Functional summary statistics for point processes on the sphere with an application to determinantal point processes

We study point processes on $\mathbb S^d$, the $d$-dimensional unit sphere $\mathbb S^d$, considering both the isotropic and the anisotropic case, and focusing mostly on the spherical case $d=2$. The first part studies reduced Palm distributions and functional summary statistics, including nearest neighbour functions, empty space functions, and Ripley's and inhomogeneous $K$-functions. The second part partly discusses the appealing properties of determinantal point process (DPP) models on the sphere and partly considers the application of functional summary statistics to DPPs. In fact DPPs exhibit repulsiveness, but we also use them together with certain dependent thinnings when constructing point process models on the sphere with aggregation on the large scale and regularity on the small scale. We conclude with a discussion on future work on statistics for spatial point processes on the sphere

Global and local synthetic descriptions of the piano soundboard

Up to around 1.1 kHz, the soundboard of the piano behaves like a homogeneous plate whereas upper in frequency, it can be described as a set of waveguides defined by the ribs. In consequence: a) The acoustical coincidence phenomenon is deeply modified in comparison with that occurring in homogeneous plates since the dispersion curve of a waveguide can present none, one, or two coincidence frequencies. This may result in a nonuniformity of the soundboard radiation in the treble range, corresponding to the so-called killer octave, where a good sustain is difficult to obtain. b) The mobility (mechanical admittance) in the direction normal to the soundboard can be synthesised with only a small number of parameters. It compares well with published measurements (Giordano, JASA, 1998), in particular the step-like falloff of the local impedance due to the localisation of the waves between ribs. c) The synthesised mobility has the same features as those which can be derived independantly, according to Skudrzyk (JASA, 1980) and Langley (JSV, 1994). This approach avoids the detailed description of the soundboard, based on a very large number of parameters. It can be used to predict global changes of the driving point mobility, and possibly of the sound radiation in the treble range, resulting from structural modifications

Vibroacoustics of the piano soundboard: Reduced models, mobility synthesis, and acoustical radiation regime

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.Comment: Research highlights: - Synthetic modelling of a piano soundboard overa broad-frequency-range (several kHz). - Quantitative agreement between predicted and measured apparent local modal density. -Modal density of a plate with non-special orthotropy and arbitrary contour. -Similar characteristic impedance between comparable pianos, over several kHz. -Extension of the subsonic regime of acoustical radiation over a wide frequency range. Journal of Sound and Vibration (2013) http://dx.doi.org/10.1016/j.jsv.2013.03.01