### Converting Any String State to Traveling Slope-Wave Components

We verified in §C.3.1 above that traveling-wave components and in Eq.(C.14) satisfy the ideal string wave equation . By definition, the physical string displacement is given by the sum of the traveling-wave components, or

Thus, given any pair of traveling waves and , we can compute a corresponding string displacement . This leads to the question whether any initial string state can be converted to a pair of equivalent traveling-wave components. If so, then d'Alembert's traveling-wave solution is

*complete*, and all solutions to the ideal string wave equation can be expressed in terms of traveling waves.

The *state* of an ideal string at
time is classically specified by its displacement and
velocity

*slope waves*as a function of an arbitrary initial slope and velocity:

*unbounded*as the string length goes to infinity. Related discussion appears in Appendix E.

It will be seen in §C.7.4 that state conversion between
physical variables and traveling-wave components is simpler
when *force* and *velocity* are chosen as
the physical state variables (as opposed to displacement and velocity
used here).

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Digital Waveguide Model

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D'Alembert Derived