March  2017, 37(3): 1159-1181. doi: 10.3934/dcds.2017048

Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers

1. 

CMLA, ENS Cachan, CNRS, Universit, Paris-Saclay, 94235 Cachan, France

2. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago 851 S Morgan St, Chicago, IL 60607, USA

3. 

Department of Mathematical Sciences, School of Natural Science Ulsan National Institute of Science and Technology, UNIST-gil 50, Ulsan 689-798, Republic of Korea

* Corresponding author

Received  November 2015 Revised  October 2016 Published  December 2016

We study the asymptotic behavior of the two dimensional Helmholtz scattering problem with high wave numbers in an exterior domain, the exterior of a circle. We impose the Dirichlet boundary condition on the obstacle, which corresponds to an incidental wave. For the outer boundary, we consider the Sommerfeld conditions. Using a polar coordinates expansion, the problem is reduced to a sequence of Bessel equations. Investigating the Bessel equations mode by mode, we find that the solution of the scattering problem converges to its limit solution at a specific rate depending on k.

Citation: Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048
References:
[1]

M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington, D. C. 1964. Google Scholar

[2]

I. Andronov, D. Bouche and F. Molinet, Asymptotic and Hybrid Methods in Electromagnetics volume 48 of IEE Electromagnetic Waves Series, Institution of Electrical Engineers (IEE), London, 2005. doi: 10.1049/PBEW051E. Google Scholar

[3]

V. M. Babič and V. S. Buldyrev, Short-wavelength Diffraction Theory volume 4 of Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1991. Asymptotic methods, Translated from the 1972 Russian original by E. F. Kuester. doi: 10.1007/978-3-642-83459-2. Google Scholar

[4]

A. H. Barnett and T. Betcke, Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains, J. Comput. Phys., 227 (2008), 7003-7026. doi: 10.1016/j.jcp.2008.04.008. Google Scholar

[5]

A. H. Barnett and T. Betcke, An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons, SIAM J. Sci. Comput., 32 (2010), 1417-1441. doi: 10.1137/090768667. Google Scholar

[6]

A. Barnett and T. Betcke, Mpspack, url: https://code.google.com/p/mpspack/.Google Scholar

[7]

A. BaylissM. Gunzburger and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math., 42 (1982), 430-451. doi: 10.1137/0142032. Google Scholar

[8]

D. Bouche, F. Molinet and R. Mittra, Asymptotic Methods in Electromagnetics Springer-Verlag, Berlin, 1997. Translated from the 1994 French original by Patricia and Daniel Gogny and revised by the authors. doi: 10.1007/978-3-642-60517-8. Google Scholar

[9]

R. N. Buchal and J. B. Keller, Boundary layer problems in diffraction theory, Comm. Pure Appl. Math., 13 (1960), 85-114. doi: 10.1002/cpa.3160130109. Google Scholar

[10]

S. N. Chandler-WildeI. G. GrahamS. Langdon and E. A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numer., 21 (2012), 89-305. doi: 10.1017/S0962492912000037. Google Scholar

[11]

S. N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 (2005), 598-618. doi: 10.1137/040615523. Google Scholar

[12]

S. N. Chandler-Wilde and B. Zhang, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math., 58 (1998), 1774-1790(electronic). doi: 10.1137/S0036139996309722. Google Scholar

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P. Cummings and X. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations, Math. Models Methods Appl. Sci., 16 (2006), 139-160. doi: 10.1142/S021820250600108X. Google Scholar

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J. Douglas JrJ. E. SantosD. Sheen and L. Schreyer Bennethum, Frequency domain treatment of one-dimensional scalar waves, Math. Models Methods Appl. Sci., 3 (1993), 171-194. doi: 10.1142/S0218202593000102. Google Scholar

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X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal., 47 (2009), 2872-2896. doi: 10.1137/080737538. Google Scholar

[20]

X. Feng and H. Wu, hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 80 (2011), 1997-2024. doi: 10.1090/S0025-5718-2011-02475-0. Google Scholar

[21]

