# Introduction to Physical Modeling

In recent years, many techniques for modelling musical instruments have been proposed, each one offering certain advantages, yet also entailing certain restrictions. In result, specific techniques are suited to deal with different problems. Techniques most widely used include:

• Finite-Element Methods (FEM). Here the geometry can be modelled to any desired precision quite easily and stationary solutions like the static buckling of violin or guitar bodies or piano sound boards can be modelled as well as eigenvalues calculations can be performed. Still, the transient calculations which are crucial for musical instruments (operating as transiently vibrating bodies) are very time-consuming and sometimes not quite accurate in higher frequency regions.

• Finite-Difference-Methods (FDM) are suited very well for transient calculations and – when formulated explicitly – are much faster than FEM models. On the other hand, it is not that easy with FDM to cover all details of the geometry. • Waveguide Methods. These methods are working in real time and are based upon the d’Alembert solution of the wave equation. Still they only show the basic behaviour of musical instruments and so lack some of the sound fluctuations associated with musical instrument behaviour.

• Lattice-Boltzmann Methods (LBM). This method is probably the most suitable for simulating flow dynamics of blown musical instruments. However, it is very slow in regard of calculation time.

The inclusion of the complete geometry of musical instruments in the model is deemed necessary. Musical instruments in fact have many degrees of freedom and so show a wide range of different behaviour under all possible playing conditions. Played notes (sound recordings) of all possible pitches have to be used for a complete judgement of the instrument quality. This implies that instruments have to be played hard and soft, and have to be tested with many styles of music. To get a concise picture of the functioning of musical instruments, and of how geometrical or material changes can influence their sound, the whole instrument body has to be considered as a compound structure.

Once this holistic approach is accepted, one would wish to have a fast or even real-time solution for simulating musical instruments with the complete geometry involved. This indeed would secure a big step forward in music production as well as in instrument building. Appropriate real-time algorithms could be used in synthesizers, sequencer software or VST-plug-ins, allowing practically infinite sound creations to be used in whatever music and other media productions today. Also instrument builders would be able to listen to the sound of their violins, guitars ect. before actually making an instrument by hand. Instead, they could perform changes in the geometry of an instrument (or certain parts thereof) with the computer, and then decide which model sounded best before starting to build the instrument according to the data obtained from the computer model.

Mathematical and computational methods for achieving this goal would be ray-tracing, geometry strain, or statistical methods. In sum, investigations in the direction outlined here certainly are of interest in regard of musical instrument acoustics. They seem also to be promising also with respect to improvements which can be of practical and economic relevance.