TRAPPING MODES IN A MEMBRANE WITH AN INHOMOGENEITY

The possible presence of a real discrete spectrum of eigenfrequencies is considered for a vibrating inhomogeneous membrane of infinite exten t. The membrane contains a distributed mass in the form of a layer of finite length oriented along the infinite membrane dimension with no s tiffness at the boundary between the membrane and the layer. In contra st to the continuous spectrum describing the waves propagating in the membrane, the discrete spectrum determines the nonpropagating modes of vibration (the trapping modes) localized near the inhomogeneity and c haracterized by an amplitude exponentially decreasing at infinity. The discrete spectrum is finite and occurs below the first cut-off freque ncy. The possibility of the extension of the discrete spectrum to the region of the continuous spectrum is analyzed: the spectral problem is considered with the added condition that determines the occurrence of a discrete spectrum above the first cut-off frequency. The approximat e solution to this problem points to the absence of a discrete spectru m in the aforementioned region.