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.(4.24), 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.

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written by 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.