Numerical Calculation of Modal Spring Reverb Parameters
In the design of real-time spring reverberation algorithms, a modal
architecture offers several advantages, including computational efficiency and parametric control flexibility. Due to the complex,
highly dispersive behavior of helical springs, computing physically accurate parameters for such a model presents specific challenges. In this paper these are addressed by applying an implicit
higher-order-in-space finite difference scheme to a two-variable
model of helical spring dynamics. A novel numerical boundary
treatment is presented, which utilises multiple centered boundary
expressions of different stencil width. The resulting scheme is unconditionally stable, and as such allows adjusting the numerical
parameters independently of each other and of the physical parameters. The dispersion relation of the scheme is shown to be
accurate in the audio range only for very high orders of accuracy
in combination with a small temporal and spatial step. The frequency, amplitude, and decay rate of the system modes are extracted from a diagonalised form of this numerical model. After
removing all modes with frequencies outside the audio range and
applying a modal amplitude correction to compensate for omitting
the magnetic transducers, a light-weight modal reverb algorithm is
obtained. Comparison with a measured impulse response shows a
reasonably good match across a wide frequency range in terms of
echo density, decay characteristics, and diffusive nature.