Strings

Figure 3: The ideal vibrating string.

The classical physical model for transverse vibrations in an ideal string, as illustrated in Fig. 3, is the following wave equation [110,111,118,180]:⁵

$$Ky''= \epsilon {\ddot y}$$

where

$$\begin{array}{rclrcl} K& \isdef & \mbox{string tension} & \... ...isdef & \frac{\partial}{\partial x}y(t,x) \nonumber \end{array}$$

(Note that `` $\isdeftext$'' means ``is defined as''.) Applying the finite difference approximation (FDA) means to replace each partial derivative by a finite difference [169], e.g.,

$${\dot y}(t,x)\approx \frac{y(t,x)-y(t-T,x)}{T}$$

and

$$y'(t,x)\approx \frac{y(t,x)-y(t,x-X)}{X}$$

where $T$ and $X$ are the time and position sampling intervals to be used, respectively. Note that the finite-difference approximations become exact, in the limit, as $T$ and $X$ approach zero. In practice, more accurate simulation is obtained with increasing sampling rates. Applying a FDA to the ideal string wave equation above yields

$$K\frac{y(t,x+X) - 2 y(t,x) + y(t,x-X)}{X^2} = \epsilon \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x)}{T^2}.$$

Normalizing $T$, $X$, and $K/\epsilon$ to 1 leads to the simplified recursion

$$y(n+1,m) = y(n,m+1) + y(n,m-1) - y(n-1,m).$$

Thus, for each time step $n$, the string position samples can be updated using all samples along the string from the previous two time steps. The FDA method for numerical string simulation was used by Pierre Ruiz in his early work on vibrating-string simulation [74,131], and it is still in use today [31,32,49].

Perhaps surprisingly, it can be shown that the above recursion is exact at the sample points, despite the apparent crudeness of the finite difference approximation at low sampling rates, provided the string initial conditions and excitations are bandlimited to less than half the sampling rate. An easy proof is based on showing its equivalence to the digital waveguide model for the ideal string [159, pp. 430-431].