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Localized Displacement Excitations

Whenever two adjacent components of $ \underline{x}_K$ are initialized with equal amplitude, only a single $ W$ -variable will be affected. For example, the initial conditions (for time $ n+1$ )

y_{n,m-1} &=& 1\\
y_{n-1,m} &=& 1

will initialize only $ y^{-}_{n,m-1}$ , a solitary left-going pulse of amplitude 1 at time $ n=0$ , as can be seen from Eq.(E.11) by adding the leftmost columns explicitly written for $ \mathbf{T}^{-1}$ . Similarly, the initialization

y_{n-1,m-2} &=& 1\\
y_{n,m-1} &=& 1

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.(E.11). The superposition of these two examples corresponds to a physical impulsive excitation at time 0 and position $ m-1$ :

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

Thus, the impulse starts out with amplitude 2 at time 0 and position $ m-1$ , and afterwards, impulses of amplitude 1 propagate away to the left and right along the string.

In summary, we see that to excite a single sample of displacement traveling in a single-direction, we must excite equally a pair of adjacent colums in $ \mathbf{T}^{-1}$ . This corresponds to equally weighted excitation of K-variable pairs the form $ (y_{n,m},y_{n-1,m\pm1})$ .

Note that these examples involved only one of the two interleaved computational grids. Shifting over an odd number of spatial samples to the left or right would involve the other grid, as would shifting time forward or backward an odd number of samples.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University