Spatial spectra of the eigenmodes of ribbed plates projected on dispersion branches

A vast literature has been devoted to the transverse vibration and the sound radiation of ribbed plates over the last decades. The present study has been motivated by the analysis of the dynamical behaviour of piano soundboards. As a rough approximation, a piano soundboard can be considered as an orthotropic ribbed plate. Our purpose is to establish condensed descriptions for their dynamics. For low frequencies, regularly ribbed plates can be considered as homogeneous plates. It is usually considered that homogenization is valid only up to a frequency corresponding roughly to the confinement of one half wave-length between the (periodically spaced) ribs. Beyond that frequency, depending on the relative characteristic mobility of the ribs and that of the base plate, the ribs may constrain transverse waves to be guided between them. We focus here on the spatial spectrum of the normal modes of the ribbed plate (2D Fourier transforms of the modal shapes). It appears that most of the peaks of each spectrum can be seen as belonging to one of a few dispersion branches in an appropriate (w; k)-plane. Interestingly, different peaks of a spectrum (of one given mode) usually "belong" to different dispersion branches. When valid, this description may prove an interesting intermediate step to derive approximations for the sound radiation of such plates.