Uniqueness of an inverse cavity scattering problem for the time-harmonic biharmonic wave equation. (English) Zbl 1542.35434
Summary: This paper addresses an inverse cavity scattering problem associated with the time-harmonic biharmonic wave equation in two dimensions. The objective is to determine the domain or shape of the cavity. The Green’s representations are demonstrated for the solution to the boundary value problem, and the one-to-one correspondence is confirmed between the Helmholtz component of biharmonic waves and the resulting far-field patterns. Two mixed reciprocity relations are deduced, linking the scattered field generated by plane waves to the far-field pattern produced by various types of point sources. Furthermore, the symmetry relations are explored for the scattered fields generated by point sources. Finally, we present two uniqueness results for the inverse problem by utilizing both far-field patterns and phaseless near-field data.
{© 2024 IOP Publishing Ltd}
{© 2024 IOP Publishing Ltd}
MSC:
| 35R30 | Inverse problems for PDEs |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35P25 | Scattering theory for PDEs |
Keywords:
biharmonic wave equation; inverse scattering problem; Green’s representation theorem; far-field pattern; phaseless data; uniquenessSoftware:
DLMFReferences:
| [1] | Bourgeois, L.; Recoquillay, A., The linear sampling method for Kirchhoff-Love infinite plates, Inverse Problems Imaging, 14, 363-84, 2020 · Zbl 1435.35415 · doi:10.3934/ipi.2020016 |
| [2] | Cheng, Z.; Dong, H., Uniqueness and reconstruction method for inverse elastic wave scattering with phaseless data, Inverse Problems Imaging, 18, 406-433, 2023 · Zbl 07815413 · doi:10.3934/ipi.2023038 |
| [3] | Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, 2013, Springer · Zbl 1266.35121 |
| [4] | Dong, H.; Lai, J.; Li, P., An inverse acoustic-elastic interaction problem with phased or phaseless far-field data, Inverse Problems, 36, 2020 · Zbl 1437.35702 · doi:10.1088/1361-6420/ab693e |
| [5] | Dong, H.; Li, P., A novel boundary integral formulation for the biharmonic wave scattering problem, J. Sci. Comput., 98, 42, 2024 · Zbl 1534.65260 · doi:10.1007/s10915-023-02429-6 |
| [6] | Evans, D. V.; Porter, R., Penetration of flexural waves through a periodically constrained thin elastic plate in vacuo and floating on water, J. Eng. Math., 58, 317-37, 2007 · Zbl 1117.74014 · doi:10.1007/s10665-006-9128-0 |
| [7] | Farhat, M.; Guenneau, S.; Enoch, S., Ultrabroadband elastic cloaking in thin plates, Phys. Rev. Lett., 103, 2009 · doi:10.1103/PhysRevLett.103.024301 |
| [8] | Hahner, P.; Hsiao, G. C., Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Problems, 9, 525-34, 1993 · Zbl 0789.35172 · doi:10.1088/0266-5611/9/5/002 |
| [9] | Isakov, V., On uniqueness in the inverse transmission scattering problem, Commun. Part. Diff. Equ., 15, 1565-87, 1990 · Zbl 0728.35148 · doi:10.1080/03605309908820737 |
| [10] | Ji, X.; Liu, X., Inverse elastic scattering problems with phaseless far field data, Inverse Problems, 35, 2019 · Zbl 1423.35449 · doi:10.1088/1361-6420/ab2a35 |
| [11] | Kirsch, A.; Kress, R., Uniqueness in inverse obstacle scattering, Inverse Problems, 9, 285-99, 1993 · Zbl 0787.35119 · doi:10.1088/0266-5611/9/2/009 |
| [12] | Olver, F. W.; Lozier, D.; Boisvert, R. F.; Clark, C. W., Nist Handbook of Mathematical Functions, 2010, Cambridge · Zbl 1198.00002 |
| [13] | Pelat, A.; Gautier, F.; Conlon, S. C.; Semperlotti, F., The acoustic black hole: a review of theory and applications, J. Sound Vib., 476, 2020 · doi:10.1016/j.jsv.2020.115316 |
| [14] | Stenger, N.; Wilhelm, M.; Wegener, M., Experiments on elastic cloaking in thin plates, Phys. Rev. Lett., 108, 2012 · doi:10.1103/PhysRevLett.108.014301 |
| [15] | Watanabe, E.; Utsunomiya, T.; Wang, C., Hydroelastic analysis of pontoon-type VLFS: a literature survey, Eng. Struct., 26, 245-56, 2004 · doi:10.1016/j.engstruct.2003.10.001 |
| [16] | Xu, X.; Zhang, B.; Zhang, H., Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency II, SIAM J. Appl. Math., 78, 3024-39, 2018 · Zbl 1403.78010 · doi:10.1137/18M1196820 |
| [17] | Xu, X.; Zhang, B.; Zhang, H., Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency, Inverse Problems Imaging, 14, 489-510, 2020 · Zbl 1508.78009 · doi:10.3934/ipi.2020023 |
| [18] | Xu, X.; Zhang, B.; Zhang, H., Uniqueness in inverse electromagnetic scattering problem with phaseless far-field data at a fixed frequency, IMA J. Appl. Math., 85, 823-39, 2020 · Zbl 1465.78004 · doi:10.1093/imamat/hxaa024 |
| [19] | Zhang, B.; Zhang, H., Recovering scattering obstacles by multi-frequency phaseless far-field data, J. Comput. Phys., 345, 58-73, 2017 · Zbl 1378.35210 · doi:10.1016/j.jcp.2017.05.022 |
| [20] | Zhang, D.; Guo, Y., Uniqueness results on phaseless inverse scattering with a reference ball, Inverse Problems, 34, 2018 · Zbl 1442.35553 · doi:10.1088/1361-6420/aac53c |
| [21] | Zhang, D.; Guo, Y.; Sun, F.; Liu, H., Unique determinations in inverse scattering problems with phaseless near-field measurements, Inverse Problems Imaging, 14, 569-82, 2020 · Zbl 1441.78017 · doi:10.3934/ipi.2020026 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
