The violin: A review of timbre and a finite difference model for bending plate waves

Abstract

Each violin generates a unique sound with a distinct set of acoustic properties. In particular, the Stradivarius violins are said to have a very unique sound. The ultimate goal of this thesis is to aid in recreating the unique sound of the Stradivarius. Though, the field of research surrounding the violin and its acoustic properties is very broad and many aspects of this research, such as subtle structural differences between violins, fall beyond the scope of this thesis. Rather, the goal of this thesis is to provide the tools to find the differences between different violins. This thesis is split into two parts. In the first part, a literature review, describing the different parts of the violin, is presented along with an overview of different experiments and finite element models, describing the plate dynamics of the violin. In the second part, a model of the violin is devised, with modelling cycles based on consecutive simplification, and implemented in Python using finite difference equations and the method of lines. This model is made with the goal to provide a framework for creating a realistic model of the violin where differences in timbre between violins can shown. In the literature review, four subjects are addressed. First, an explanation on timbre is given with an illustration of its perception. Additionally the timbre of historical and contemporary violins is compared. Second, the radiated sound of the violin is discussed. In specific, the problem with directivity of radiation is brought to attention. Third, an overview of the bridge is given with a strong emphasis on the influence of the bridge admittance on the radiated sound. Last, the corpus is dissected into different parts which are investigated with optical sensors and modelled with the software packages Comsol and Abaqus. The model in the second part of the thesis is based on principle of consecutive simplification. To set the groundwork for further iterations of the model, the most important assumptions are kept track off. The method of lines is then used to simulate the vibrations of a wooden square plate using the flexural wave equation as the governing equation of motion. In this simulation, the forward Euler method is used for the numerical time integration. The steady state solutions of the numerical model are coherent with the analytical solutions to the flexural wave equation. However, the time dependent results diverge from the results shown in the literature study. This is either due numerical time integration method being unstable or due to the physical properties of the wood not being translated by the stiffness matrix in the model.