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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).
\includegraphics{eps/fdlsuncoupled}

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.


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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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