December  2015, 4(4): 447-491. doi: 10.3934/eect.2015.4.447

Mathematics of nonlinear acoustics

1. 

Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

Received  May 2015 Revised  October 2015 Published  November 2015

The aim of this paper is to highlight some recent developments and outcomes in the mathematical analysis of partial differential equations describing nonlinear sound propagation. Here the emphasis lies on well-posedness and decay results, first of all for the classical models of nonlinear acoustics, later on also for some higher order models. Besides quoting results, we also try to give an idea on their derivation by showning some of the crucial energy estimates. A section is devoted to optimization problems arising in the practical use of high intensity ultrasound.
    While this review puts a certain focus on results obtained in the context of the mentioned FWF project, we also provide some important additional references (although certainly not all of them) for interesting further reading.
Citation: Barbara Kaltenbacher. Mathematics of nonlinear acoustics. Evolution Equations & Control Theory, 2015, 4 (4) : 447-491. doi: 10.3934/eect.2015.4.447
References:
[1]

Numer. Funct. Anal. Optim., 19 (1998), 697-704. doi: 10.1080/01630569808816854.  Google Scholar

[2]

Monographs in Mathematics, 89, Birkhäuser, Boston, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

Jindřich Nečas Center for Mathematical Modeling Lecture Notes, 6, Matfyzpress, 2009.  Google Scholar

[4]

International Journal of Engineering Science, 74 (2014), 190-206. doi: 10.1016/j.ijengsci.2013.09.005.  Google Scholar

[5]

SIAM J. Numer. Anal., 34 (1997), 603-639. doi: 10.1137/S0036142994261518.  Google Scholar

[6]

J. Comput. Appl. Math., 152 (2003), 17-34. doi: 10.1016/S0377-0427(02)00694-5.  Google Scholar

[7]

Tech Report GD/E Report GD-1463-52 General Dynamics Corp., Rochester, New York, 1963. Google Scholar

[8]

Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[9]

SIAM J. Contr. Opt., 27 (1989), 446-455. doi: 10.1137/0327023.  Google Scholar

[10]

Springer, New York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[11]

J. Math. Anal. Appl., 433 (2016), 1037-1054. doi: 10.1016/j.jmaa.2015.07.046.  Google Scholar

[12]

R. Brunnhuber and P. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption,, submitted., ().   Google Scholar

[13]

Discrete and Continuous Dynamical Systems - A, 34 (2014), 4515-4535. doi: 10.3934/dcds.2014.34.4515.  Google Scholar

[14]

Evolution Equations and Control Theory, 3 (2014), 595-626. doi: 10.3934/eect.2014.3.595.  Google Scholar

[15]

PhD thesis, Alpen-Adria-Universität Klagenfurt, 2015. Google Scholar

[16]

R. Brunnhuber and S. Meyer, Optimal regularity and exponential stability for the Blackstock-Crighton equation in $L_p$-spaces with Dirichlet and Neumann boundary conditions,, , ().   Google Scholar

[17]

Springer, Berlin, New York, 1974. doi: 10.1007/978-94-010-1745-9.  Google Scholar

[18]

Control and Cybernetics, 31 (2002), 695-712.  Google Scholar

[19]

Physics Letters A, 373 (2009), 1037-1043. doi: 10.1016/j.physleta.2009.01.042.  Google Scholar

[20]

Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15.  Google Scholar

[21]

Mechanics Research Communications, 36 (2009), 481-486. doi: 10.1016/j.mechrescom.2008.11.003.  Google Scholar

[22]

The Quarterly Journal of Mechanics and Applied Mathematics, 60 (2007), 473-495. doi: 10.1093/qjmam/hbm017.  Google Scholar

[23]

Evolution Equations and Control Theory, 2 (2013), 281-300. doi: 10.3934/eect.2013.2.281.  Google Scholar

[24]

J. Math. Anal. Appl., 356 (2009), 738-751. doi: 10.1016/j.jmaa.2009.03.043.  Google Scholar

[25]

A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore-Gibson- Thompson equation,, Applied Mathematics & Information Sciences., ().   Google Scholar

[26]

Annual Review of Fluid Mechanics, 11 (1979), 11-33. Google Scholar

[27]

