RUI: Musical Acoustics: Coupled Oscillators, Mandolin Bridges, and Holographic Interferometry

Lewis And Clark College, Portland OR

Investigators

Abstract

This project is an experimental investigation of the acoustic properties of string instruments, in particular of the mandolin, which provides research experiences and training for undergraduate students. A musical instrument can be thought of as a system of coupled oscillators. In the case of the mandolin, the strings are set into oscillation with a plectrum and contain energy in a harmonic series of frequencies. The bridge of the instrument is set into vibration by the strings and in turn conveys the energy to the front plate of the mandolin. The front plate radiates sound but also couples to the back plate through the ribs of the instrument; both surfaces set the air inside the body of the instrument into oscillation. The instrument is thus modeled as a collection of harmonic oscillators coupled to each other through the bridge, ribs, or by direct contact between plate and air. The PI is studying the coupling between two tuned strings, their coupled interaction with the bridge and soundboard, and the two-slope decay. He is measuring the input of mechanical impedance at the bridge and the structural modal shapes of the body and the coupling between the plates and the air cavity. The ultimate objective of the proposed work is to understand how the mechanical properties of the instrument and its construction determine the character of the musical chords and identify potential improvements in bridge design for the mandolin, and other string instruments. The theory of coupled oscillators and the phenomenon of resonance is a central and broadly applicable subject in dynamics. The mandolin provides a rich and fascinating experimental arena for the application of dynamics and one that is accessible to measurement with instrumentation available to undergraduate physics departments. This project will provide transformative experiences to undergraduate students by captivating their interest in hands-on research and then providing rigorous training in its methods. A significant outcome of this project is to invest in the next generation of our nation's STEM workforce. The students will be exposed to a wide variety of experimental techniques and scientific instruments as well as advanced theoretical concepts, all broadly applicable throughout physics and engineering. In investigating the acoustics of the mandolin, the PI and his students will carry out a detailed study of the coupling of the doubled strings using high-speed video, and the musical implications of these interactions will be elucidated. These results will also shed light on other musical instruments with doubled strings, such as the lute, oud, and 12-string guitar. Measurements of the sound spectrum and bridge impedance combine to characterize the transfer of mechanical energy from string motions through the bridge to the motions of the instrument body that ultimately produce sound. Experiments to understand in detail the connection between the mechanical properties of the bridge and the resulting sound spectrum aim to identify potential improvements in bridge design. Details of the resulting body motions, the modes of vibration, will be studied using holographic interferometry. A study of the coupling of low-frequency plate modes of the mandolin?s front and back surfaces with the Helmholtz modes due to oscillations of air within the body will be compared to the classic studies of guitars.