Considerations on travelling waves in the horn equation and energetic aspects

The digital waveguide synthesis of wind resonators and of the vocal tract is based on decompositions into travelling waves. Typical ones are planar waves in straight pipes and spherical waves in conical pipes. However, approximating a bore by cascading such basic segments introduce unrealistic discontinuities on the radius R or the slope R' (with acoustic consequences). It also can generate artificial instabilities in time-domain simulations, e.g. for non convex junctions of cones. In this paper, we investigate the case of the "conservative curvilinear horn equation" for segments such that the flaring parameter R''/R is constant, with which smooth profiles can be built. First, acoustic states that generalize planar waves and spherical waves are studied. Examining the energy balance and the passivity for these travelling waves allows to characterize stability domains. Second, two other definitions of travelling waves are studied: (a) one locally diagonalizes the wave propagation operator, (b) one diagonalizes the transfer matrix of a segment. The propagators obtained for (a) are known to efficiently factorize computations in simulations but are not stable if the flaring parameter is negative. A study in the Laplace domain reveals that propagators (b) are stable for physically meaningful configurations.