E. Giladi, Asymptotically derived boundary elements for the Helmholtz equation in high frequencies, J. Comput. Appl. Math., 198 (2007), 52-74. doi: 10.1016/j.cam.2005.11.024. Google Scholar

[22]

E. Giladi and J. B. Keller, A hybrid numerical asymptotic method for scattering problems, J. Comput. Phys., 174 (2001), 226-247. doi: 10.1006/jcph.2001.6903. Google Scholar

[23]

M. J. Grote and J. B. Keller, On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), 231-243. doi: 10.1006/jcph.1995.1210. Google Scholar

[24]

G.-M. Gie and J. P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differential Equations, 253 (2012), 1862-1892. doi: 10.1016/j.jde.2012.06.008. Google Scholar

[25]

G.-M. Gie and C.-Y. Jung, Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition, Asymptot. Anal., 84 (2013), 17-33. Google Scholar

[26]

Y. HongC.-Y. Jung and R. Temam, On the numerical approximations of stiff convection-diffusion equations in a circle, Numerische Mathematik, 127 (2014), 291-313. doi: 10.1007/s00211-013-0585-x. Google Scholar

[27]

F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM, Comput. Math. Appl., 30 (1995), 9-37. doi: 10.1016/0898-1221(95)00144-N. Google Scholar

[28]

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering volume 132 of Applied Mathematical Sciences Springer-Verlag, New York, 1998. doi: 10.1007/b98828. Google Scholar

[29]

C.-Y. JungM. Petcu and R. Temam, Singular perturbation analysis on a homogeneous ocean circulation model, Anal. Appl. (Singap.), 9 (2011), 275-313. doi: 10.1142/S0219530511001832. Google Scholar

[30]

J. B. Keller, Diffraction by a convex cylinder, Div. Electromag. Res., Inst. Math. Sci., New York Univ., Res. Rep. No. EM-94 (1956), 10 pp. Also: Trans. I.R.E., AP-4 (1956), 312-321. Google Scholar

[31]

J. B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Amer., 52 (1962), 116-130. doi: 10.1364/JOSA.52.000116. Google Scholar

[32]

J. B. Keller and D. Givoli, Exact nonreflecting boundary conditions, J. Comput. Phys., 82 (1989), 172-192. doi: 10.1016/0021-9991(89)90041-7. Google Scholar

[33]

J. B. Keller, R. M. Lewis and B. D. Seckler, Asymptotic Solution of Some Diffraction Problems Res. Rep. No. EM-81. Div. Electromag. Res. , Inst. Math. Sci. , New York Univ. , 1955. Google Scholar

[34]

L. J. Landau, Bessel functions: Monotonicity and bounds, J. London Math. Soc. (2), 61 (2000), 197-215. doi: 10.1112/S0024610799008352. Google Scholar

[35]

N. N. Lebedev, Special Functions and Their Applications Revised English edition. Translated and edited by Richard A. Silverman. Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1965. Google Scholar

[36]

A. Lechleiter and S. Ritterbusch, A variational method for wave scattering from penetrable rough layers, IMA J. Appl. Math., 75 (2010), 366-391. doi: 10.1093/imamat/hxp040. Google Scholar

[37]

J. M. Melenk, On Generalized Finite-Element Methods, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph. D. )-University of Maryland, College Park. Google Scholar

[38]

R. B. Melrose and M. E. Taylor, The radiation pattern of a diffracted wave near the shadow boundary, Comm. Partial Differential Equations, 11 (1986), 599-672. doi: 10.1080/03605308608820439. Google Scholar

[39]

J. R. Ockendon and R. H. Tew, Thin-layer solutions of the Helmholtz and related equations, SIAM Rev., 54 (2012), 3-51. doi: 10.1137/090761641. Google Scholar

[40]

F. W. J. Olver, Asymptotics and Special Functions AKP Classics. A K Peters Ltd. , Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. Google Scholar

[41]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, editors, NIST Handbook Of Mathematical Functions U. S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. Google Scholar

[42]

