## String Coupling Effects

It turns out that a single digital waveguide provides a relatively
static-sounding string synthesizer. This is because several
*coupling* mechanisms exist in natural string instruments. The
overtones in coupled strings exhibit more interesting amplitude
envelopes over time. Coupling between different strings is not the
only important coupling phenomenon. In a real string, there are
*two* orthogonal planes of transverse vibration which
are intrinsically coupled to each other [181]. There is also
intrinsic coupling between transverse vibrational waves and
longitudinal waves (see §B.6).

### Horizontal and Vertical Transverse Waves

The transverse waves considered up to now represent string vibration
only in a single two-dimensional plane. One such plane can be chosen
as being perpendicular to the top plate of a stringed musical instrument. We
will call this the plane and refer to it as the *vertical
plane of polarization* for transverse waves on a string (or simply the
*vertical component* of the transverse vibration). To more fully
model a real vibrating string, we also need to include transverse waves
in the plane, *i.e.*, a *horizontal plane of polarization* (or
*horizontal component* of vibration). Any polarization for transverse
traveling waves can be represented as a linear combination of
horizontal and vertical polarizations, and general transverse string
vibration in 3D can be expressed as a linear superposition of
vibration in any two distinct polarizations.

If string terminations were perfectly rigid, the horizontal
polarization would be largely *independent* of the vertical
polarization, and an accurate model would consist of two identical,
uncoupled, filtered delay loops (FDL), as depicted in
Fig.6.19. One FDL models vertical force waves while the other
models horizontal force waves. This model neglects the small degree
of nonlinear coupling between horizontal and vertical traveling waves
along the length of the string--valid when the string slope is much
less than unity (see §B.6).

Note that the model for two orthogonal planes of vibration on a single string is identical to that for a single plane of vibration on two different strings.

### 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.

### Asymmetry of Horizontal/Vertical Terminations

It is common in real stringed instruments that horizontal and vertical
transverse waves are *transduced differently* at the bridge. For
example, the bridge on a guitar is typically easier to ``push'' into
the top plate than it is to ``shear'' sidewise along the top plate.
In terms of Eq.(6.16), we have
(at most frequencies). This
unequal terminating impedance causes the horizontal component of
string vibration to *decay slower* than the vertical component of
vibration. We can say that this happens because the vertical bridge
admittance is much greater than the horizontal admittance, giving rise
to a faster rate of energy transfer from the vertical string
polarization into the bridge--in other words, the bridge is more
``yielding'' in the vertical direction. The audible consequence of
this unequal rate of decay is a *two-stage amplitude envelope*.
The initial fast decay gives a strong onset to the note, while the
slower late decay provides a long-lasting sustain--two normally
opposing but desirable features.

### Coupled Strings

We have just discussed the coupling between vertical and horizontal planes of vibration along a single string. There is also important coupling among different strings on the same instrument. For example, modern pianos are constructed having up to three physical strings associated with each key. These strings are slightly mistuned in order to sculpt the shape of the decay envelope, including its beating characteristics and two-stage decay. A two-stage decay is desired in piano strings in order to provide a strong initial attack followed by a long-sustaining ``aftersound'' [543], [18, Weinreich chapter].

A simple approximation to the effect of coupled strings is obtained by simply summing two or more slightly detuned strings. While this can provide a realistic beating effect in the amplitude envelope, it does not provide a true two-stage decay. A more realistic simulation of coupling requires signal to flow from each coupled string into all others.

When the bridge moves in response to string vibrations, traveling waves are generated along all other strings attached to the bridge. In the simplest case of a bridge modeled as a rigid body, the generated wave is identical on all strings. In §C.13, an efficient scattering formulation of string coupling at a bridge is derived for this case [439]. It can be seen as a simplification of the general coupling matrix shown in Fig.6.20 for the two-string (or two-polarization) case. Additionally, an eigenanalysis of the coupling matrix is performed, thereby extending the analysis of §6.12.2 above.

### Longitudinal Waves

In addition to transverse waves on a string, there are always
*longitudinal waves* present as well. In fact, longitudinal
waves hold all of the *potential energy* associated with the
transverse waves, and they carry the forward *momentum* in the
direction of propagation associated with transverse traveling waves
[122,391]. Longitudinal waves in a
string typically travel an order of magnitude faster than transverse
waves on the same string and are only weakly affected by changes in
string tension.

Longitudinal waves are often neglected, *e.g.*, in violin acoustics,
because they couple inefficiently to the body through the bridge, and
because they are ``out of tune'' anyway. However, there exist
stringed instruments, such as the Finnish Kantele [231], in
which longitudinal waves are too important to neglect. In the piano,
longitudinal waves are quite audible; to bring this out in a striking
way, sound example 5 provided in [18, Conklin chapter]
plays *Yankee Doodle* on the longitudinal modes of three piano
strings all tuned to the same (transversal) pitch. The nonlinear
nature of the coupling from transverse to longitudinal waves has been
demonstrated in [163]. Longitudinal waves have
been included in some piano synthesis models
[30,28,24,23].

**Next Section:**

Nonlinear Elements

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Loop Filter Identification