Virtual analog modelling is aimed not at simulating musical instruments, but at classic analog effects. These include purely electronic devices such as vacuum tube amplifiers, etc., but also electromechanical effects such as plate and spring reverberation. These latter effects are particularly well-matched by time domain difference methods.
The spring reverb was originally intended as a low-cost artifical reverberation device, but like so many other analog effects,
A typical spring reverb set-up involves a number of springs, each of which may be described, mathematically, by the equations of motion of a helix
The physics of the helix is surprisingly complex, even if it is modelled in 1D (i.e., the spring is of negligible thickness, which is indeed the case in typical spring reverb units. The interesting features of the response result from the twist of the spring, which leads to a partial cutoff above a given wavelength (in fact, the wavelength corresponding to one twist of the helix). This partial cutoff normally occurs in the range between 2 and 5 kHz.
This is perhaps easiest to see from the spectrogram of a measured response given below (thanks to Julian Parker, at the Helsinki University of Technology, for supplying this):
Discrete echoes are visible below 2kHz in this rate, eventually losing coherence…above the cutoff, there is a distinct set of echoes, with a different rate of repetition.
A spring reverb unit is driven through a magnet attached to one end of the spring, usually such that when excited by an external EMF, it moves so as to twist the spring…readout is taken at the other end through the use of another magnet.
Difference schemes do a good job of reproducing the response of the combined driver/spring/readout system…here is a comparison of spectrograms of measured (bottom) and synthetic (top) responses:
There are some differences still…namely that the synthetic echoes below the cutoff are “too coherent”! As always, there is more work to be done.
In any case, here are a couple of synthetic spring responses…
For much more on this, see the article which will soon appear in the IEEE Transactions on Audio Speech and Language Processing (May, 2010), written with Julian Parker, and recent paper at DAFX, 2009.