### Coupled Horizontal and Vertical Waves

No vibrating string in musical acoustics is truly rigidly terminated,
because such a string would produce no sound through the body of the
instrument.^{7.15}Yielding terminations result in *coupling* of the horizontal and
vertical planes of vibration. In typical acoustic stringed
instruments, nearly all of this coupling takes place at the
*bridge* of the instrument.

Figure 6.20 illustrates the more realistic case of
two planes of vibration which are *linearly coupled* at one end
of the string (the ``bridge''). Denoting the traveling force waves
entering the bridge from the vertical and horizontal vibration
components by and , respectively, the outgoing
waves in each plane are given by

as shown in the figure.

In physically symmetric situations, we expect . That is, the transfer function from horizontal to vertical waves is normally the same as the transfer function from vertical to horizontal waves.

If we consider a single frequency , then the coupling matrix
with
is a constant (generally complex) matrix (where
denotes the sampling interval as usual). An *eigenanalysis*
of this matrix gives information about the *modes* of the coupled
system and the *damping* and *tuning* of these modes
[543].

As a simple example, suppose the coupling matrix at some frequency has the form

^{7.16}

The eigenvector
corresponds to ``in phase'' vibration
of the two string endpoints, *i.e.*,
,
while
corresponds to ``opposite phase'' vibration, for
which
. If it happens to be the case
that

More generally, the two *eigenvectors* of the coupling
frequency-response matrix

*decoupled polarization planes*. That is, at each frequency there are two

*eigenpolarizations*in which incident vibration reflects in the same plane. In general, the eigenplanes change with frequency. A related analysis is given in [543].

By definition of the eigenvectors of , we have

*eigenvalue*of the coupling-matrix at frequency , where . Since the eigenvector holds the Fourier transform of the incoming waves for mode of the coupled-string system, we see that the eigenvalues have the following interpretation:

The th eigenvalue of the coupling matrix equals theIn particular, thefrequency responseseen by the th eigenpolarization.

*modulus*of the eigenvalue gives the reflectance magnitude (affecting mode

*damping*), and the

*angle*of the eigenvalue is the phase shift of the reflection, for that mode (affecting

*tuning*of the mode). Use of coupling matrix eigenanalysis to determine mode damping and tuning is explored further in §C.13.

**Next Section:**

Asymmetry of Horizontal/Vertical Terminations

**Previous Section:**

Horizontal and Vertical Transverse Waves