June  2021, 10(2): 229-247. doi: 10.3934/eect.2020063

Periodic solutions and multiharmonic expansions for the Westervelt equation

Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

Received  August 2019 Revised  March 2020 Published  June 2020

In this paper we consider nonlinear time periodic sound propagation according to the Westervelt equation, which is a classical model of nonlinear acoustics and a second order quasilinear strongly damped wave equation exhibiting potential degeneracy. We prove existence, uniqueness and regularity of solutions with time periodic forcing and time periodic initial-end conditions, on a bounded domain with absorbing boundary conditions. In order to mathematically recover the physical phenomenon of higher harmonics, we expand the solution as a superposition of contributions at frequencies that are multiples of a fundamental excitation frequency. This multiharmonic expansion is proven to converge, in appropriate function spaces, to the periodic solution in time domain.

Citation: Barbara Kaltenbacher. Periodic solutions and multiharmonic expansions for the Westervelt equation. Evolution Equations & Control Theory, 2021, 10 (2) : 229-247. doi: 10.3934/eect.2020063
References:
[1]

A. AnvariF. Forsberg and A. E. Samir, A primer on the physical principles of tissue harmonic imaging, RadioGraphics, 35 (2015), 1955-1964.  doi: 10.1148/rg.2015140338.  Google Scholar

[2]

L. Bjørnø, Characterization of biological media by means of their non-linearity, Ultrasonics, 24 (1986), 254-259.   Google Scholar

[3]

H. Brèzis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math., 31 (1978), 1–30. doi: 10.1002/cpa.3160310102.  Google Scholar

[4]

R. Brunnhuber and P. M. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption, International Journal of Non-Linear Mechanics, 78 (2016), 131-132.  doi: 10.1016/j.ijnonlinmec.2015.10.008.  Google Scholar

[5]

R. BrunnhuberB. Kaltenbacher and P. Radu, Relaxation of regularity for the Westervelt equation by nonlinear damping with application in acoustic-acoustic and elastic-acoustic coupling, Evol. Equ. Control Theory, 3 (2014), 595-626.  doi: 10.3934/eect.2014.3.595.  Google Scholar

[6] J. Burgers, A mathematical model illustrating the theory of turbulence, in Advances in Applied Mechanics, Academic Press, Inc., 1948.   Google Scholar
[7]

V. BurovI. GurinovichO. Rudenko and E. Tagunov, Reconstruction of the spatial distribution of the nonlinearity parameter and sound velocity in acoustic nonlinear tomography, Acoustical Physics, 40 (1994), 816-823.   Google Scholar

[8]

C. A. Cain, Ultrasonic reflection mode imaging of the nonlinear parameter B/A: I. A theoretical basis, The Journal of the Acoustical Society of America, 80 (1986), 28-32.  doi: 10.1109/ULTSYM.1985.198640.  Google Scholar

[9]

A. Celik and M. Kyed, Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech, 98 (2018), 412-430.  doi: 10.1002/zamm.201600280.  Google Scholar

[10]

A. Celik and M. Kyed, Nonlinear acoustics: Blackstock-Crighton equations with a periodic forcing term, J. Math. Fluid Mech., 21 (2019), no. 3, Paper No. 45, 12 pp. doi: 10.1007/s00021-019-0451-4.  Google Scholar

[11]

T. Christopher, Finite amplitude distortion-based inhomogeneous pulse echo ultrasonic imaging, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 44 (1997), 125-139.  doi: 10.1109/58.585208.  Google Scholar

[12]

R. D. Fay, Plane sound waves of finite amplitude, J. Acoust. Soc. Amer., 3 (1931), 222-241.  doi: 10.1121/1.1915557.  Google Scholar

[13]

A. S. Fokas and J. T. Stuart, The time periodic solution of the Burgers equation on the half-line and an application to steady streaming, J. Nonlinear Math. Phys., 12 (2006), 302-314.  doi: 10.2991/jnmp.2005.12.s1.24.  Google Scholar

[14]

M. Fontes and O. Verdier, Time-periodic solutions of the Burgers equation, J. Math. Fluid Mech., 11 (2009), 303-323.  doi: 10.1007/s00021-007-0260-z.  Google Scholar

[15]

E. Fubini, Anomalies in the propagation of acoustic waves at great amplitude, Alta Frequenza, 4 (1935), 530-581.   Google Scholar

[16]

