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, 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]

Do ng ZhangXi ufen Gong and Sh igong 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]

Do ng ZhangXi ufen Gong and Sh igong 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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