In §C.3.6, an arbitrary string state was converted to traveling displacement-wave components to show that the traveling-wave representation is complete, i.e., that any physical string state can be expressed as a pair of traveling-wave components. In this section, we revisit this topic using force and velocity waves.
By definition of the traveling-wave decomposition, we have
Using Eq.(C.46), we can eliminate
and
,
giving, in matrix form,
Thus, the string state (in terms of force and velocity) is expressed as a linear transformation of the traveling force-wave components. Using the Ohm's law relations to eliminate instead
To convert an arbitrary initial string state
That is, the wave impedance must be a positive, finite number. This restriction makes good physical sense because one cannot propagate a finite-energy wave in either a zero or infinite wave impedance.
Carrying out the inversion to obtain force waves
from
yields
Similarly, velocity waves