mathrm{d}r=\int_{0}^{b}\left[p_{C}^{0}+\sum_{m=1}^{\infty}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)\right]r\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)r\mathrm{d}r=\frac{b^{2}}{2}p_{C}^{0}+\sum_{m=1}^{\infty}\frac{bc}{\alpha_{m}}C_{m}J_{1}\left(\alpha_{m}\rho\right)$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Such that +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +\frac{b^{2}}{2}p_{B}^{0}=\frac{b^{2}}{2}p_{C}^{0}+\sum_{m=1}^{\infty}\frac{bc}{\alpha_{m}}C_{m}J_{1}\left(\alpha_{m}\rho\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Divide by +\begin_inset Formula $\frac{b^{2}}{2}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +p_{B}^{0}=p_{C}^{0}+2\sum_{m=1}^{\infty}\frac{J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}C_{m} +\] + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +p_{B}^{0}=p_{C}^{0}+2\sum_{m=1}^{\infty}\frac{J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}C_{m} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\int_{0}^{b}\left[p_{B}^{0}J_{0}\left(\alpha_{p}\frac{r}{b}\right)r+\sum_{n=1}^{\infty}B_{n}J_{0}\left(\alpha_{n}\frac{r}{b}\right)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\right]\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{n=1}^{\infty}B_{n}\int_{0}^{b}J_{0}\left(\alpha_{n}\frac{r}{b}\right)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Setting +\begin_inset Formula $p=n$ +\end_inset + + en +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +\int J_{0}(C_{1}x)^{2}x\mathrm{d}x=\frac{1}{2}x^{2}\left(J_{0}(C_{1}x)^{2}+J_{1}(C_{1}x)^{2}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $C_{1}=\frac{\alpha_{n}}{b}$ +\end_inset + + en +\begin_inset Formula $x=b$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{B}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=B_{n}\frac{1}{2}b^{2}J_{0}(\alpha_{n})^{2}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +– Zelfde voor integraal voor +\begin_inset Formula $p_{C}$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\int_{0}^{b}\left[P_{C}^{0}+\sum_{m=1}^{\infty}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)\right]J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{m=1}^{\infty}C_{m}\int_{0}^{b}J_{0}\left(\alpha_{m}\frac{r}{c}\right)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Gebruik de regel: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +\int J_{0}(C_{1}x)J_{0}(C_{2}x)x\mathrm{d}x=x\frac{C_{1}J_{1}(C_{1}x)J_{0}(C_{2}x)-C_{2}J_{0}\left(C_{1}x\right)J_{1}(C_{2}x)}{C_{1}^{2}-C_{2}^{2}} +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Waarbij: +\begin_inset Formula $C_{1}=\frac{\alpha_{m}}{c}$ +\end_inset + + , +\begin_inset Formula $C_{2}=\frac{\alpha_{p}}{b}$ +\end_inset + + ; +\begin_inset Formula $x=b$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{m=1}^{\infty}C_{m}b\frac{\frac{\alpha_{m}}{c}J_{1}(\frac{\alpha_{m}}{c}b)J_{0}(\frac{\alpha_{p}}{b}b)-\frac{\alpha_{p}}{b}J_{0}\left(\frac{\alpha_{m}}{c}x\right)J_{1}(\frac{\alpha_{p}}{b}b)}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{p}}{b}\right)^{2}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{m=1}^{\infty}C_{m}\frac{\rho\alpha_{m}}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{p}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{p})$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Zet +\begin_inset Formula $p=n$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\int_{0}^{b}p_{C}(z=0)J_{0}\left(\alpha_{p}\frac{r}{b}\right)r\mathrm{d}r=\sum_{m=1}^{\infty}C_{m}\frac{\rho\alpha_{m}}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n})$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Zodat: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +B_{n}\frac{1}{2}b^{2}J_{0}(\alpha_{n})^{2}=\sum_{m=1}^{\infty}C_{m}\frac{\rho\alpha_{m}}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n}) +\] + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +B_{n}\frac{1}{2}b^{2}J_{0}(\alpha_{n})^{2}=\sum_{m=1}^{\infty}C_{m}\frac{\rho\alpha_{m}}{\left(\frac{\alpha_{m}}{c}\right)^{2}-\left(\frac{\alpha_{n}}{b}\right)^{2}}J_{1}(\alpha_{m}\rho)J_{0}(\alpha_{n}) +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Deel linker en rechterzijde door +\begin_inset Formula $\frac{1}{2}b^{2}$ +\end_inset + +: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +B_{n}J_{0}(\alpha_{n})^{2}=2\sum_{m=1}^{\infty}\rho^{-1}C_{m}\frac{\alpha_{m}}{\alpha_{m}^{2}-\frac{\alpha_{n}^{2}}{\rho^{2}}}J_{0}(\alpha_{n})J_{1}(\alpha_{m}\rho) +\] + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Oftewel: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula +\[ +B_{n}J_{0}(\alpha_{n})^{2}=\frac{2}{\rho}\sum_{m=1}^{\infty}T_{mn}C_{m} +\] + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +B_{n}J_{0}(\alpha_{n})^{2}=\frac{2}{\rho}\sum_{m=1}^{\infty}T_{mn}C_{m} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $B_{n}=\frac{2}{\rho J_{0}(\alpha_{n})^{2}}\sum_{q=1}^{\infty}T_{qn}C_{q}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{1}{\alpha_{m}}+\sum_{n=1}^{\infty}Y_{B,n}T_{mn}\frac{2}{\rho J_{0}(\alpha_{n})^{2}}\sum_{q=1}^{\infty}T_{qn}C_{q}=Y_{C,m}\frac{1}{2\rho}J_{0}(\alpha_{m})^{2}C_{m}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\sum_{n=1}^{\infty}\frac{2Y_{B,n}}{J_{0}(\alpha_{n})^{2}}T_{mn}\sum_{q=1}^{\infty}T_{qn}C_{q}-\frac{1}{2}Y_{C,m}J_{0}(\alpha_{m})^{2}C_{m}=-u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{m}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +—————Setting ——- +\begin_inset Formula $C_{m}=ikbu_{B}^{0}z_{0}D_{m}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\sum_{n=1}^{\infty}\frac{2Y_{B,n}}{J_{0}(\alpha_{n})^{2}}ikbu_{B}^{0}z_{0}T_{mn}\sum_{q=1}^{\infty}T_{qn}D_{q}-\frac{1}{2}Y_{C,m}ikbD_{m}u_{B}^{0}z_{0}J_{0}(\alpha_{m})^{2}D_{m}=-u_{B}^{0}J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{q}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Using: +\begin_inset Formula $z_{0}Y_{B,n}=\frac{i\beta_{n}}{k}$ +\end_inset + + and +\begin_inset Formula $z_{0}Y_{C,m}=-\frac{i\gamma_{m}}{k}$ +\end_inset + + and , +\begin_inset Formula $\gamma_{m}=\sqrt{\left(\frac{\alpha_{m}}{c}\right)^{2}-k^{2}}$ +\end_inset + + and +\begin_inset Formula $\beta_{n}=\sqrt{\left(\frac{\alpha_{n}}{b}\right)^{2}-k^{2}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\sum_{n=1}^{\infty}\frac{2}{J_{0}(\alpha_{n})^{2}}\sqrt{\left(\frac{\alpha_{n}}{bk}\right)^{2}-1}kbT_{mn}\sum_{q=1}^{\infty}T_{qn}D_{q}+\sqrt{\left(\frac{\alpha_{m}}{kc}\right)^{2}-1}\frac{1}{2}kbD_{m}J_{0}(\alpha_{m})^{2}D_{m}=+J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{m}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +When +\begin_inset Formula $kc\sim kb\ll1$ +\end_inset + +, this can be rewritten to: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\sum_{n=1}^{\infty}\frac{2\alpha_{n}}{J_{0}(\alpha_{n})^{2}}T_{mn}\sum_{q=1}^{\infty}T_{qn}D_{q}+\frac{\alpha_{m}\rho}{2}D_{m}J_{0}(\alpha_{m})^{2}D_{m}=J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{m}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\sum_{n=1}^{\infty}\frac{2\alpha_{n}}{J_{0}(\alpha_{n})^{2}}T_{mn}\sum_{q=1}^{\infty}T_{qn}D_{q}+\frac{1}{2}\rho\alpha_{m}J_{0}(\alpha_{m})^{2}D_{m}=J_{1}(\alpha_{m}\rho)\frac{\rho}{\alpha_{m}},\label{eq:D_meq} +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{equation} +D_{m}=\frac{C_{m}}{ikbu_{B}^{0}z_{0}} +\end{equation} + +\end_inset + +Eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:D_meq" + +\end_inset + + is a set of infinite equations in terms of an infinite number of unknowns + for +\begin_inset Formula $D_{m}$ +\end_inset + +. + In matrix algebra for a finite set, this can be written as +\begin_inset Formula +\begin{equation} +(\boldsymbol{M}_{1}\cdot\boldsymbol{M}_{2}+\boldsymbol{K})\cdot\boldsymbol{D}=\boldsymbol{R} +\end{equation} + +\end_inset + +where +\begin_inset Formula +\begin{align} +M_{1,ij} & =\frac{2\alpha_{j}}{J_{0}(\alpha_{j})^{2}}T_{ij}\\ +M_{2,ij} & =T_{ji}\\ +K_{ij} & =\frac{1}{2}\rho\alpha_{j}J_{0}(\alpha_{j})^{2} & ;\quad i=j\\ +K_{ij} & =0 & ;\quad i\neq j\\ +R_{i} & =J_{1}(\alpha_{i}\rho)\frac{\rho}{\alpha_{q}} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +Finally, the added acoustic mass, +\begin_inset Formula +\begin{equation} +p_{C}^{0}=p_{B}^{0}-i\omega M_{A}U_{B}, +\end{equation} + +\end_inset + +can be computed as +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $p_{B}^{0}=p_{C}^{0}+\sum_{m=1}^{\infty}\frac{2J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}C_{m}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $p_{B}^{0}=p_{C}^{0}+ikbu_{B}^{0}z_{0}\sum_{m=1}^{\infty}\frac{2J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}D_{m}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Filling in: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $p_{C}^{0}=p_{B}^{0}-i\omega M_{A}U_{B}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +Then: +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $p_{B}^{0}=p_{C}^{0}+i\omega M_{A}U_{B}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +or: +\begin_inset Formula $i\omega M_{A}U_{B}=ikbu_{B}^{0}z_{0}\sum_{m=1}^{\infty}\frac{2J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}D_{m}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +And since: +\begin_inset Formula $M_{A}=\chi(\alpha)\frac{8\rho_{0}}{3\pi^{2}a_{L}}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $\chi(\alpha)=\frac{3\pi}{4}\sum_{m=1}^{\infty}\frac{J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}D_{m}$ +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\rho_{0}\sum_{m=1}^{\infty}\frac{2}{\pi b}\frac{J_{1}\left(\alpha_{m}\rho\right)}{\rho\alpha_{m}}D_{m} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +For a given velocity +\begin_inset Formula $u_{C,0}$ +\end_inset + + the velocity profile at +\begin_inset Formula $z=0$ +\end_inset + + is +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $u_{C}(z)=u_{C}^{0}(z)+\sum_{m=1}^{\infty}Y_{C,m}C_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)e^{-\gamma_{m}z}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\lang english +\begin_inset Formula $u_{C}=u_{C}^{0}+u_{B}^{0}\sum_{m=1}^{\infty}\gamma_{m}bD_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right)$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +u_{C}=u_{C}^{0}+bu_{B}^{0}\sum_{m=1}^{\infty}\gamma_{m}D_{m}J_{0}\left(\alpha_{m}\frac{r}{c}\right) +\end{equation} + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/tex/preamble.tex b/tex/preamble.tex new file mode 100644 index 0000000..63528e0 --- /dev/null +++ b/tex/preamble.tex @@ -0,0 +1,264 @@ + +% The format should be: +% height: 240mm +% width: 170 mm +% \showtrimsoff + + + +% This code is for stock size a4 and small crop signs +% a4 stock size +% \stockaiv +% a4 is 297 hoog bij 210 breed. We zetten de trim op (297-240)/2=28.5mm +% bij (210-170)/2=20 +% \setpagecc{240mm}{170mm}%Setting trimmed page centered on stock +%Do this for final to press +% \trimXmarks +% \trimLmarks +% For preview PDF +% \trimFrame + +% This code is for the digital version +%\setstocksize{240mm}{170mm} +%\settrimmedsize{240mm}{170mm}{*} +%\showtrimsoff +% End code for digital version + +\usepackage{fancyhdr} +\fancyhead[RO,LE]{ \includegraphics[width=0.2\textwidth]{img_default/ascee_beeldmerk.pdf} } +\fancyhead[RE,LO]{} +% \fancyfoot[RO,LE]{\textbf{\footnotesize{Confidential}}} + +% Copyright sign \textcopyright, \textregistered +\usepackage{textcomp} + + +\usepackage{xcolor} +%\definecolor{asceegray}{RGB}{79,76,77} +\definecolor{asceegray}{HTML}{4f4c4d} + +\usepackage{lipsum} +\date{\today} +%\date{Vul datum in in preamble} + + +% Use SI unitx package +\usepackage{siunitx} +\sisetup{sticky-per = true}% +\sisetup{inter-unit-product = \ensuremath{ {} \!\cdot\!{} } } +% We use engineering scientific notation +%\sisetup{scientific-notation=engineering} % +\sisetup{exponent-product=\!\cdot\!} + + +% Margins +% \setlrmarginsandblock{2.25cm}{1.75cm}{*} +% \setulmarginsandblock{3cm}{2.5cm}{*} %Top and bottom margin +% \checkandfixthelayout + + +% \usepackage[dutch,english]{babel} +% \usepackage[T1]{fontenc} + +%\usepackage{pifont} +\usepackage{import} + +\usepackage{xcolor} + +% If we use non-tex fonts we need this for biblatex +% \usepackage{polyglossia} +% \setdefaultlanguage{english} + +\usepackage[% +giveninits=true, +doi=false, +url=false, +isbn=false, +%% natbib=true, +%% date=year, +bibencoding=utf8, +% style=numeric-comp, +backend=biber, +% refsection=chapter, % If we want bibliographies per chapter +]{biblatex} + + +\DeclareFieldFormat*{url}{} +\DeclareFieldFormat[misc]{url}{\mkbibacro{URL}\addcolon\space\url{#1}} +\DeclareFieldFormat[report]{url}{\mkbibacro{URL}\addcolon\space\url{#1}} +\DeclareFieldFormat*{urldate}{} +\DeclareFieldFormat[misc]{urldate}{\mkbibparens{\bibstring{urlseen}\space#1}} +\DeclareFieldFormat[report]{urldate}{\mkbibparens{\bibstring{urlseen}\space#1}} +\renewbibmacro{in:}{} + +% We need absolute path to the bibliography here, or it should be in +% the same directory +\addbibresource{lrftubes.bib} + +% Separate the items in the bibliography somewhat +% \setlength{\bibsep}{4pt} + +\newcommand{\lrftubes}{\textbf{\texttt{LRFTubes}}} +\newcommand{\lrftubess}{\lrftubes\ } +% Customize the caption of the figures and tables +\usepackage[margin=10pt,font={footnotesize},labelfont=bf,labelsep=endash]{caption} + +%Chapter style +% Set chapter style demo with sans serif font family +%% \chapterstyle{bianchi} +%% \chapterstyle{madsen} +% \chapterstyle{thatcher} +% ABSTRACT ---------------------------------------------------------- +% \setlength\absrightindent{0pt} +% \setlength\absleftindent{0pt} +% \renewcommand{\abstractname}{} +% \addto{\captionsenglish}{\renewcommand{\abstractname}{}} +% \renewcommand{\abstracttextfont}{\normalfont\small\itshape} +% \abstractrunin +% Make table of contents and other lists wrap words for long chapter +% titles + +% \addtocontents{lof}{\protect\sloppy} +% \listoffigures +% \addtocontents{lot}{\protect\sloppy} +% \listoftables + +% Some fancy header settins +% \nouppercaseheads +% \pagestyle{headings} % customized: +% \makeevenhead{headings}{\thepage}{}{\small\leftmark} +% \makeoddhead{headings}{\small\rightmark}{}{\thepage} +% \makeevenfoot{plain}{}{\thepage}{} +% \makeoddfoot{plain}{}{\thepage}{} +% \makeheadrule{headings}{\textwidth}{\normalrulethickness} + +% % Blind