Free acoustic oscillations in cylindrical tube coupled with annular sector cavity

An analytical solution of two-dimensional problem of finding frequencies and forms of natural oscillations of a cylindrical resonator having special form is constructed. The cross-section of a resonator consists of circle with adjacent concentric truncated sector. Two outside round and radial boundaries of the system are ideally rigid. The arc wall between two regions is a thin elastic cylindrical shell with the clamped edges. The adjacent cavities of the resonator are filled with various ideal compressible irrotational fluids. The effect of a natural frequencies lowering under diminution of sector radial size is confirmed.