Page  00000001 TOWARDS THE SYNTHESIS OF WAVEFRONT EVOLUTION IN 2-D Georg Essl gessl@cs.princeton.edu* ABSTRACT The dynamics of drums can be described by wavefronts and their wake on a bundle of rays. Using the ray picture in their construction preserves the structure of the wave propagation, which in general is dispersed by traditional meshed methods. The idea is based on emphasizing local dynamical descriptions and impulse responses of continuous solutions in the plane and translating these properties into a computational model. 1. INTRODUCTION This work is part of a larger program trying to understand if we can find efficient computational structure in the 2 -dimensional wave-equation'. The previous results in this program can be found in [4, 6] and discuss geometric features of wave fronts rendered with the algorithm to be described here. Aspects of numerical behavior as well as the nature and impact of wakes - originally to be presented here - have been moved to an alternative publication [5] due to space restrictions. In [4] I described the basic algorithm and related the results of the algorithm to known features of ray dynamics on a circular domain. In [6] the domain was altered to an ellipse and a stadium (two semicircles connected with straight lines) to give an argument relating the observed wave fronts to the whispering gallery phenomenon. Here I would like to bring the argument on the one hand close to the full acoustics of drums in the plane. On the other hand I want to give a first rendering of this algorithm for sound synthesis. For this, a number of pieces are missing in [4, 6] to arrive at a realistic simulation for sound synthesis. Most notably dissipation and radiation has not been modeled so far. Hence we are adding new structure and information to the algorithm to complete the picture. This is however still not an exact numerical integration of the wave equation as wakes have been omitted. See [5] for details. Digital waveguides [14] have proven to be an extraordinarily successful way of simulating the wave equation in one dimension. This has to do with the particular structure of the solution space of the wave equation, when written as traveling waves (d'Alembert's original solution). This form allows for a computational structure to be utilized that is, in a sense, better than a naive estimation of *The author is currently without affiliation. Please use the listed email for correspondence. SThroughout this paper dimensions refer to spatial dimensions of the domain. The additional temporal dimension is implied. a meshed method to integrate the wave equation. By requiring only constant, rather than order of spatial samples, waveguides are optimal with respect to spatial integration. The second main advantage is that the traveling waves are not, or only mildly disturbed by propagation. This allows for copy-propagation rather than arithmetic propagation. Hence rounding errors are avoided or kept limited to few operations thus giving waveguides highly desirable numerical properties as well. At the same time waveguides are exact with respect to the continuous equations on sampling points. We are still far from having the same properties for the 2-dimensional case. A number of methods have been employed to simulate drums. These are all predominantly meshed methods. These include spring-based methods [2], standard finite differencing methods, and waveguide meshes [16, 8] and hybrids between waveguide and finite differencing methods [1, 11]. All these share that the dynamics is discretized on mesh points and different directions of propagation are coupled at these mesh points. The difference either consists in the motivation or implementation. The mesh itself is responsible for inexactness in the solution known as mesh dispersion and various attempts have been proposed to reduce this effect for various situations [12, 9]. Another approach is using functional transforms[15]. This is really a modal methods, in which the modes-strengths are calculated from the equation once transform functions have been picked. As these transform functions usually constitute a support that ranges over the whole domain of solution, local aspects of the problem become approximated by truncated infinite series of these transform functions. Banded waveguides have also been proposed for 2-dimensional structures such as drums and cymbals [7]. This is a modal method which tries to retain some notion of spatial information. However this information is asymptotic and hence incomplete. Secondly, it is very difficult to recover the meaningful spatial positions of various waveguide bands crossing. Recently a related method, which employs details about the circular symmetry have been proposed [17] as well as hybrids with waveguide meshes[13]. 2. APPROACH The method to be described here does not solve the full problem. It rather is an attempt to understand the structure better. It differs from earlier work in that it does not