April  2019, 13(2): 263-284. doi: 10.3934/ipi.2019014

On finding the surface admittance of an obstacle via the time domain enclosure method

Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan

Received  June 2017 Revised  August 2018 Published  January 2019

Fund Project: The author was partially supported by Grant-in-Aid for Scientific Research (C)(No 17K05331) of Japan Society for the Promotion of Science.

An inverse obstacle scattering problem for the electromagnetic wave governed by the Maxwell system over a finite time interval is considered. It is assumed that the wave satisfies the Leontovich boundary condition on the surface of an unknown obstacle. The condition is described by using an unknown positive function on the surface of the obstacle which is called the surface admittance. The wave is generated at the initial time by a volumetric current source supported on a very small ball placed outside the obstacle and only the electric component of the wave is observed on the same ball over a finite time interval. It is shown that from the observed data one can extract information about the value of the surface admittance and the curvatures at the points on the surface nearest to the center of the ball. This shows that a single shot contains a meaningful information about the quantitative state of the surface of the obstacle.

Citation: Masaru Ikehata. On finding the surface admittance of an obstacle via the time domain enclosure method. Inverse Problems & Imaging, 2019, 13 (2) : 263-284. doi: 10.3934/ipi.2019014
References:
[1]

N. G. Alexopoulos and G. A. Tadler, Accuracy of the Leontovich boundary condition for continuous and discontinuous surface impedances, J. Appl. Phys., 46 (2008), 3326-3332.  doi: 10.1063/1.322058.  Google Scholar

[2]

C. A. Balanis, Antenna Theory, Analysis and Design, 3$^{rd}$ edition, Wiley-Interscience, Hoboken, New Jersey, 2005. Google Scholar

[3]

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover, New York, 1986.  Google Scholar

[4]

M. Cheney and R. Borden, Fundamentals of Radar Imaging, CBMS-NSF, Regional Conference Series in Applied Mathematics, 79, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719291.  Google Scholar

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D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[6]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Spectral Theory and Applications, Vol. 3, Springer-Verlag, Berlin, 1990.  Google Scholar

[7]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edn, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[8]

M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308.  Google Scholar

[9]

M. Ikehata, Extracting the geometry of an obstacle and a zeroth-order coefficient of a boundary condition via the enclosure method using a single reflected wave over a finite time interval, Inverse Problems, 30 (2014), 045011 (24pp). doi: 10.1088/0266-5611/30/4/045011.  Google Scholar

[10]

M. Ikehata, New development of the enclosure method for inverse obstacle scattering, Chapter 6 in Inverse Problems and Computational Mechanics (eds. L. Marin, L. Munteanu, V. Chiroiu), Vol. 2,123–147, Editura Academiei, Bucharest, Romania, 2016. Google Scholar

[11]

M. Ikehata, The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain, Inverse Problems and Imaging, 10 (2016), 131-163.  doi: 10.3934/ipi.2016.10.131.  Google Scholar

[12]

M. Ikehata, On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method, Inverse Problems and Imaging, 11 (2017), 99-123.  doi: 10.3934/ipi.2017006.  Google Scholar

[13]

M. Ikehata, A remark on finding the coefficient of the dissipative boundary condition via the enclosure method in the time domain, Math. Meth. Appl. Sci., 40 (2017), 915-927.  doi: 10.1002/mma.4021.  Google Scholar

[14]

B. V. Kapitonov, On exponential decay as $t\longrightarrow\infty$ of solutions of an exterior boundary value problem for the Maxwell system, Math. USSR Sbornik, 66 (1990), 475-498.  doi: 10.1070/SM1990v066n02ABEH001318.  Google Scholar

[15]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-harmononic Maxwell's Equations, Expansion-, Integral-, and Variational Methods, Springer, 2015. doi: 10.1007/978-3-319-11086-8.  Google Scholar

[16]

S. G. Krein and I. M. Kulikov, The Maxwell-Leontovich operator, Differentsial'nye Uravneniya, 5 (1969), 1275-1282.   Google Scholar

[17]

