### 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,

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

*nonsingular*. Requiring both determinants to be nonzero yields the condition

Carrying out the inversion to obtain force waves from yields

**Next Section:**

Power Waves

**Previous Section:**

Wave Impedance