L. Prandtl, Verber fl¨ussigkeiten bei sehr kleiner reibung, Verk. Ⅲ Intem. Math. Kongr. Heidelberg, pages 484-491,1905.Google Scholar

[43]

M. LebentalJ. S. LauretJ. ZyssC. Schmit and E. Bogomolny, Directional emission of stadium-shaped microlasers, Physical Review A, 75 (2007), 033806. doi: 10.1103/PhysRevA.75.033806. Google Scholar

[44]

L. Prandtl, Gesammelte Abhandlungen zur angewandten Mechanik, Hydro-und Aerodynamik Herausgegeben von Walter Tollmien, Hermann Schlichting, Henry Görtler. Schriftleitung: F. W. Riegels. In 3 Teilen. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. Google Scholar

[45]

J. Shen and L.-L. Wang, Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 43 (2005), 623-644. doi: 10.1137/040607332. Google Scholar

[46]

J. Shen and L.-L. Wang, Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains, SIAM J. Numer. Anal., 45 (2007), 1954-1978. doi: 10.1137/060665737. Google Scholar

[47]

A. Sommerfeld, Partial Differential Equations in Physics Academic Press, Inc. , New York, N. Y. , 1949. Translated by Ernst G. Straus. Google Scholar

[48]

A. Sommerfeld, Optics. Lectures on Theoretical Physics, Vol. Ⅳ Academic Press Inc. , New York, 1954. Translated by O. Laporte and P. A. Moldauer. Google Scholar

[49]

W. A. Strauss, Partial Differential Equations John Wiley & Sons, Inc. , New York, 1992. An introduction. Google Scholar

[50]

R. H. TewS. J. ChapmanJ. R. KingJ. R. Ockendon and I. Zafarullah, Scalar wave diffraction by tangent rays, Wave Motion, 32 (2000), 363-380. doi: 10.1016/S0165-2125(00)00051-2. Google Scholar

[51]

G. N. Watson, A Treatise on the Theory of Bessel Functions Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington, D. C. 1964. Google Scholar

[2]

I. Andronov, D. Bouche and F. Molinet, Asymptotic and Hybrid Methods in Electromagnetics volume 48 of IEE Electromagnetic Waves Series, Institution of Electrical Engineers (IEE), London, 2005. doi: 10.1049/PBEW051E. Google Scholar

[3]

V. M. Babič and V. S. Buldyrev, Short-wavelength Diffraction Theory volume 4 of Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1991. Asymptotic methods, Translated from the 1972 Russian original by E. F. Kuester. doi: 10.1007/978-3-642-83459-2. Google Scholar

[4]

A. H. Barnett and T. Betcke, Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains, J. Comput. Phys., 227 (2008), 7003-7026. doi: 10.1016/j.jcp.2008.04.008. Google Scholar

[5]

A. H. Barnett and T. Betcke, An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons, SIAM J. Sci. Comput., 32 (2010), 1417-1441. doi: 10.1137/090768667. Google Scholar

[6]

A. Barnett and T. Betcke, Mpspack, url: https://code.google.com/p/mpspack/.Google Scholar

[7]

A. BaylissM. Gunzburger and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math., 42 (1982), 430-451. doi: 10.1137/0142032. Google Scholar

[8]

D. Bouche, F. Molinet and R. Mittra, Asymptotic Methods in Electromagnetics Springer-Verlag, Berlin, 1997. Translated from the 1994 French original by Patricia and Daniel Gogny and revised by the authors. doi: 10.1007/978-3-642-60517-8. Google Scholar

[9]

R. N. Buchal and J. B. Keller, Boundary layer problems in diffraction theory, Comm. Pure Appl. Math., 13 (1960), 85-114. doi: 10.1002/cpa.3160130109. Google Scholar

[10]

S. N. Chandler-WildeI. G. GrahamS. Langdon and E. A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numer., 21 (2012), 89-305. doi: 10.1017/S0962492912000037. Google Scholar

[11]

S. N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 (2005), 598-618. doi: 10.1137/040615523. Google Scholar

[12]