D. Givoli, Non-reflecting boundary conditions, J. Comput. Phys., 94 (1991), 1-29.  doi: 10.1016/0021-9991(91)90135-8.  Google Scholar

[17]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 8 (1999), 47-106.  doi: 10.1017/S0962492900002890.  Google Scholar

[18]

N. IchidaT. Sato and M. Linzer, Imaging the nonlinear ultrasonic parameter of a medium, Ultrasonic Imaging, 5 (1983), 295-299.  doi: 10.1177/016173468300500401.  Google Scholar

[19]

H. R. Jauslin, H. O. Kreiss, and J. Moser, On the forced Burgers equation with periodic boundary conditions, In Differential Equations: La Pietra 1996 (Florence), Amer. Math. Soc., Providence, RI, 1999,133–153. doi: 10.1090/pspum/065/1662751.  Google Scholar

[20]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst., Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar

[21]

P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910–2009, Mechanics Research Communications, 73 (2016), 127-139.  doi: 10.1016/j.mechrescom.2016.02.014.  Google Scholar

[22]

B. Kaltenbacher, Mathematics of Nonlinear Acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.  Google Scholar

[23]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst., Ser. S, 2 (2009), 503-523.  doi: 10.3934/dcdss.2009.2.503.  Google Scholar

[24]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, in Discrete Contin. Dyn. Syst. 2011, 8th AIMS Conference. Suppl. Vol. II, 2011,763–773.  Google Scholar

[25]

B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321.  doi: 10.1002/mana.201000007.  Google Scholar

[26]

B. Kaltenbacher, I. Lasiecka and M. A. Pospieszalska, Wellposedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352.  Google Scholar

[27]

B. Kaltenbacher and M. Thalhammer, Fundamental models in nonlinear acoustics part I. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.  doi: 10.1142/S0218202518500525.  Google Scholar

[28]

N. N. Kochina, On periodic solutions of Burgers' equation, J. Appl. Math. Mech., 25 (1962), 1597-1607.  doi: 10.1016/0021-8928(62)90138-7.  Google Scholar

[29]

P. Kokocki, Effect of resonance on the existence of periodic solutions for strongly damped wave equation, Nonlinear Anal., 125 (2015), 167-200.  doi: 10.1016/j.na.2015.05.012.  Google Scholar

[30]

N. Krylová, Periodic solutions of hyperbolic partial differential equation with quadratic dissipative term, Czechoslovak Math. J., 20 (1970), 375-405.   Google Scholar

[31]

V. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics-Acoustics, 16 (1971), 467-470.   Google Scholar

[32]

M. B. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528.  doi: 10.1017/S0022112068000303.  Google Scholar

[33]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271.  doi: 10.1007/s00245-011-9138-9.  Google Scholar

[34]

S. Meyer and M. Wilke, Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.  doi: 10.3934/eect.2013.2.365.  Google Scholar

[35]

A. Parker, On the periodic solution of the Burgers equation: A unified approach, Proc. Roy. Soc. London Ser. A, 438 (1992), 113-132.  doi: 10.1098/rspa.1992.0096.  Google Scholar

[36]

A. Rozanova, The Khokhlov-Zabolotskaya-Kuznetsov equation, C. R. Math. Acad. Sci. Paris, 344 (2007), 337-342.  doi: 10.1016/j.crma.2007.01.010.  Google Scholar

[37]

I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation, J. Comput. Phys., 302 (2015), 200-221.  doi: 10.1016/j.jcp.2015.08.051.  Google Scholar

[38]

G. Simonett and M. Wilke, Well-posedness and longtime behavior for the Westervelt equation with absorbing boundary conditions of order zero, J. Evol. Equ., 17 (2017), 551-571.  doi: 10.1007/s00028-016-0361-3.  Google Scholar

[39]

G. Teschl, Ordinary differential equations and dynamical systems, Graduate Studies in Mathematics, Vol. 140, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/140.  Google Scholar

[40]

S. Tsynkov, Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math., 27 (1998), 465-532.  doi: 10.1016/S0168-9274(98)00025-7.  Google Scholar

[41]

F. VarrayO. BassetP. Tortoli and C. Cachard, Extensions of nonlinear B/A parameter imaging methods for echo mode, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 58 (2011), 1232-44.  doi: 10.1109/TUFFC.2011.1933.  Google Scholar

[42]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America, 35 (1963), 535-537.  doi: 10.1121/1.1918525.  Google Scholar

[43]

