9.4.1 Including a body resonance in a reflection function

In order to use the digital waveguide simulation approach to study the wolf note, we need a body reflection function that includes the influence of the body vibration. We can deduce an appropriate form from the analysis described in section 5.1.2. We derived the following approximate expression for the reflection coefficient $R$ for an ideal string with impedance $Z_0$ attached to a body with bridge admittance $Y(\omega)$:

$$R \approx -1 + 2 Z_0 Y(\omega) \tag{1}$$

from equation (5) of section 5.1.2.

To turn this into an approximate expression for the reflection function, we simply need the inverse Fourier transform. We already know the answer to that: the admittance $Y(\omega)$ is the Fourier transform of the impulse response $g(t)$ of the bridge; the velocity response to a unit impulse (delta function of applied force). So the reflection function $r(t)$ is given by

$$r(t) \approx -\delta(t-2\beta L /c) + 2Z_0 g(t-2\beta L /c) \tag{2}$$

where $2\beta L/c$ is the time delay for a wave on the string to travel to the bridge and back at the wave speed $c$, $L$ being the string length and $\beta$ being the relative position of the bowed point.

For the simplest wolf model, we would include just one body mode. We can represent the resonance responsible for the wolf as a mass-spring-dashpot oscillator with mass $m$, stiffness $k$ and dashpot strength $c$. We can deduce the impulse response of such an oscillator from the analysis in section 2.2.7: provided the damping is small, it is given by

$$g(t) \approx \dfrac{1}{m} \cos (\omega_b t) e^{-\omega_b \eta_b t/2} \tag{3}$$

where the resonance frequency is $\omega_b = \sqrt{k/m}$ and the modal loss factor is defined by $\omega_b \eta_b \approx c/m$. It is easy to see that this has the expected form for the velocity response to an impulse: at $t=0$ the impulse provides a unit jump in momentum, so that the velocity immediately after the impulse must be $1/m$. But there is no residual acceleration immediately after the impulse, so the free vibration of the oscillator must be in the cosine phase.

The expression (3) relates to an ideal string, so it includes the infinitely sharp delta function. In the spirit of the simple model discussed in section 9.2, we can spread this out into a narrow pulse with finite width. The reflection function then looks like the example shown in Fig. 1. Now, it is obvious that the tail of this reflection function goes on for ever. Does that mean we have to do convolutions of indefinite length in order to represent the effect? Fortunately, the answer is no: there is a computational trick whereby this infinite tail can be represented exactly while only requiring a few arithmetic operations for each successive time sample. This trick, known as an “IIR digital filter”, will be described in section 9.5.2.

Figure 1. Example of the reflection function from equation (3), incorporating the decaying oscillation of a single body mode.