footnotes at Chapters +% \newcommand\blfootnote[1]{% +% \begingroup +% \renewcommand\thefootnote{}\footnote{#1}% +% \addtocounter{footnote}{-1}% +% \endgroup +% } + +% Replace the colored links with just black ones +% \definecolor{green}{cmyk}{1,1,1,1} +% \definecolor{red}{cmyk}{1,1,1,1} +% \definecolor{magenta}{cmyk}{1,1,1,1} + + +% Roman font for URL in Bibliography +\urlstyle{rm} + + +%% References to plot lines +% \newcommand{\plotref}[1]{\protect\ref{#1}} +%% \newcommand{\plotref}[1]{#1} + +% Some hacks to get proper names and ordering in the bibliography + +% Use in the bibtex file: +% @PREAMBLE{ {\providecommand{\noopsort}[1]{}} } +% To get the "Lord" of Lord Rayleigh as a prefix instead of a abbreviated first +% name. In the bibtex item the author is provided as: +% author = {{\noopsort{Rayleigh}}{Lord Rayleigh}}, + +% To get nice copyright symbol +\usepackage{textcomp} + +% To get the 'draft' watermark +\usepackage{watermark} + +% New definition of square root: +% it renames \sqrt as \oldsqrt +% \let\oldsqrt\sqrt +% it defines the new \sqrt in terms of the old one +% \def\sqrt{\mathpalette\DHLhksqrt} +% \def\DHLhksqrt#1#2{% +% \setbox0=\hbox{$#1\oldsqrt{#2\,}$}\dimen0=\ht0 +% \advance\dimen0-0.2\ht0 +% \setbox2=\hbox{\vrule height\ht0 depth -\dimen0}% +% {\box0\lower0.4pt\box2}} +% End code to redefine sqrt + +% \let\originalleft\left +% \let\originalright\right +% \def\left#1{\mathopen{}\originalleft#1} +% \def\right#1{\originalright#1\mathclose{}} + + + +%Nice tables +\usepackage{booktabs} +%More space between rows: +\renewcommand{\arraystretch}{1.2} %(or 1.3) + +% The fixmath package is to make uppercase greek letters appear in italic, not straight up +\usepackage{fixmath} + +% For psfrag +%\usepackage{psfrag} + +% Define color yellow +\definecolor{yellow}{RGB}{211,211,0} +\newcommand{\hl}[1]{\colorbox{yellow}{#1}} + + +% Nomenclature +\usepackage{nomencl} +\renewcommand{\nomgroup}[1]{% +\ifthenelse{\equal{#1}{A}}{\item[\textbf{Roman symbols}]}{% +\ifthenelse{\equal{#1}{G}}{\item[\textbf{Greek symbols}]}{% +\ifthenelse{\equal{#1}{O}}{\item[\textbf{Abbreviations and acronyms}]}{% +\ifthenelse{\equal{#1}{C}}{\item[\textbf{Calligraphic Symbols}]}{% +\ifthenelse{\equal{#1}{B}}{\item[\textbf{Abbreviations}]}{% +\ifthenelse{\equal{#1}{S}}{\item[\textbf{Sub- and superscripts}]}{% +\ifthenelse{\equal{#1}{D}}{\item[\textbf{Decorators}]}{% +\ifthenelse{\equal{#1}{M}}{\item[\textbf{Miscellaneous symbols and operators}]} +{} +}%Decorators +}% matches mathematical symbols +} +} +}% matches Subscripts +}% matches Abbreviations +}% matches Greek Symbols +}% matches Roman Symbols + + +%\newcommand{\nomunit}[1]{% +%\renewcommand{\nomentryend}{\hspace*{\fill}#1}} +% This one is with fill dots +\newcommand{\nomunit}[1]{% +\renewcommand{\nomentryend}{\dotfill[#1]}} +\newcommand{\nonomunit}{% +\renewcommand{\nomentryend}{\dotfill}} + +\renewcommand\nomname{List of symbols} + +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%Item separation in itemize and enumerate + +\let\oldenumerate=\enumerate +\def\enumerate{ +\oldenumerate +\setlength{\itemsep}{0pt} +} +%% Itemize spacing +\let\olditemize=\itemize +\def\itemize{ +\olditemize +\setlength{\itemsep}{0pt} +} + + +%% For the caption of graphics + +% Define lines for use in captions \lXYZ +% X = black(b), dark(d), light(l) +% Y = solid(s), dashed(da), dash-dot (dd) +% Z = normal(), thick(t) + +% \input{tex/hyphenation.tex} +%% This package does not work together with hyphenation advice? +%% \usepackage[english=usenglishmax]{hyphsubst} +