Abstract
Scattering of sound waves in two stepped cylindrical duct which walls are coated with different acoustically absorbent materials is investigated by using Wiener-Hopf technique directly and by determining scattering matrices. First, by using Fourier transform technique we obtain a couple of modified Wiener-Hopf equations whose solutions involve four sets of infinitely many unknown expansion coefficients providing systems of linear algebraic equations. Then we determine scattering matrices of the problem and we state the total transmitted field by using generalized scattering matrix method. Numerical results are compared for different parameters.Keywords:
Wiener-Hopf, scattering matrices, step discontinuity, impedance discontinuity, ductReferences
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34. Tiryakioglu B., Demir A. (2019), Sound Wave Radiation from Partially Lined Duct, Archives of Acoustics, 44(2): 239–249, https://doi.org/10.24425/aoa.2019.128487
35. Utsumi M. (1999), An efficient method for sound transmission in non-uniform circular ducts, Journal of Sound and Vibration, 227(4): 735–748, https://doi.org/10.1006/jsvi.1999.2381
36. VanBlaricum G.F., Mittra R. (1969), A modified residue-calculus technique for solving a class of boundary value problems. Part I: Waveguide discontinuities, IEEE Transactions on Microwave Theory and Techniques, 17(6): 302–309, https://doi.org/10.1109/TMTT.1969.1126961
37. Wang X., Sun X. (2011), A new segmentation approach for sound propagation in non-uniform lined ducts with mean flow, Journal of Sound and Vibration, 330(10): 2369–2387, https://doi.org/10.1016/j.jsv.2010.11.022
38. Warren D.P., Lawrie J.B., Mohamed I.M. (2002), Acoustic scattering in waveguides that are discontinuous in geometry and material property, Wave Motion, 36(2): 119–142, https://doi.org/10.1016/S0165-2125%2802%2900005-7
2. Alfredson R.J. (1972), The propagation of sound in circular duct of continuous varying cross-sectional area, Journal of Sound and Vibration, 23(4): 433–442, https://doi.org/10.1016/0022-460X%2872%2990502-0
3. Campos L.M.B.C. (1984), Some general properties of the exact acoustic fields in horns and baffles, Journal of Sound and Vibration, 95(2): 177–201, https://doi.org/10.1016/0022-460X%2884%2990541-8
4. Campos L.M.B.C., Oliveira J.M.G.S. (2013), On sound generation in cylindrical flow duct with non-uniform wall impedance, International Journal of Aeroacoustics, 12(4): 309–347, https://doi.org/10.1260/1475-472X.12.4.309
5. Demir A. (2016), Sound transmission in a duct with sudden area expansion, extended inlet and lined walls in overlapping region, Advances in Acoustic and Vibration, 2016: 9485163, https://doi.org/10.1155/2016/9485163
6. Demir A. (2017), Scattering matrices in nonuniformly lined ducts, Zeitschrift für angewandte Mathematik und Physik, 68: 8, https://doi.org/10.1007/s00033-016-0754-8
7. Gupta V.H., Easwaran V., Munjal M.L. (1995), A modified segmentation approach for analyzing plane wave propagation in non-uniform ducts with mean flow, Journal of Sound and Vibration, 182(5): 697–707, https://doi.org/10.1006/jsvi.1995.0227
8. Homentcovschi D., Miles R.N. (2010), A reexpansion method for determining the acoustical impedance and the scattering matrix for the waveguide discontinuity problem, Journal of the Acoustical Society of America, 128(2): 628–638, https://doi.org/10.1121/1.3455836
9. Hudde H., Letens U. (1985), Scattering matrix of a discontinuity with a non-rigid wall in a lossless circular duct, Journal of the Acoustical Society of America, 78(5): 1826–1837, https://doi.org/10.1121/1.392769
10. Karal F.C. (1953), The analogous acoustical impedance for discontinuities of circular cross section, Journal of the Acoustical Society of America, 25(2): 327– 334, https://doi.org/10.1121/1.1907041
11. Kergomard J. (1991), Calculation of discontinuities in waveguides using mode-matching method: an alternative to the scattering matrix approach, Journal d’Acoustique (Les Ulis), 4(2): 111–138.
12. Kergomard J., Garcia A. (1987), Simple discontinuities in acoustic waveguides at low frequencies: Critical analysis and formulae, Journal of Sound and Vibration, 114(3): 465–479, https://doi.org/10.1016/S0022-460X%2887%2980017-2
13. Levine H., Schwinger J. (1948), On the radiation of sound from an unflanged circular pipe, Physical Review, 73(4): 383–406, https://doi.org/10.1103/PhysRev.73.383
14. McApline A., Astley R.J., Hii V.J.T., Baker N.J., Kempton A.J. (2006), Acoustic scattering by an 520 Archives of Acoustics – Volume 45, Number 3, 2020 axially-segmented turbofan inlet duct, Journal of Sound and Vibration, 294(4–5): 780–806, https://doi.org/10.1016/j.jsv.2005.12.039
15. Miles J. (1944), The reflection of sound due to a change in cross section of a circular tube, Journal of the Acoustical Society of America, 16(1): 14–19, https://doi.org/10.1121/1.1916257
16. Miles J.W. (1945a), The analysis of plane discontinuities in cylindrical tubes: part I, Journal of the Acoustical Society of America, 17(3): 259–271, https://doi.org/10.1121/1.1916327
17. Miles J.W. (1945b), The analysis of plane discontinuities in cylindrical tubes: part II, Journal of the Acoustical Society of America, 17(3): 272–284, https://doi.org/10.1121/1.1916328
