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
Next Section: Power WavesPrevious Section: Wave Impedance