S. N. Chandler-Wilde and B. Zhang, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math., 58 (1998), 1774-1790(electronic). doi: 10.1137/S0036139996309722. Google Scholar

[13]

D. Colton, Partial Differential Equations Dover Publications, Inc. , Mineola, NY, 2004. An introduction, Corrected reprint of the 1988 original. Google Scholar

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory volume 93 of Applied Mathematical Sciences, Springer, New York, third edition, 2013. doi: 10.1007/978-1-4614-4942-3. Google Scholar

[15]

P. Cummings and X. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations, Math. Models Methods Appl. Sci., 16 (2006), 139-160. doi: 10.1142/S021820250600108X. Google Scholar

[16]

J. Douglas JrJ. E. SantosD. Sheen and L. Schreyer Bennethum, Frequency domain treatment of one-dimensional scalar waves, Math. Models Methods Appl. Sci., 3 (1993), 171-194. doi: 10.1142/S0218202593000102. Google Scholar

[17]

W. Eckhaus and E. M. de Jager, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type, Arch. Rational Mech. Anal., 23 (1966), 26-86. doi: 10.1007/BF00281135. Google Scholar

[18]

K. Feng, Finite element method and natural boundary reduction, In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) pages 1439-1453. PWN, Warsaw, 1984. Google Scholar

[19]

X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal., 47 (2009), 2872-2896. doi: 10.1137/080737538. Google Scholar

[20]

X. Feng and H. Wu, hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 80 (2011), 1997-2024. doi: 10.1090/S0025-5718-2011-02475-0. Google Scholar

[21]

E. Giladi, Asymptotically derived boundary elements for the Helmholtz equation in high frequencies, J. Comput. Appl. Math., 198 (2007), 52-74. doi: 10.1016/j.cam.2005.11.024. Google Scholar

[22]

E. Giladi and J. B. Keller, A hybrid numerical asymptotic method for scattering problems, J. Comput. Phys., 174 (2001), 226-247. doi: 10.1006/jcph.2001.6903. Google Scholar

[23]

M. J. Grote and J. B. Keller, On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), 231-243. doi: 10.1006/jcph.1995.1210. Google Scholar

[24]

G.-M. Gie and J. P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differential Equations, 253 (2012), 1862-1892. doi: 10.1016/j.jde.2012.06.008. Google Scholar

[25]

G.-M. Gie and C.-Y. Jung, Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition, Asymptot. Anal., 84 (2013), 17-33. Google Scholar

[26]

Y. HongC.-Y. Jung and R. Temam, On the numerical approximations of stiff convection-diffusion equations in a circle, Numerische Mathematik, 127 (2014), 291-313. doi: 10.1007/s00211-013-0585-x. Google Scholar

[27]

F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM, Comput. Math. Appl., 30 (1995), 9-37. doi: 10.1016/0898-1221(95)00144-N. Google Scholar

[28]

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering volume 132 of Applied Mathematical Sciences Springer-Verlag, New York, 1998. doi: 10.1007/b98828. Google Scholar

[29]

C.-Y. JungM. Petcu and R. Temam, Singular perturbation analysis on a homogeneous ocean circulation model, Anal. Appl. (Singap.), 9 (2011), 275-313. doi: 10.1142/S0219530511001832. Google Scholar

[30]

J. B. Keller, Diffraction by a convex cylinder, Div. Electromag. Res., Inst. Math. Sci., New York Univ., Res. Rep. No. EM-94 (1956), 10 pp. Also: Trans. I.R.E., AP-4 (1956), 312-321. Google Scholar

[31]

J. B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Amer., 52 (1962), 116-130. doi: 10.1364/JOSA.52.000116. Google Scholar

[32]

J. B. Keller and D. Givoli, Exact nonreflecting boundary conditions, J. Comput. Phys., 82 (1989), 172-192. doi: 10.1016/0021-9991(89)90041-7. Google Scholar

[33]

J. B. Keller, R. M. Lewis and B. D. Seckler, Asymptotic Solution of Some Diffraction Problems Res. Rep. No. EM-81. Div. Electromag. Res. , Inst. Math. Sci. , New York Univ. , 1955. Google Scholar