SIAM, 2001.  Google Scholar

[28]

in Shape Optimization and Free Boundaries (eds. M. Delfour and G. Sabidussi), NATO ASI Series, 380, Springer Netherlands, 1992, 35-111. doi: 10.1007/978-94-011-2710-3_2.  Google Scholar

[29]

Memoirs Amer. Math. Soc., 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[30]

in Proceedings of the IEEE Ultrasonics Symposium, IEEE, {2}, 2000, 1239-1242. doi: 10.1109/ULTSYM.2000.921547.  Google Scholar

[31]

Fluid Mechanics and Its Applications, Springer Netherlands, 2006. Available from: https://books.google.at/books?id=GuHpBwAAQBAJ. Google Scholar

[32]

Graduate Studies in Mathematics, American Mathematical Society, Providence, R.I., 1998.  Google Scholar

[33]

Int. J. Numer. Meth. Engng., 67 (2006), 1791-1810. doi: 10.1002/nme.1669.  Google Scholar

[34]

Academic Press, 1998. Google Scholar

[35]

SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718690.  Google Scholar

[36]

IEEE Transactions on Ultrasonics, Ferroelectics, and Frequency Control, 48 (2001), 779-786. doi: 10.1109/58.920712.  Google Scholar

[37]

Journal of Mathematical Analysis and Applications, 314 (2006), 126-149. doi: 10.1016/j.jmaa.2005.03.100.  Google Scholar

[38]

ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 517-539. doi: 10.1051/cocv:2008002.  Google Scholar

[39]

Discrete Contin. Dyn. Syst., Ser. B, 19 (2014), 2189-2205. doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar

[40]

J. Acoust. Soc. Am., 124 (2008), P2491. doi: 10.1121/1.4782790.  Google Scholar

[41]

Phys. Lett. A, 326 (2004), 77-84. doi: 10.1016/j.physleta.2004.03.067.  Google Scholar

[42]

in Proceedings of the OCEANS 2009 MTS/IEEE BILOXI Conference & Exhibition, 2009. Google Scholar

[43]

Applied Mathematics and Optimization, 62 (2010), 381-410. doi: 10.1007/s00245-010-9108-7.  Google Scholar

[44]

Discrete and Continuous Dynamical Systems (DCDS), 2 (2009), 503-523. doi: 10.3934/dcdss.2009.2.503.  Google Scholar

[45]

Discrete and Continuous Dynamical Systems Supplement, 2 (2011), 763-773.  Google Scholar

[46]

Mathematische Nachrichten, 285 (2012), 295-321. doi: 10.1002/mana.201000007.  Google Scholar

[47]

Control and Cybernetics, 40 (2011), 971-988. Google Scholar

[48]

M3AS, 22 (2012), 1250035, 34pp. doi: 10.1142/S0218202512500352.  Google Scholar

[49]

in Parabolic problems: Progr. Nonlinear Differential Equations Appl., 80, Birkhäuser/Springer Basel AG, Basel, 2011, 357-387. doi: 10.1007/978-3-0348-0075-4_19.  Google Scholar

[50]

IMA J. Numer. Anal., 35 (2015), 1092-1124. doi: 10.1093/imanum/dru029.  Google Scholar

[51]

B. Kaltenbacher and G. Peichl, Sensitivity analysis for a shape optimization problem in lithotripsy,, submitted., ().   Google Scholar

[52]

B. Kaltenbacher and I. Shevchenko, Well-posedness of the Westervelt equation with higher order absorbing boundary conditions,, submitted., ().   Google Scholar

[53]

European Journal of Applied Mathematics, 22 (2011), 21-43. doi: 10.1017/S0956792510000276.  Google Scholar

[54]

3rd edition, Springer, Berlin, 2015. Google Scholar

[55]

Wave Motion, 48 (2011), 782-790. doi: 10.1016/j.wavemoti.2011.04.013.  Google Scholar

[56]

Research in Applied Mathematics, Masson Chichester New York Brisbane, Paris, Milan, Barcelona, 1994.  Google Scholar

[57]

Physics Letters A, 374 (2010), 2011-2016. doi: 10.1016/j.physleta.2010.02.067.  Google Scholar