E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams, Soviet Physics-Acoustics, 15 (1969), 35-40.   Google Scholar

[44]

D. ZhangX. Chen and X. F. Gong, Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source–Theoretical analysis and computer simulations, The Journal of the Acoustical Society of America, 109 (2001), 1219-1225.  doi: 10.1121/1.1344160.  Google Scholar

[45]

Dong ZhangXiufen Gong and Shigong Ye, Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics, The Journal of the Acoustical Society of America, 99 (1996), 2397-2402.  doi: 10.1121/1.415427.  Google Scholar

show all references

References:
[1]

A. AnvariF. Forsberg and A. E. Samir, A primer on the physical principles of tissue harmonic imaging, RadioGraphics, 35 (2015), 1955-1964.  doi: 10.1148/rg.2015140338.  Google Scholar

[2]

L. Bjørnø, Characterization of biological media by means of their non-linearity, Ultrasonics, 24 (1986), 254-259.   Google Scholar

[3]

H. Brèzis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math., 31 (1978), 1–30. doi: 10.1002/cpa.3160310102.  Google Scholar

[4]

R. Brunnhuber and P. M. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption, International Journal of Non-Linear Mechanics, 78 (2016), 131-132.  doi: 10.1016/j.ijnonlinmec.2015.10.008.  Google Scholar

[5]

R. BrunnhuberB. Kaltenbacher and P. Radu, Relaxation of regularity for the Westervelt equation by nonlinear damping with application in acoustic-acoustic and elastic-acoustic coupling, Evol. Equ. Control Theory, 3 (2014), 595-626.  doi: 10.3934/eect.2014.3.595.  Google Scholar

[6] J. Burgers, A mathematical model illustrating the theory of turbulence, in Advances in Applied Mechanics, Academic Press, Inc., 1948.   Google Scholar
[7]

V. BurovI. GurinovichO. Rudenko and E. Tagunov, Reconstruction of the spatial distribution of the nonlinearity parameter and sound velocity in acoustic nonlinear tomography, Acoustical Physics, 40 (1994), 816-823.   Google Scholar

[8]

C. A. Cain, Ultrasonic reflection mode imaging of the nonlinear parameter B/A: I. A theoretical basis, The Journal of the Acoustical Society of America, 80 (1986), 28-32.  doi: 10.1109/ULTSYM.1985.198640.  Google Scholar

[9]

A. Celik and M. Kyed, Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech, 98 (2018), 412-430.  doi: 10.1002/zamm.201600280.  Google Scholar

[10]

A. Celik and M. Kyed, Nonlinear acoustics: Blackstock-Crighton equations with a periodic forcing term, J. Math. Fluid Mech., 21 (2019), no. 3, Paper No. 45, 12 pp. doi: 10.1007/s00021-019-0451-4.  Google Scholar

[11]

T. Christopher, Finite amplitude distortion-based inhomogeneous pulse echo ultrasonic imaging, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 44 (1997), 125-139.  doi: 10.1109/58.585208.  Google Scholar

[12]

R. D. Fay, Plane sound waves of finite amplitude, J. Acoust. Soc. Amer., 3 (1931), 222-241.  doi: 10.1121/1.1915557.  Google Scholar

[13]

A. S. Fokas and J. T. Stuart, The time periodic solution of the Burgers equation on the half-line and an application to steady streaming, J. Nonlinear Math. Phys., 12 (2006), 302-314.  doi: 10.2991/jnmp.2005.12.s1.24.  Google Scholar

[14]

M. Fontes and O. Verdier, Time-periodic solutions of the Burgers equation, J. Math. Fluid Mech., 11 (2009), 303-323.  doi: 10.1007/s00021-007-0260-z.  Google Scholar

[15]

E. Fubini, Anomalies in the propagation of acoustic waves at great amplitude, Alta Frequenza, 4 (1935), 530-581.   Google Scholar

[16]

D. Givoli, Non-reflecting boundary conditions, J. Comput. Phys., 94 (1991), 1-29.  doi: 10.1016/0021-9991(91)90135-8.  Google Scholar

[17]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 8 (1999), 47-106.  doi: 10.1017/S0962492900002890.  Google Scholar

[18]

N. IchidaT. Sato and M. Linzer, Imaging the nonlinear ultrasonic parameter of a medium, Ultrasonic Imaging, 5 (1983), 295-299.  doi: 10.1177/016173468300500401.  Google Scholar

[19]