[34]

L. J. Landau, Bessel functions: Monotonicity and bounds, J. London Math. Soc. (2), 61 (2000), 197-215. doi: 10.1112/S0024610799008352. Google Scholar

[35]

N. N. Lebedev, Special Functions and Their Applications Revised English edition. Translated and edited by Richard A. Silverman. Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1965. Google Scholar

[36]

A. Lechleiter and S. Ritterbusch, A variational method for wave scattering from penetrable rough layers, IMA J. Appl. Math., 75 (2010), 366-391. doi: 10.1093/imamat/hxp040. Google Scholar

[37]

J. M. Melenk, On Generalized Finite-Element Methods, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph. D. )-University of Maryland, College Park. Google Scholar

[38]

R. B. Melrose and M. E. Taylor, The radiation pattern of a diffracted wave near the shadow boundary, Comm. Partial Differential Equations, 11 (1986), 599-672. doi: 10.1080/03605308608820439. Google Scholar

[39]

J. R. Ockendon and R. H. Tew, Thin-layer solutions of the Helmholtz and related equations, SIAM Rev., 54 (2012), 3-51. doi: 10.1137/090761641. Google Scholar

[40]

F. W. J. Olver, Asymptotics and Special Functions AKP Classics. A K Peters Ltd. , Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. Google Scholar

[41]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, editors, NIST Handbook Of Mathematical Functions U. S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. Google Scholar

[42]

L. Prandtl, Verber fl¨ussigkeiten bei sehr kleiner reibung, Verk. Ⅲ Intem. Math. Kongr. Heidelberg, pages 484-491,1905.Google Scholar

[43]

M. LebentalJ. S. LauretJ. ZyssC. Schmit and E. Bogomolny, Directional emission of stadium-shaped microlasers, Physical Review A, 75 (2007), 033806. doi: 10.1103/PhysRevA.75.033806. Google Scholar

[44]

L. Prandtl, Gesammelte Abhandlungen zur angewandten Mechanik, Hydro-und Aerodynamik Herausgegeben von Walter Tollmien, Hermann Schlichting, Henry Görtler. Schriftleitung: F. W. Riegels. In 3 Teilen. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. Google Scholar

[45]

J. Shen and L.-L. Wang, Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 43 (2005), 623-644. doi: 10.1137/040607332. Google Scholar

[46]

J. Shen and L.-L. Wang, Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains, SIAM J. Numer. Anal., 45 (2007), 1954-1978. doi: 10.1137/060665737. Google Scholar

[47]

A. Sommerfeld, Partial Differential Equations in Physics Academic Press, Inc. , New York, N. Y. , 1949. Translated by Ernst G. Straus. Google Scholar

[48]

A. Sommerfeld, Optics. Lectures on Theoretical Physics, Vol. Ⅳ Academic Press Inc. , New York, 1954. Translated by O. Laporte and P. A. Moldauer. Google Scholar

[49]

W. A. Strauss, Partial Differential Equations John Wiley & Sons, Inc. , New York, 1992. An introduction. Google Scholar

[50]

R. H. TewS. J. ChapmanJ. R. KingJ. R. Ockendon and I. Zafarullah, Scalar wave diffraction by tangent rays, Wave Motion, 32 (2000), 363-380. doi: 10.1016/S0165-2125(00)00051-2. Google Scholar

[51]

G. N. Watson, A Treatise on the Theory of Bessel Functions Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. Google Scholar

Figure 1.  Scattering by a circular object
Figure 2.  Sound-soft scattering from a circular object as in Figure 1 at (Top) $k=50$, (Bottom) $k=500$: (a) $\Re (u_{inc}^{(k)}) = \cos(ikx)$: incident wave traveling to the right, (b) $\Re u_s^{(k)}$: scattered wave, (c) $\Re u^{(k)} = \Re u^{(k)}_{inc} + \Re u^{(k)}_s$: total solution. This simulation is generated using the MPSPACK (see [6])
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