Fredholm property of boundary-contact problems of acoustics. (English. Russian original) Zbl 0585.76114

Belinskij, B. P.

Sov. Phys., Dokl. 29, 797-799 (1984); translation from Dokl. Akad. Nauk SSSR 278, 1090-1094 (1984).

The description of the vibrations of a thin plate or shell reinforced with stiffness ribs in a fluid poses an important applied problem. The corresponding boundary-value problem for the Helmholtz equation is called a boundary-contact problem (BCP) of acoustics. For a BCP in an arbitrary planar domain we establish the law of conservation of energy, discuss a variational formulation of the nonstationary problem, give a definition of a weak solution of the stationary problem, derive a Fredholm equation for the latter in the appropriate space, establish the completeness of the system of eigenfunctions of the BCP, and discuss the nature of the smoothness of the solution.

MSC:

76Q05	Hydro- and aero-acoustics
45B05	Fredholm integral equations

Keywords:

vibrations of a thin plate; boundary-value problem; Helmholtz equation; boundary-contact problem; planar domain; conservation of energy; variational formulation; nonstationary problem; weak solution; stationary problem; Fredholm equation; completeness; eigenfunctions; smoothness of the solution