Eigenvalues of radially symmetric modes in composite spherical domains with a very small concentric cavity

Abstract

In his study, Wang proved that the fundamental frequency coefficient of a circular annular membrane fixed at the outer radius ´b´ and at the inner radius ´a´, is the same eigenvalue as in the case of a solid circular membrane, when the inner radius of the annular membrane approaches zero. Related studies showed that the same rather unexpected conclusions holds true in the case of higher modes of vibrations and also in the case of composite membranes. The present work demonstrates that from a mathematical viewpoint, the same property holds when solving a Helmholtz differential-type system in the case of composite spherical domain when a/c approaches zero.