Initialization

Scheme (6.33), like the wave equation itself, must be initialized with two sets of values. From the explicit for of the recursion (6.34), it should be clear that one must specify values of the grid function $u_{l}^{n}$ at time steps $n=0$ and $n=1$, namely $u_{l}^{0}$ and $u_{l}^{1}$. Supposing that the initial displacement $u_{l}^{0}$ is given, and that one also has a grid function $v_{l}^{0}$ representing the initial velocity, one may then write for $u_{l}^{1}$:

$$u_{l}^{1}=u_{l}^{0}+kv_{l}^{0}$$ (6.36)

which is exactly the same procedure as in the case of the simple harmonic oscillator.