[58]

Soviet Physics-Acoustics, 16 (1971), 467-470. Google Scholar

[59]

I. Lasiecka and W. X., Moore-Gibson-Thompson equation with memory, part I: Exponential decay of energy,, submitted., ().   Google Scholar

[60]

Journal of Differential Equations, 259 (2015), 7610-7635. doi: 10.1016/j.jde.2015.08.052.  Google Scholar

[61]

Journal of Fluid Mechanics, 31 (1968), 501-528. doi: 10.1017/S0022112068000303.  Google Scholar

[62]

PhD thesis, University of Karlsruhe, 2006. Google Scholar

[63]

Journal of Inverse and Ill-Posed Problems, 21 (2013), 825-869. doi: 10.1515/jip-2012-0096.  Google Scholar

[64]

Acustica, Acta Acustica, 82 (1996), 579-606. Google Scholar

[65]

Acustica, Acta Acustica, 83 (1997), 197-222. Google Scholar

[66]

Acustica, Acta Acustica, 83 (1997), 827-846. Google Scholar

[67]

Mathematical Methods in the Applied Sciences, 35 (2012), 1896-1929. doi: 10.1002/mma.1576.  Google Scholar

[68]

Applied Mathematics and Optimization, 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.  Google Scholar

[69]

Evolution Equations and Control Theory, 2 (2013), 365-378. doi: 10.3934/eect.2013.2.365.  Google Scholar

[70]

Inverse Problems, 28 (2012), 093001, 35pp. doi: 10.1088/0266-5611/28/9/093001.  Google Scholar

[71]

J. Math. Anal. Appl., 427 (2015), 1131-1167. doi: 10.1016/j.jmaa.2015.02.076.  Google Scholar

[72]

PhD thesis, Alpen-Adria-University Klagenfurt, 2015. Google Scholar

[73]

V. Nikolić and B. Kaltenbacher, On higher regularity for the Westervelt equation with strong nonlinear damping,, submitted and , ().   Google Scholar

[74]

V. Nikolić and B. Kaltenbacher, Sensitivity analysis for shape optimization of a focusing acoustic lens in lithotripsy,, submitted and , ().   Google Scholar

[75]

Meccanica, 36 (2001), 297-321. doi: 10.1023/A:1013911407811.  Google Scholar

[76]

Texts in Applied Mathematics, Springer, New York, 2004.  Google Scholar

[77]

Acoustical Society of America, 1989. Available from: https://books.google.at/books?id=D8GqhULfKfAC. Google Scholar

[78]

A. Rasmussen, M. Sørensen, Y. Gaididei and P. Christiansen, Analytical and numerical modeling of front propagation and interaction of fronts in nonlinear thermoviscous fluids including dissipation,, , ().   Google Scholar

[79]

Acta Acustica United with Acustica, 99 (2013), 607-614. Google Scholar

[80]

Comptes Rendus Mathematique, 344 (2007), 337-342. doi: 10.1016/j.crma.2007.01.010.  Google Scholar

[81]

Appl. Anal., 89 (2010), 391-408. doi: 10.1080/00036810903569440.  Google Scholar

[82]

Commun. Math. Sci., 7 (2009), 679-718. doi: 10.4310/CMS.2009.v7.n3.a9.  Google Scholar

[83]

Journal of Computational Physics, 302 (2015), 200-221. doi: 10.1016/j.jcp.2015.08.051.  Google Scholar

[84]

in Mathematical Modeling of Wave Phenomena, AIP Conference Proceedings, Amer. Inst. Physics, 834, 2006, 214-221. doi: 10.1063/1.2205805.  Google Scholar

[85]

Springer, Berlin, New York, 2012. Google Scholar

[86]

Physics Letters A, 374 (2010), 2667-2669. doi: 10.1016/j.physleta.2010.04.054.  Google Scholar

[87]

Theory, Methods and Applications, vol. 112, Translated from the 2005 German original by Jürgen Sprekels, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar

[88]

Wave Motion, 58 (2015), 180-195. doi: 10.1016/j.wavemoti.2015.05.006.  Google Scholar

[89]

