**Résumé** |
The violin has been the topic of a lot of research over the past few hundred years. Its characteristic sound has, during live performances, not a omnidirectional radiation, but a radiation pattern dependent on the frequency. Those rays of sound, for higher frequencies, 'enlighten' the acoustical properties of the surrounding, giving an extra dimension to the live performance. The goal of this research is to study this frequency dependent sound radiation and to synthesize this directivity behaviour. The model of the violin is a 70 year old violin, whose sound eld is measured. This is not done the direct way, where the bridge is excited and its radiation is measured, but the inverse way, where an external source excites the violin from a certain direction. The vibrations of the violin are then captured by a calibrated sensor inside the bridge. Those measurements use the reciprocity assumption, which states that if the body vibrates strongly due to a sound wave with given frequency and direction, it will radiate sound waves with the same power and direction as this excitation. Those measurements are done at the anechoic chambre at IRCAM, over a large frequency range and over a whole sphere surrounding the violin. From the liturature is known, that the violin will have standing wave solutions of the vibrations of its physical structure. The analysis consists of nding the peaks in the measured power spectrum, which are assumed to appear at the modal frequencies of the body. The width of the peak de nes the damping coefficient of this mode. Per mode, the radiation pattern is decomposed into the zeroth and rst order spherical harmonics, yielding an estimated directivity of the sound. The reproduction is based on the propagation laws. If we have two sources whose sound eld are equal at a surrounding surface, the sound elds outside this surface are the same according to the propagation laws. In our case, the rst source is the violin and the second is a unit which can generate elementary directivity patterns using the zeroth and rst spherical harmonics. Now, the decomposition of the radiation patterns can be used directly as multiplication factors in the eld reproduction. To reproduce the sound of the violin, a modal-based synthesis program is used, which decomposes complex structures in simple structures having vibrating modes. Those modes interact together when excited, and the velocity of the modes at a given position in the structure is the output sound. This program is extended with a directivity processing function. In this research the measurements, analysis and reproduction are all done. There are three main problems encountered. The rst is, that the peak and damping nding algorithm are too straightforward, which result in large di erences between the simulated and real frequency spectrum. The second is, that high frequencies dominate the power spectrum in the measurements. These give very sharp sounds during the reproduction phase. The last is, that the Fourier decomposition returns complex factors, meaning phase shift information. In a signal processing sense, these phase shifts result in lter operations. This is left for further research. |