The similarity analysis of vibrating membrane with its applications

Abstract The similarity analysis is applied to the study of vibrating membrane with variable density. In this paper, the design of the eigenvalue problems of the vibrating mode (Helmholtz equation) by the infinitesimal transformation groups is exhibited. The exact similarity solutions of the eigenvalue problems of Helmholtz equation with variable weighting function under the spiral and inversion groups are constructed in the regions of spiral, concentric circles, eccentric circles, and upper-half plane with circle removed. The similitude and scaling law are also built under the study and manipulation of the differential equations that are known to govern the behavior of the eigenvalue problems of Helmholtz equation. Without the requirement of geometrically similitude or mapping, the eigenvalues of a complicated domain can be obtained directly from a given simple region through the scaling law and they are verified and checked by numerical calculation and the Rayleigh–Ritz method.