J. Comput. Acoust., 15 (2007), 353-375. doi: 10.1142/S0218396X0700338X.  Google Scholar

[90]

International Journal of Non-Linear Mechanics, 48 (2013), 72-77. doi: 10.1016/j.ijnonlinmec.2012.07.006.  Google Scholar

[91]

The Journal of the Acoustic Society of America, 35 (1963), 535-537. doi: 10.1121/1.1918525.  Google Scholar

[92]

Soviet Physics-Acoustics, 15 (1969), 35-40. Google Scholar

show all references

References:
[1]

Numer. Funct. Anal. Optim., 19 (1998), 697-704. doi: 10.1080/01630569808816854.  Google Scholar

[2]

Monographs in Mathematics, 89, Birkhäuser, Boston, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[3]

Jindřich Nečas Center for Mathematical Modeling Lecture Notes, 6, Matfyzpress, 2009.  Google Scholar

[4]

International Journal of Engineering Science, 74 (2014), 190-206. doi: 10.1016/j.ijengsci.2013.09.005.  Google Scholar

[5]

SIAM J. Numer. Anal., 34 (1997), 603-639. doi: 10.1137/S0036142994261518.  Google Scholar

[6]

J. Comput. Appl. Math., 152 (2003), 17-34. doi: 10.1016/S0377-0427(02)00694-5.  Google Scholar

[7]

Tech Report GD/E Report GD-1463-52 General Dynamics Corp., Rochester, New York, 1963. Google Scholar

[8]

Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[9]

SIAM J. Contr. Opt., 27 (1989), 446-455. doi: 10.1137/0327023.  Google Scholar

[10]

Springer, New York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[11]

J. Math. Anal. Appl., 433 (2016), 1037-1054. doi: 10.1016/j.jmaa.2015.07.046.  Google Scholar

[12]

R. Brunnhuber and P. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption,, submitted., ().   Google Scholar

[13]

Discrete and Continuous Dynamical Systems - A, 34 (2014), 4515-4535. doi: 10.3934/dcds.2014.34.4515.  Google Scholar

[14]

Evolution Equations and Control Theory, 3 (2014), 595-626. doi: 10.3934/eect.2014.3.595.  Google Scholar

[15]

PhD thesis, Alpen-Adria-Universität Klagenfurt, 2015. Google Scholar

[16]

R. Brunnhuber and S. Meyer, Optimal regularity and exponential stability for the Blackstock-Crighton equation in $L_p$-spaces with Dirichlet and Neumann boundary conditions,, , ().   Google Scholar

[17]

Springer, Berlin, New York, 1974. doi: 10.1007/978-94-010-1745-9.  Google Scholar

[18]

Control and Cybernetics, 31 (2002), 695-712.  Google Scholar

[19]

Physics Letters A, 373 (2009), 1037-1043. doi: 10.1016/j.physleta.2009.01.042.  Google Scholar

[20]

Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15.  Google Scholar

[21]

Mechanics Research Communications, 36 (2009), 481-486. doi: 10.1016/j.mechrescom.2008.11.003.  Google Scholar

[22]

The Quarterly Journal of Mechanics and Applied Mathematics, 60 (2007), 473-495. doi: 10.1093/qjmam/hbm017.  Google Scholar

[23]

Evolution Equations and Control Theory, 2 (2013), 281-300. doi: 10.3934/eect.2013.2.281.  Google Scholar

[24]

J. Math. Anal. Appl., 356 (2009), 738-751. doi: 10.1016/j.jmaa.2009.03.043.  Google Scholar

[25]

A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore-Gibson- Thompson equation,, Applied Mathematics & Information Sciences., ().   Google Scholar

[26]

Annual Review of Fluid Mechanics, 11 (1979), 11-33. Google Scholar

[27]

SIAM, 2001.  Google Scholar

[28]

in Shape Optimization and Free Boundaries (eds. M. Delfour and G. Sabidussi), NATO ASI Series, 380, Springer Netherlands, 1992, 35-111. doi: 10.1007/978-94-011-2710-3_2.  Google Scholar

[29]

Memoirs Amer. Math. Soc., 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[30]

in Proceedings of the IEEE Ultrasonics Symposium, IEEE, {2}, 2000, 1239-1242. doi: 10.1109/ULTSYM.2000.921547.  Google Scholar

