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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 $ xy$ 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 $ xz$ 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.

Figure 6.19: Digital waveguide model of a rigidly terminated string vibrating transversally in three-dimensional space (two uncoupled planes of vibration).

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.7.15Yielding 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: Digital waveguide model of a string in which vertical and horizontal planes of vibration are coupled linearly at the bridge.

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 $ F_v^+(z)$ and $ F_h^+(z)$, respectively, the outgoing waves in each plane are given by

$\displaystyle \left[\begin{array}{c} F_v^-(z) \\ [2pt] F_h^-(z) \end{array}\rig...
...] \left[\begin{array}{c} F_v^+(z) \\ [2pt] F_h^+(z) \end{array}\right] \protect$ (7.16)

as shown in the figure.

In physically symmetric situations, we expect $ H_{vh}(z) = H_{hv}(z)$. 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 $ \omega $, then the coupling matrix with $ z = e^{j\omega T}$ is a constant (generally complex) matrix (where $ T$ 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 $ \mathbf{H}(e^{j\omega T})$ at some frequency has the form

$\displaystyle \mathbf{H}(e^{j\omega T}) = \left[\begin{array}{cc} A & B \\ [2pt] B & A \end{array}\right]

where $ A$ and $ B$ are any complex numbers. This means both string terminations are identical, and the coupling is symmetric (the simplest case in practice). The eigenvectors are easily calculated to be7.16

$\displaystyle \underline{e}_1 = \left[\begin{array}{c} 1 \\ [2pt] 1 \end{array}...
\underline{e}_2 = \left[\begin{array}{c} 1 \\ [2pt] -1 \end{array}\right],

and the eigenvalues are $ A+B$ and $ A-B$, respectively.

The eigenvector $ \underline{e}_T=[1, 1]$ corresponds to ``in phase'' vibration of the two string endpoints, i.e., $ F_v(e^{j\omega T}) = F_h(e^{j\omega T})$, while $ \underline{e}_T=[1, -1]$ corresponds to ``opposite phase'' vibration, for which $ F_v(e^{j\omega T}) = -F_h(e^{j\omega T})$. If it happens to be the case that

$\displaystyle \vert A+B\vert<\vert A-B\vert

then the in-phase vibration component will decay faster than the opposite-phase vibration. This situation applies to coupled piano strings [543], as discussed further below.

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

$\displaystyle \mathbf{H}(e^{j\omega}) \isdef \left[\begin{array}{cc} H_{vv}(e^{...
...\omega}) \\ [2pt] H_{hv}(e^{j\omega}) & H_{hh}(e^{j\omega}) \end{array}\right]

correspond to two 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 $ \mathbf{H}(e^{j\omega T})$, we have

$\displaystyle \mathbf{H}(e^{j\omega T}) \underline{e}_i(e^{j\omega T}) = \lambda_i(e^{j\omega T})\underline{e}_i(e^{j\omega T})

where $ \lambda_i(e^{j\omega T})$ denotes the $ i$th eigenvalue of the coupling-matrix $ \mathbf{H}(e^{j\omega T})$ at frequency $ \omega $, where $ i=1,2$. Since the eigenvector $ \underline{e}_i(e^{j\omega T})$ holds the Fourier transform of the incoming waves for mode $ i$ of the coupled-string system, we see that the eigenvalues have the following interpretation:
The $ i$th eigenvalue of the coupling matrix equals the frequency response seen by the $ i$th eigenpolarization.
In particular, the 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 $ \left\vert H_{hh}(e^{j\omega})\right\vert\gg\left\vert H_{vv}(e^{j\omega})\right\vert$ (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].

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