Tubular Bells: A Physical and Algorithmic Model

Tubular bells are geometrically simple representatives of three-dimensional vibrating structures. Under certain assumptions, a tubular bell can be modeled as a rectangular plate with different types of homogeneous boundary conditions. Suitable functional transformations with respect to time and space turn the corresponding initial-boundary value problem into a two-dimensional transfer function. An algorithmic model follows according to the functional transformation method in digital sound synthesis. As with simpler vibrating structures (strings, membranes) the synthesis algorithms consist of a parallel arrangement of second-order sections. Their coefficients are obtained by simple analytic expressions directly from the physical parameters of the tubular bell.