Waveguide Resonators

The Karplus-Strong model of a recursive waveguide with filter has long been regarded as an efficient synthesis technique for plucked string sounds (Karplus & Strong, 1983). The basic model for the waveguide uses a delay line of p samples which determines the resonant frequency of the string, a low-pass filter which simulates the energy loss caused by the reflection of the wave, and the feedback of the sample back into the delay line. The initial energy input is simulated by initializing the delay line with random values in order to simulate the "pluck", that is, introducing a noise burst whose spectrum decays to a sine wave at a rate proportional to the length of the delay line.

The resonator sounds so realistic because it is in fact a physical model of the string, with the delay line corresponding to the length of the string, along with the filter and feedback creating the basic resonance. In the last examples below we substitute a noisy excitation function (bowing on the bridge of an actual cello) for the pluck, and then, just like the instrument, when the resonator is tuned to an open string on the cello, the result is striking similar to a bowed note on the same string.

The model applies equally to a string fixed at both ends or a tube open at both ends, at least in terms of the resonant frequencies all being harmonics of the fundamental. If the sample is negated before being fed back into the delay line, the resulting change of phase models a tube closed at one end, which results in only the odd harmonics being resonant, and lowers the fundamental frequency by an octave, since the negation effectively doubles the length of the delay line. For the basic model, the fundamental resonance equals

SR / (p + 1/2)

where SR is the sampling rate, and p is the length of the delay line.

  Plucked string with doubling of the delay line length

  Plucked string with negation (odd harmonics, 1 octave lower) with doubling of the delay line length

  Plucked string improvisation showing influence of different initial random values

  Bowing on bridge as an excitation function

  Bowing on bridge resonated on C, G, D, ...

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References:

Karplus, K. & A. Strong, 1983. Digital synthesis of plucked string and drum timbres. Computer Music Journal, 7(2).

Truax, B. 1994. Discovering inner complexity: Time-shifting and transposition with a real-time granulation technique. Computer Music Journal, 18(2), 38-48.