Radiation of thin complex plates estimated with the landscape of localisation theory

Abstract

The landscape of localisation is a practical tool that enables the prediction of the geographical localisation of localized modes and helps us to understand the transition between localized and delocalised states. Moreover, approximations based on the Rayleigh quotient and on a variant of Weyl’s law are employed to predict the eigenfrequencies for the Schro ¨ dinger operator in quantum mechanics, but they are also valid for the Laplace and biharmonic operators, which characterize the behaviour of most dynamical systems in acoustics and vibrations. When studying the acoustic radiation from a vibrating structure, three global parameters are key indicators: the mean squared velocity, the acoustic radiated power, and the radiation efficiency. The literature on this subject is very vast for the plate case, where for simple geometries, it is still possible to derive analytical solutions or, at least, very useful approximations. For more complex structures, numerical simulations seem to be appropriate for lack of a simpler solution. In this context, this work aims to give some light to create a direct relationship between these global parameters and the landscape of localisation function, based on the multipolar radiation behaviour presented by localized modes and estimated by geometrical means.