[31]

Fluid Mechanics and Its Applications, Springer Netherlands, 2006. Available from: https://books.google.at/books?id=GuHpBwAAQBAJ. Google Scholar

[32]

Graduate Studies in Mathematics, American Mathematical Society, Providence, R.I., 1998.  Google Scholar

[33]

Int. J. Numer. Meth. Engng., 67 (2006), 1791-1810. doi: 10.1002/nme.1669.  Google Scholar

[34]

Academic Press, 1998. Google Scholar

[35]

SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718690.  Google Scholar

[36]

IEEE Transactions on Ultrasonics, Ferroelectics, and Frequency Control, 48 (2001), 779-786. doi: 10.1109/58.920712.  Google Scholar

[37]

Journal of Mathematical Analysis and Applications, 314 (2006), 126-149. doi: 10.1016/j.jmaa.2005.03.100.  Google Scholar

[38]

ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 517-539. doi: 10.1051/cocv:2008002.  Google Scholar

[39]

Discrete Contin. Dyn. Syst., Ser. B, 19 (2014), 2189-2205. doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar

[40]

J. Acoust. Soc. Am., 124 (2008), P2491. doi: 10.1121/1.4782790.  Google Scholar

[41]

Phys. Lett. A, 326 (2004), 77-84. doi: 10.1016/j.physleta.2004.03.067.  Google Scholar

[42]

in Proceedings of the OCEANS 2009 MTS/IEEE BILOXI Conference & Exhibition, 2009. Google Scholar

[43]

Applied Mathematics and Optimization, 62 (2010), 381-410. doi: 10.1007/s00245-010-9108-7.  Google Scholar

[44]

Discrete and Continuous Dynamical Systems (DCDS), 2 (2009), 503-523. doi: 10.3934/dcdss.2009.2.503.  Google Scholar

[45]

Discrete and Continuous Dynamical Systems Supplement, 2 (2011), 763-773.  Google Scholar

[46]

Mathematische Nachrichten, 285 (2012), 295-321. doi: 10.1002/mana.201000007.  Google Scholar

[47]

Control and Cybernetics, 40 (2011), 971-988. Google Scholar

[48]

M3AS, 22 (2012), 1250035, 34pp. doi: 10.1142/S0218202512500352.  Google Scholar

[49]

in Parabolic problems: Progr. Nonlinear Differential Equations Appl., 80, Birkhäuser/Springer Basel AG, Basel, 2011, 357-387. doi: 10.1007/978-3-0348-0075-4_19.  Google Scholar

[50]

IMA J. Numer. Anal., 35 (2015), 1092-1124. doi: 10.1093/imanum/dru029.  Google Scholar

[51]

B. Kaltenbacher and G. Peichl, Sensitivity analysis for a shape optimization problem in lithotripsy,, submitted., ().   Google Scholar

[52]

B. Kaltenbacher and I. Shevchenko, Well-posedness of the Westervelt equation with higher order absorbing boundary conditions,, submitted., ().   Google Scholar

[53]

European Journal of Applied Mathematics, 22 (2011), 21-43. doi: 10.1017/S0956792510000276.  Google Scholar

[54]

3rd edition, Springer, Berlin, 2015. Google Scholar

[55]

Wave Motion, 48 (2011), 782-790. doi: 10.1016/j.wavemoti.2011.04.013.  Google Scholar

[56]

Research in Applied Mathematics, Masson Chichester New York Brisbane, Paris, Milan, Barcelona, 1994.  Google Scholar

[57]

Physics Letters A, 374 (2010), 2011-2016. doi: 10.1016/j.physleta.2010.02.067.  Google Scholar

[58]

Soviet Physics-Acoustics, 16 (1971), 467-470. Google Scholar

[59]

I. Lasiecka and W. X., Moore-Gibson-Thompson equation with memory, part I: Exponential decay of energy,, submitted., ().   Google Scholar

[60]

Journal of Differential Equations, 259 (2015), 7610-7635. doi: 10.1016/j.jde.2015.08.052.  Google Scholar

[61]

