zbMATH — the first resource for mathematics

Boundary-contact problem of acoustics in an infinite domain. (English. Russian original) Zbl 0637.76082
Sov. Phys., Dokl. 30, No. 7, 560-562 (1985); translation from Dokl. Akad. Nauk SSSR 283, No. 1, 77-81 (1985).
The boundary-contact problem of acoustics (BCP) is a boundary value problem for the Helmholtz equation in a domain on part of the boundary which is used to position a flexible oscillating plate. To distinguish a unique solution of the BCP, is is necessary, in addition to the boundary conditions, to fix conditions at individual points or lines of the boundary (boundary-contact conditions - BCC). Although an entire series of exact analytic solutions of model BCP in comparatively simple domains has been constructed, the question of the existence of a solution in an arbitrary infinite domain remains open. We consider the BCP in a plane domain which is the union of an arbitrary finite domain with a piecewise smooth boundary and an infinite waveguide - a half strip. Partial radiation conditions, are formulated. These conditions make it possible to reduce the BCP in an infinite domain to a problem in a finite domain. The latter is formulated in the form of a Fredholm equation of second kind in a suitable Hilbert space, which makes is possible to prove a theorem on conditional solvability of the BCP. The BCP in a domain, which is the union of a finite domain and a sector, can be studied in the same way. The principle of limiting absorption is justified. A simple example is given of a BCP, in which an eigenvalue lies on the continuous spectrum. Completeness of the system of eigenfunctions of the cross section of the waveguide, one of whose walls is a plate, is established by the technique of operator pencils. A theorem on expansion in these eigenfunctions is discussed.
76Q05 Hydro- and aero-acoustics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
45B05 Fredholm integral equations