Spatial Modes in Complex Geometries

Working all of these paths out gets more and more difficult for higher order modes (except for the simple (N, N) harmonics). The important thing to realize here is that each mode has a unique path length, which could be modeled by a delay line in a banded waveguide network. 

Prayer bowls are made from a mixture of many metals. The makers and users believe each metal possesses some particular spiritual power, and the 

One more topie to briefly diseuss relates to objeets with very many highly resonant modes. These objeets, when struek, ean makes sounds that vibrate in odd ways. For example, the struek prayer bowl exhibits beating, whieh is the interferenee of modes that are elose together in frequeney. For fun, add sin(7i0 and sin(l. 01 TiO and look at the result. What you aetually end up seeing instead of the addition of two sine waves is instead sin(1.0057iO eos(O.Ol7i05 the 

The good news from this chapter is that all of the techniques we have used in this book so far are applicable in two and three dimensions. The bad news is that things are generally more complex in higher dimensions. Meshes are the most general way to model the spatial aspects of vibrations, and modes are the most general way to handle the pure spectral behavior. Banded waveguides are applicable in some systems, especially where clear, closed wavepaths can be identified. The next chapter will depart entirely from these types of models, and look instead at the statistical behavior of random type sounds. 

Claude Cadoz, Annie Luciani and Jean-Loup Florens. CORDIS-ANIMA A Modeling and Simulation System for Sound Image Synthesis-The General Formalization. Computer Music Journal 17(1) 21-29(1993). 

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