Geodesic
Distribution of natural oscillations models in a plate imbedded into absolutely rigid half-space
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 53, Tome 521 (2023), pp. 154-199

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We study oscillations of a thin isotropic cylindrical plate imbedded into a notch and attached to its absolutely rigid surface. We demonstrate that only in the case of sufficiently deep notch, in particular, completely submerged plate, its natural oscillations are described by the two-dimensional model, that is, elasticity theory plane problem in the longitudinal cross-section with the Dirichlet conditions at its boundary. In other cases we establish the exponential decay of eigenmodes at a distance of the lateral side of the plate. Moreover, a formal asymptotic analysis leads to other models of reduced dimension in the low-frequency range of the spectrum, namely multifarious ordinary differential equations while the corresponding modes of natural oscillations concentrate near the whole lateral side or some points on it. 
@article{ZNSL_2023_521_a9,
     author = {S. A. Nazarov},
     title = {Distribution of natural oscillations models in a plate imbedded into absolutely rigid half-space},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {154--199},
     publisher = {mathdoc},
     volume = {521},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_521_a9/}
}
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S. A. Nazarov. Distribution of natural oscillations models in a plate imbedded into absolutely rigid half-space. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 53, Tome 521 (2023), pp. 154-199. http://geodesic.mathdoc.fr/item/ZNSL_2023_521_a9/