P. D. Lax and R. S. Phillips, The scattering of sound waves by an obstacle, Comm. Pure and Appl. Math., 30 (1977), 195-233.  doi: 10.1002/cpa.3160300204.  Google Scholar

[18]

J.-C. Nédélec, Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Springer, New York, 2001. doi: 10.1007/978-1-4757-4393-7.  Google Scholar

[19]

B. O'Neill, Elementary Differential Geometry, Revised, 2nd Edition, Academic Press, Amsterdam, 2006.  Google Scholar

[20]

K. Yosida, Functional Analysis, Third Edtition, Springer, New York, 1971. Google Scholar

show all references

References:
[1]

N. G. Alexopoulos and G. A. Tadler, Accuracy of the Leontovich boundary condition for continuous and discontinuous surface impedances, J. Appl. Phys., 46 (2008), 3326-3332.  doi: 10.1063/1.322058.  Google Scholar

[2]

C. A. Balanis, Antenna Theory, Analysis and Design, 3$^{rd}$ edition, Wiley-Interscience, Hoboken, New Jersey, 2005. Google Scholar

[3]

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover, New York, 1986.  Google Scholar

[4]

M. Cheney and R. Borden, Fundamentals of Radar Imaging, CBMS-NSF, Regional Conference Series in Applied Mathematics, 79, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719291.  Google Scholar

[5]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[6]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Spectral Theory and Applications, Vol. 3, Springer-Verlag, Berlin, 1990.  Google Scholar

[7]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edn, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[8]

M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308.  Google Scholar

[9]

M. Ikehata, Extracting the geometry of an obstacle and a zeroth-order coefficient of a boundary condition via the enclosure method using a single reflected wave over a finite time interval, Inverse Problems, 30 (2014), 045011 (24pp). doi: 10.1088/0266-5611/30/4/045011.  Google Scholar

[10]

M. Ikehata, New development of the enclosure method for inverse obstacle scattering, Chapter 6 in Inverse Problems and Computational Mechanics (eds. L. Marin, L. Munteanu, V. Chiroiu), Vol. 2,123–147, Editura Academiei, Bucharest, Romania, 2016. Google Scholar

[11]

M. Ikehata, The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain, Inverse Problems and Imaging, 10 (2016), 131-163.  doi: 10.3934/ipi.2016.10.131.  Google Scholar

[12]

M. Ikehata, On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method, Inverse Problems and Imaging, 11 (2017), 99-123.  doi: 10.3934/ipi.2017006.  Google Scholar

[13]

M. Ikehata, A remark on finding the coefficient of the dissipative boundary condition via the enclosure method in the time domain, Math. Meth. Appl. Sci., 40 (2017), 915-927.  doi: 10.1002/mma.4021.  Google Scholar

[14]

B. V. Kapitonov, On exponential decay as $t\longrightarrow\infty$ of solutions of an exterior boundary value problem for the Maxwell system, Math. USSR Sbornik, 66 (1990), 475-498.  doi: 10.1070/SM1990v066n02ABEH001318.  Google Scholar

[15]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-harmononic Maxwell's Equations, Expansion-, Integral-, and Variational Methods, Springer, 2015. doi: 10.1007/978-3-319-11086-8.  Google Scholar

[16]

S. G. Krein and I. M. Kulikov, The Maxwell-Leontovich operator, Differentsial'nye Uravneniya, 5 (1969), 1275-1282.   Google Scholar

[17]

P. D. Lax and R. S. Phillips, The scattering of sound waves by an obstacle, Comm. Pure and Appl. Math., 30 (1977), 195-233.  doi: 10.1002/cpa.3160300204.  Google Scholar

[18]

J.-C. Nédélec, Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Springer, New York, 2001. doi: 10.1007/978-1-4757-4393-7.  Google Scholar

[19]

B. O'Neill, Elementary Differential Geometry, Revised, 2nd Edition, Academic Press, Amsterdam, 2006.  Google Scholar

[20]

K. Yosida, Functional Analysis, Third Edtition, Springer, New York, 1971. Google Scholar

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