Control of 3D sound scattering based on decomposition into spherical harmonic components

Abstract

This thesis explores the active control of sound scattering from finite 3D structures using series of spherical harmonic components to describe the involved physical quantities. The investigation is theoretically focused and attempts to establish a framework for modelling complicated 3D sound scattering behaviour. An idealized approach is first explored where all variations of relevant physical quantities are known at all moments in time. This involves solving partial differential equations for an exterior impedance boundary-value problem and is done for two different models surrounded by an idealized fluid. One model is the uniform, locally-reacting impedance sphere, which is characterized by a radial wave impedance on the separation boundary that does not change with frequency, does not exhibit modes of dynamic motion, and is independent at each point on the boundary from all other points on the boundary. The other model is the thin, uniform, empty, elastic spherical shell based on Love`s first approximation and is characterized by a wave impedance on the separation boundary that has the opposite three properties of the previous model. Idealized active control approaches based on spherical harmonic components are explored for the two scenarios when using a small number of secondary control sources to suppress the scattered sound. For simplicity, the incidence that produces the scattering is chosen as a single monochromatic plane-wave. The common constraint of ‘least mean squares’ minimization is used for the scattered sound power. Predictions are realized for the scattering behaviour before and after power minimization using the spherical harmonic components, specifically at low frequencies, where the chosen active control produces the most significant suppression. These predictions confirm that the minimization with one or two control sources is similar to completely cancelling the first and, respectively, first two most dominant spherical harmonic components of the scattered sound in which the secondary also couples the best. The next component in the sequence is left as a residual after control, either enhanced or suppressed. With more control parameters to optimize, increasingly more of the dominant components in the scattering can be suppressed, depending on the order in the sequence of which is most dominant and coupled best into. A practical-oriented control approach is also explored, which relies on measurements of the incidence plus scattering at discrete locations, but on no prior knowledge of the scattering. This takes the form of velocity feedback control and is based on Bobrovnitskii`s method for modelling scattering, but translated to modal components, i.e. the spherical harmonics, rather than elemental components in space. Such active control is found to change the loaded impedance of the shell when excited with forces and measured with velocity sensors. The first few structural resonances can be dampened using this approach, which results in some reduction of the scattered sound at these frequencies.

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ORCID for Mihai Oriță: orcid.org/0000-0001-7307-0497

ORCID for Mihai Oriță: orcid.org/0000-0001-7307-0497

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