H. R. Jauslin, H. O. Kreiss, and J. Moser, On the forced Burgers equation with periodic boundary conditions, In Differential Equations: La Pietra 1996 (Florence), Amer. Math. Soc., Providence, RI, 1999,133–153. doi: 10.1090/pspum/065/1662751.  Google Scholar

[20]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst., Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar

[21]

P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910–2009, Mechanics Research Communications, 73 (2016), 127-139.  doi: 10.1016/j.mechrescom.2016.02.014.  Google Scholar

[22]

B. Kaltenbacher, Mathematics of Nonlinear Acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.  Google Scholar

[23]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst., Ser. S, 2 (2009), 503-523.  doi: 10.3934/dcdss.2009.2.503.  Google Scholar

[24]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, in Discrete Contin. Dyn. Syst. 2011, 8th AIMS Conference. Suppl. Vol. II, 2011,763–773.  Google Scholar

[25]

B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321.  doi: 10.1002/mana.201000007.  Google Scholar

[26]

B. Kaltenbacher, I. Lasiecka and M. A. Pospieszalska, Wellposedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352.  Google Scholar

[27]

B. Kaltenbacher and M. Thalhammer, Fundamental models in nonlinear acoustics part I. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.  doi: 10.1142/S0218202518500525.  Google Scholar

[28]

N. N. Kochina, On periodic solutions of Burgers' equation, J. Appl. Math. Mech., 25 (1962), 1597-1607.  doi: 10.1016/0021-8928(62)90138-7.  Google Scholar

[29]

P. Kokocki, Effect of resonance on the existence of periodic solutions for strongly damped wave equation, Nonlinear Anal., 125 (2015), 167-200.  doi: 10.1016/j.na.2015.05.012.  Google Scholar

[30]

N. Krylová, Periodic solutions of hyperbolic partial differential equation with quadratic dissipative term, Czechoslovak Math. J., 20 (1970), 375-405.   Google Scholar

[31]

V. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics-Acoustics, 16 (1971), 467-470.   Google Scholar

[32]

M. B. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528.  doi: 10.1017/S0022112068000303.  Google Scholar

[33]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271.  doi: 10.1007/s00245-011-9138-9.  Google Scholar

[34]

S. Meyer and M. Wilke, Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.  doi: 10.3934/eect.2013.2.365.  Google Scholar

[35]

A. Parker, On the periodic solution of the Burgers equation: A unified approach, Proc. Roy. Soc. London Ser. A, 438 (1992), 113-132.  doi: 10.1098/rspa.1992.0096.  Google Scholar

[36]

A. Rozanova, The Khokhlov-Zabolotskaya-Kuznetsov equation, C. R. Math. Acad. Sci. Paris, 344 (2007), 337-342.  doi: 10.1016/j.crma.2007.01.010.  Google Scholar

[37]

I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation, J. Comput. Phys., 302 (2015), 200-221.  doi: 10.1016/j.jcp.2015.08.051.  Google Scholar

[38]

G. Simonett and M. Wilke, Well-posedness and longtime behavior for the Westervelt equation with absorbing boundary conditions of order zero, J. Evol. Equ., 17 (2017), 551-571.  doi: 10.1007/s00028-016-0361-3.  Google Scholar

[39]

G. Teschl, Ordinary differential equations and dynamical systems, Graduate Studies in Mathematics, Vol. 140, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/140.  Google Scholar

[40]

S. Tsynkov, Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math., 27 (1998), 465-532.  doi: 10.1016/S0168-9274(98)00025-7.  Google Scholar

[41]

F. VarrayO. BassetP. Tortoli and C. Cachard, Extensions of nonlinear B/A parameter imaging methods for echo mode, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 58 (2011), 1232-44.  doi: 10.1109/TUFFC.2011.1933.  Google Scholar

[42]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America, 35 (1963), 535-537.  doi: 10.1121/1.1918525.  Google Scholar

[43]

E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams, Soviet Physics-Acoustics, 15 (1969), 35-40.   Google Scholar

[44]

D. ZhangX. Chen and X. F. Gong, Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source–Theoretical analysis and computer simulations, The Journal of the Acoustical Society of America, 109 (2001), 1219-1225.  doi: 10.1121/1.1344160.  Google Scholar

[45]

Dong ZhangXiufen Gong and Shigong Ye, Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics, The Journal of the Acoustical Society of America, 99 (1996), 2397-2402.  doi: 10.1121/1.415427.  Google Scholar

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