  In string musical instruments, the sound is radiated by the soundboard,
subject to the strings excitation. This vibration of this rather complex
structure is described here with models which need only a small number of
parameters. Predictions of the models are compared with results of experiments
that have been presented in Ege et al. [Vibroacoustics of the piano soundboard:
(Non)linearity and modal properties in the low- and mid- frequency ranges,
Journal of Sound and Vibration 332 (5) (2013) 1288-1305]. The apparent modal
density of the soundboard of an upright piano in playing condition, as seen
from various points of the structure, exhibits two well-separated regimes,
below and above a frequency flim that is determined by the wood characteristics
and by the distance between ribs. Above flim, most modes appear to be
localised, presumably due to the irregularity of the spacing and height of the
ribs. The low-frequency regime is predicted by a model which consists of
coupled sub-structures: the two ribbed areas split by the main bridge and, in
most cases, one or two so-called cut-off corners. In order to assess the
dynamical properties of each of the subplates (considered here as homogeneous
plates), we propose a derivation of the (low-frequency) modal density of an
orthotropic homogeneous plate which accounts for the boundary conditions on an
arbitrary geometry. Above flim, the soundboard, as seen from a given excitation
point, is modelled as a set of three structural wave-guides, namely the three
inter-rib spacings surrounding the excitation point. Based on these low- and
high-frequency models, computations of the point-mobility and of the apparent
modal densities seen at several excitation points match published measurements.
The dispersion curve of the wave-guide model displays an acoustical radiation
scheme which differs significantly from that of a thin homogeneous plate. It
appears that piano dimensioning is such that the subsonic regime of acoustical
radiation extends over a much wider frequency range than it would be for a
homogeneous plate with the same low-frequency vibration. One problem in piano
manufacturing is examined in relationship with the possible radiation schemes
induced by the models.
