Geodesic
The Dirichlet–Neumann conditions and orthogonality of three-dimensional natural waves in layered solids
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 9, pp. 1598-1612

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Orthogonality relations for homogeneous waves in layered plates are obtained, and they are generalized to the case of a contact with fluid layers. For layers of infinite thickness, it is shown that the homogeneous waves of the discrete spectrum are orthogonal to each other and to the waves of the continuous spectrum. For finite-size sources, exact formulas are derived for the coefficients multiplying the modes. Based on the orthogonality relations, a nonlocal radiation principle is proposed such that the infinite domain in the numerical solution of diffraction problems for layered plates can be replaced by a virtual cylinder. 
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     author = {D. D. Zakharov},
     title = {The {Dirichlet{\textendash}Neumann} conditions and orthogonality of three-dimensional natural waves in layered solids},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1598--1612},
     publisher = {mathdoc},
     volume = {50},
     number = {9},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_9_a4/}
}
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D. D. Zakharov. The Dirichlet–Neumann conditions and orthogonality of three-dimensional natural waves in layered solids. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 9, pp. 1598-1612. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_9_a4/