Functional Transformation Method

This chapter introduces a new physical modeling method for the vibration simulation of musical instruments. Similar to the methods described in the previous chapter the functional transformation method (FTM) first subdivides the musical instrument into different parts that are analyzed independently from each other as described in section 2. In contrast to the DWG and the MS, the FTM does not approximate the resulting sound or measured vibrational characteristics at discretized points on real instruments but it directly solves the underlying PDE. In contrast to the FDM where the solutions of the PDEs are approximated, the FTM solves the initial-boundary-value problems given in form of PDEs analytically. This is advantageous since the solution comprehends all geometrical properties and material parameters of the vibrating structure. Principally the FTM is, similar to the MS, a frequency-based and thus a perception-oriented physical modeling method. But since it solves the physical models analytically, the time signals and thus the sound production mechanisms are also simulated accurately. Furthermore, efficient realizations can be found with the FTM that allow real-time simulations of several vibrating structures.