#### DW Displacement Inputs

We define general DW inputs as follows:

(E.33) | |||

(E.34) |

The th block of the input matrix driving state components and multiplying is then given by

Typically, input signals are injected equally to the left and right along the string, in which case

To show that the directly obtained FDTD and DW state-space models correspond to the same dynamic system, it remains to verify that . It is somewhat easier to show that

A straightforward calculation verifies that the above identity holds, as expected. One can similarly verify , as expected. The relation provides a recipe for translating any choice of input signals for the FDTD model to equivalent inputs for the DW model, or vice versa. For example, in the scalar input case (), the DW input-weights become FDTD input-weights according to

Finally, when and for all , we obtain the result familiar from Eq.(E.23):

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DW Non-Displacement Inputs

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Proof that the Third-Order Time Derivative is Ill Posed