Physical modeling of strings using multidimensional transfer functions

Physical modeling of strings using multidimensional transfer functions

Description of string vibrations by PDEs

String vibrations of acoustical instruments can be classified into longitudinal and transversal vibrations. Physical analysis of the string vibrations by applying the basic laws of physics leads to partial differential equations (PDEs). These PDEs are two-dimensional since they depend on one temporal and one spatial variable. For the complete description also boundary conditions and initial conditions have to be defined. The boundary conditions depend on the fixing of the string and the initial conditions depend on the initial shape of the string deflection. The only assumptions that are made for the string here are that the string is elastic, homogeneous and isotropic. Furthermore, the smoothness of its surface shall not permit stress concentration. These PDEs describe the string vibrations in an analytical way, but they cannot be implemented in the computer directly due to the inherent differetial terms.

Functional transformation method

As other methods the functional transformation method (FTM) discretizes the PDEs for computer implementation. But instead of approximating the solution, the FTM solves the PDEs analytically. This is done by applying functional transformations with repect to time and space. Then, a multidimensional transfer function model is obtained, representing the analytical solution in the temporal and spatial frequency domain. There the initial and boundary conditions are included as additive terms.
To obtain the analytical solution in the time and space domain the inverse functional transformations have to be applied to the multidimensional transfer function. This continuous solution is then discretized for computer implementation with well known discretization schemes. This guaranties the preservation of the inherent stability of the continuous model also in the discetized solution. Also for nonlinear excitations, as they occur in piano synthesis and bowed strings, the stabilty of the discrete system can be ensured. In addition inherent nonlinearities, as they occur at high deflections, can be simulated in a stable way.

Sound Examples

Sounds of transversal vibrating strings produced with the functional transformation method give an impression of the possibilities of this method.

Electric bass guitar,
Slapped bass guitar,
transformation from a guitar string to a xylophone,
lowest piano note,
polyphonial spinet.

Publications on string synthesis with FTM

In the following there is a list of publications dealing with string synthesis based on FTM. There the functional transformations are described in detail and they are explained on transversal vibrating, disperive strings with frequency dependent losses.

doc ppt	L. Trautmann, R. Rabenstein Multidimensional String Models In First OnLine Symposium for Electronic Engineers (OSEE), www, July, 2000.
pdf ps	L. Trautmann, R. Rabenstein Sound Synthesis with Tension Modulated Nonlinearities Based on Functional Transformations In Acoustics and Music: Theory and Applications (AMTA), pp. 444-449, N.E. Mastorakis (ed.), WSES, Jamaica, December, 2000.
pdf ps	L. Trautmann, R. Rabenstein Transfer Function Models with Nonlinear Excitations for Digital Sound Synthesis In Proc. X European Signal Processing Conference (EUSIPCO 2000), vol. 4, pp. 2217-2220, M. Gabbouj (ed.), EURASIP, Tampere, Finland, September, 2000.
pdf ps	L. Trautmann, R. Rabenstein Digital Sound Synthesis Based on Transfer Function Models In IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA 99), pp. 83-86, IEEE, New Paltz, New York, October, 1999.

Lutz Trautmann

Last modified: Mon Jul 9 13:39:45 MET 2001