Initialization

The difference equation (3.20) must be initialized with two values, typically $u^{0}$ and $u^{1}$. This pair of values is slightly different from what one would normally use to initialize the continuous time harmonic oscillator, namely the values $u(0)$ and $\frac{du}{dt}\vert _{t=0}$. In the distributed case, especially under striking conditions, it is often the initial velocity which is of interest. Supposing that one has, instead of the two values $u^{0}$ and $u^{1}$, an initial value $u^{0}$ and an initial velocity condition (call it $v^{0}$), a very simple way of proceeding is to write

$$\delta_{t+}u^{0} = v_{0}\qquad\rightarrow\qquad u^{1} = u^{0}+kv_{0}$$ (3.21)

Though this approximation is only first-order accurate, it is wholly sufficient for most sound synthesis applications, especially if one is operating at a typical elevated audio sampling rate (i.e., for a small value of $k$).