###
State Conversions

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

and

,
we obtain

To convert an arbitrary initial string state

to either a
traveling force-wave or velocity-wave simulation, we simply must be
able to

*invert* the appropriate two-by-two matrix above. That
is, the matrix must be

*nonsingular*. Requiring both

determinants to be nonzero yields the condition

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

are prepared from

according to

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