An elastic membrane with a mass concentrated in a small region, the diameter of which is of the order of a small parameter \(\epsilon\), is considered. The surface density of the membrane in this region is of order \(\epsilon^{-m}\) for some \(m>0\). The asymptotic behaviour of the eigenvalue problem in this region is considered for \(\epsilon>0\). The asymptotic behaviour exhibits local vibrations (which are significant only at distances of order \(\epsilon\) from the concentrated mass) and global vibrations of the membrane. In the case \(m>2\) and \(m<2\) the asymptotic behaviour is described and the conveyance is proved. In the case \(m=2\), the vibrations are described using a formal asymptotic expansion.
The paper has mathematical rigor and is of interest to applied mathematicians.