Abstract
A previous paper [Woodhouse et al., Acta Acustica 5, 15 (2021) https://doi.org/10.1051/aacus/2021009] showed acoustical measurements of an American 5-string banjo alongside similar measurements on a guitar, revealing a strong contrast in bridge admittance. Theoretical and numerical modelling is now presented to probe the physics behind this contrast. Without the bridge and strings, the banjo membrane has a rising trend of admittance associated with its modal density, and it has a distinctive pattern of sound radiation because an ideal membrane has no critical frequency. When the bridge and strings are added to the banjo, three formants shape the amplitude envelope of the admittance. One is associated with local effects of mass and stiffness near the bridge, and is sensitive to bridge mass and the break angle of the strings over the bridge. The other two formants are associated with dynamical behaviour of the bridge, analogous to the “bridge hill” in the violin.
Highlights
In a preceding paper [1], measurements of vibrational response were presented for an American 5-string banjo, and contrasted with corresponding results for a steel-string guitar
The lowest one or two peaks are consistently predicted too high in amplitude, typically by 10–15 dB, even though the frequencies and damping factors are generally well matched by the Finite Elements and Boundary Elements (FE/Boundary Element (BE)) calculations
The results show an opposite pattern to those in Figure 11a: as would be expected, changing the stiffness has no effect at high frequency, but a strong and systematic effect at the lowest frequencies
Summary
In a preceding paper [1], measurements of vibrational response were presented for an American 5-string banjo (a Deering Eagle II [2]), and contrasted with corresponding results for a steel-string guitar (by Martin Woodhouse [3]). The agenda for the present paper is to investigate the physics lying behind the contrast in the admittance functions, and to understand the origin of the three banjo formants. Included is the admittance measured directly on the head membrane of the banjo, with the strings and bridge removed, at approximately the same position. The three curves in this figure show remarkably different trends and levels: the admittance of the complete banjo lies typically 20–30 dB above that of the guitar over most of the range, and the admittance of the bare membrane lies a further 20–30 dB higher for frequencies above about 2 kHz. Two of the three formants can be seen in the red curve in this plot: a broad feature peaking around 500– 800 Hz, and a smaller peak around 3.5 kHz. Some parameter.
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