18. Mittra R., Lee S.W. (1971), Analytical Techniques in the Theory of Guided Waves, The Macmillian Company, New York.
19. Morse P.M. (1948), Vibration and Sound, McGraw Hill, New York, NY.
20. Morse P.M., Feschbach H. (1953), Methods of Theoretical Physics, Vol. 1–2, McGraw Hill, New York, NY.
21. Nilsson B., Brander O. (1980a), The Propagation of sound in cylindrical ducts with mean flow and bulk-reacting lining. I. Modes in an infinite duct, IMA Journal of Applied Mathematics, 26(3): 269–298, https://doi.org/10.1093/imamat/26.3.269
22. Nilsson B., Brander O. (1980b), The Propagation of sound in cylindrical ducts with mean flow and bulk-reacting lining. II. Bifurcated ducts, IMA Journal of Applied Mathematics, 26(4): 381–410, https://doi.org/10.1093/imamat/26.4.381
23. Nilsson B., Brander O. (1980c), The Propagation of sound in cylindrical ducts with mean flow and bulk-reacting lining. III. Step discontinuities, IMA Journal of Applied Mathematics, 27(1): 105–131, https://doi.org/10.1093/imamat/27.1.105
24. Nilsson B., Brander O. (1980d), The Propagation of sound in cylindrical ducts with mean flow and bulkreacting lining. IV. Several interacting discontinuities, IMA Journal of Applied Mathematics, 27(3): 263–290, https://doi.org/10.1093/imamat/27.3.263
25. Noble B. (1969), Methods based on the Wiener-Hopf Technique for the solution of partial equations, Pergamon Press Inc.
26. Pace J., Mittra R. (1964), Generalized scattering matrix analysis of waveguide discontinuity problems, Quasi-Optics, XIV: 177–197.
27. Pagneux V., Amir N., Kergomard J. (1996), A study of wave propagation in varying cross-section waveguides by modal decomposition. Part I: theory and validation, Journal of the Acoustical Society of America, 100(4): 2034–2048, https://doi.org/10.1121/1.417913
28. Papadopoulos V.M. (1957), The scattering effect of a junction between two circular waveguides, The Quarterly Journal of Mechanics and Applied Mathematics, 10(2): 191–209, https://doi.org/10.1093/qjmam/10.2.191
29. Peake N., Abrahams I.D. (2020), Sound radiation from a semi-infinite lined duct, Wave Motion, 92: 102407, 10.1016/j.wavemoti.2019.102407.
30. Pierce A.D. (1981), Acoustics: An Introduction to its Physical Principles and Applications, McGraw Hill, New York, NY.
31. Rayleigh J.W.S. (1945), The Theory of Sound, Vol. 1–2, Dover, New York.
32. Rienstra S.W. (2007), Acoustic scattering at a hardsoft lining transition in a flow duct, Journal of Engineering Mathematics, 59(4): 451–475, https://doi.org/10.1007/s10665-007-9193-z
33. Snakowska A., Jurkiewicz J., Gorazd L. (2017), A hybrid method for determination of the acoustic impedance of an unflanged cylindrical duct for multimode wave, Journal of Sound and Vibration, 396: 325–339, https://doi.org/10.1016/j.jsv.2017.02.040
34. Tiryakioglu B., Demir A. (2019), Sound Wave Radiation from Partially Lined Duct, Archives of Acoustics, 44(2): 239–249, https://doi.org/10.24425/aoa.2019.128487
35. Utsumi M. (1999), An efficient method for sound transmission in non-uniform circular ducts, Journal of Sound and Vibration, 227(4): 735–748, https://doi.org/10.1006/jsvi.1999.2381
36. VanBlaricum G.F., Mittra R. (1969), A modified residue-calculus technique for solving a class of boundary value problems. Part I: Waveguide discontinuities, IEEE Transactions on Microwave Theory and Techniques, 17(6): 302–309, https://doi.org/10.1109/TMTT.1969.1126961
37. Wang X., Sun X. (2011), A new segmentation approach for sound propagation in non-uniform lined ducts with mean flow, Journal of Sound and Vibration, 330(10): 2369–2387, https://doi.org/10.1016/j.jsv.2010.11.022
38. Warren D.P., Lawrie J.B., Mohamed I.M. (2002), Acoustic scattering in waveguides that are discontinuous in geometry and material property, Wave Motion, 36(2): 119–142, https://doi.org/10.1016/S0165-2125%2802%2900005-7
