Boundary Conditions

The relations of the previous section do not hold exactly when the string length is finite. A finite-length string forces consideration of boundary conditions. In this section, we will introduce boundary conditions as perturbations of the state transition matrix. In addition, we will use the DW-FDTD equivalence to obtain physically well behaved boundary conditions for the FDTD method.

Consider an ideal vibrating string with $M=8$ spatial samples. This is a sufficiently large number to make clear most of the repeating patterns in the general case. Introducing boundary conditions is most straightforward in the DW paradigm. We therefore begin with the order 8 DW model, for which the state vector (for the 0 th subgrid) will be

$$\underline{x}_W(n) = \left[\! \begin{array}{l} y^{+}_{n,0}\\ y^{-}_{n,0}\\ y^{+}_{n,2}\\ y^{-}_{n,2}\\ y^{+}_{n,4}\\ y^{-}_{n,4}\\ y^{+}_{n,6}\\ y^{-}_{n,6}\\ \end{array}\!\right].$$

The displacement output matrix is given by

$$\mathbf{C}_W= \left[\! \begin{array}{ccccccccccc} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{array}\!\right]$$

and the input matrix ${\mathbf{B}_W}$ is an arbitrary $M\times 2q$ matrix. We will choose a scalar input signal $u(n)$ driving the displacement of the second spatial sample with unit gain:

$${\mathbf{B}_W} = \left[\! \begin{array}{cc} 0 & 0 \\ 0 & 0 \\ 1/2 & 1/2 \\ 1/2 & 1/2 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\!\right]$$

The state transition matrix $\mathbf{A}_W$ is obtained by reducing Eq.(E.32) to finite order in some way, thereby introducing boundary conditions.