Geodesic
On frequencies of free oscillations of a truncated spherical cone covered with a thin elastic spherical shell
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 31, Tome 285 (2002), pp. 124-134

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The analytical solution of a problem on a determination of frequencies and forms of free axisymmetric oscillations of the truncated spherical cone filled with an ideal compressible fluid is constructed. Spherical wall of a smaller radius and radial wall of a sector are absolutely rigid. On the spherical boundary of the greater radius the thin elastic shell is placed. The shell edge is clamped in a radial wall. The outside shell surface bounds with vacuum. The effect of anomalous lowering of natural frequency at rapprochement of spherical walls is revealed. There is constructed and numerically tested an approximate formula for search of the lowest natural frecuensies which are approximately proportional to the square root of difference of radiuses of spherical walls for small significances of this difference. 
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     author = {Yu. A. Lavrov and V. D. Luk'yanov},
     title = {On frequencies of free oscillations of a truncated spherical cone covered with a thin elastic spherical shell},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {124--134},
     publisher = {mathdoc},
     volume = {285},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2002_285_a9/}
}
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Yu. A. Lavrov; V. D. Luk'yanov. On frequencies of free oscillations of a truncated spherical cone covered with a thin elastic spherical shell. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 31, Tome 285 (2002), pp. 124-134. http://geodesic.mathdoc.fr/item/ZNSL_2002_285_a9/