Journal of Fluid Mechanics, 31 (1968), 501-528. doi: 10.1017/S0022112068000303.  Google Scholar

[62]

PhD thesis, University of Karlsruhe, 2006. Google Scholar

[63]

Journal of Inverse and Ill-Posed Problems, 21 (2013), 825-869. doi: 10.1515/jip-2012-0096.  Google Scholar

[64]

Acustica, Acta Acustica, 82 (1996), 579-606. Google Scholar

[65]

Acustica, Acta Acustica, 83 (1997), 197-222. Google Scholar

[66]

Acustica, Acta Acustica, 83 (1997), 827-846. Google Scholar

[67]

Mathematical Methods in the Applied Sciences, 35 (2012), 1896-1929. doi: 10.1002/mma.1576.  Google Scholar

[68]

Applied Mathematics and Optimization, 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.  Google Scholar

[69]

Evolution Equations and Control Theory, 2 (2013), 365-378. doi: 10.3934/eect.2013.2.365.  Google Scholar

[70]

Inverse Problems, 28 (2012), 093001, 35pp. doi: 10.1088/0266-5611/28/9/093001.  Google Scholar

[71]

J. Math. Anal. Appl., 427 (2015), 1131-1167. doi: 10.1016/j.jmaa.2015.02.076.  Google Scholar

[72]

PhD thesis, Alpen-Adria-University Klagenfurt, 2015. Google Scholar

[73]

V. Nikolić and B. Kaltenbacher, On higher regularity for the Westervelt equation with strong nonlinear damping,, submitted and , ().   Google Scholar

[74]

V. Nikolić and B. Kaltenbacher, Sensitivity analysis for shape optimization of a focusing acoustic lens in lithotripsy,, submitted and , ().   Google Scholar

[75]

Meccanica, 36 (2001), 297-321. doi: 10.1023/A:1013911407811.  Google Scholar

[76]

Texts in Applied Mathematics, Springer, New York, 2004.  Google Scholar

[77]

Acoustical Society of America, 1989. Available from: https://books.google.at/books?id=D8GqhULfKfAC. Google Scholar

[78]

A. Rasmussen, M. Sørensen, Y. Gaididei and P. Christiansen, Analytical and numerical modeling of front propagation and interaction of fronts in nonlinear thermoviscous fluids including dissipation,, , ().   Google Scholar

[79]

Acta Acustica United with Acustica, 99 (2013), 607-614. Google Scholar

[80]

Comptes Rendus Mathematique, 344 (2007), 337-342. doi: 10.1016/j.crma.2007.01.010.  Google Scholar

[81]

Appl. Anal., 89 (2010), 391-408. doi: 10.1080/00036810903569440.  Google Scholar

[82]

Commun. Math. Sci., 7 (2009), 679-718. doi: 10.4310/CMS.2009.v7.n3.a9.  Google Scholar

[83]

Journal of Computational Physics, 302 (2015), 200-221. doi: 10.1016/j.jcp.2015.08.051.  Google Scholar

[84]

in Mathematical Modeling of Wave Phenomena, AIP Conference Proceedings, Amer. Inst. Physics, 834, 2006, 214-221. doi: 10.1063/1.2205805.  Google Scholar

[85]

Springer, Berlin, New York, 2012. Google Scholar

[86]

Physics Letters A, 374 (2010), 2667-2669. doi: 10.1016/j.physleta.2010.04.054.  Google Scholar

[87]

Theory, Methods and Applications, vol. 112, Translated from the 2005 German original by Jürgen Sprekels, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar

[88]

Wave Motion, 58 (2015), 180-195. doi: 10.1016/j.wavemoti.2015.05.006.  Google Scholar

[89]

J. Comput. Acoust., 15 (2007), 353-375. doi: 10.1142/S0218396X0700338X.  Google Scholar

[90]

International Journal of Non-Linear Mechanics, 48 (2013), 72-77. doi: 10.1016/j.ijnonlinmec.2012.07.006.  Google Scholar

[91]

The Journal of the Acoustic Society of America, 35 (1963), 535-537. doi: 10.1121/1.1918525.  Google Scholar

[92]

Soviet Physics-Acoustics, 15 (1969), 35-40. Google Scholar

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