Electroacoustic absorbers for the low-frequency modal equalization of a room ::what is the optimal target impedance for maximum modal damping, depending on the total area of absorbers?

Electroacoustic absorbers consist of loudspeakers, shunted with an electrical circuit designed to obtain a given acoustic impedance. Their design and principles are now well understood. Unidimensional experiments, using plane waves under normal incidence in ducts, agree perfectly with simulations and confirm the validity of the model: the low-frequency absorption is maximal (and total) when the acoustic impedance at the diaphragm of the loudspeaker is purely resistive and equals the characteristic impedance of the medium. However, the working principle of an absorber in a 3D acoustic domain is somehow different: the absorber should bring a maximum of additional damping to each eigenmode, in order to realize a modal equalization of the domain. The target acoustic impedance value of the absorber can therefore differ from the characteristic impedance of the medium. Thus, extension of 1D results to the tridimensional case brings up several challenges, such as dealing with non trivial geometry, the finite size of the absorber, as well as the absence of simple formula to determine eigenmodes and their damping in a relatively general framework. In order to define the optimal target value for the acoustical impedance of an absorber in a realistic case, and to sense how this optimal value depends on some parameters, this paper proposes to address the following cases: a 1D case, under normal incidence, where the absorber has a sensibly smaller area than the cross-section of the duct, another 1D case, where the absorber is parietal, that is to say, under grazing incidence, and a 3D case, where absorbers have a given orientation and position, but a varying area