Localized Displacement Excitations

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Localized Displacement Excitations

Whenever two adjacent components of $\underline{x}_K$ are initialized with equal amplitude, only a single

-variable will be affected. For example, the initial conditions (for time

)

$\begin{eqnarray*} y_{n,m-1} &=& 1\\ y_{n-1,m} &=& 1 \end{eqnarray*}$

will initialize only $y^{-}_{n,m-1}$ , a solitary left-going pulse of amplitude 1 at time , as can be seen from Eq. $\,$ (P.11) by adding the leftmost columns explicitly written for $\mathbf{T}^{-1}$ . Similarly, the initialization

$\begin{eqnarray*} y_{n-1,m-2} &=& 1\\ y_{n,m-1} &=& 1 \end{eqnarray*}$

gives rise to an isolated right-going pulse $y^{+}_{n,m-1}$ , corresponding to the leftmost column of $\mathbf{T}^{-1}$ plus the first column on the left not explicitly written in Eq. $\,$ (P.11). The superposition of these two examples corresponds to a physical impulsive excitation at time 0 and position :

$\displaystyle y_{n-1,m-2}$	$\displaystyle =$	$\displaystyle 1$
$\displaystyle y_{n,m-1}$	$\displaystyle =$	$\displaystyle 2$
$\displaystyle y_{n-1,m}$	$\displaystyle =$	$\displaystyle 1$	(P.12)