### 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.

*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}

*i.e.*, , while corresponds to ``opposite phase'' vibration, for which . If it happens to be the case that

*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

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Horizontal and Vertical Transverse Waves