The unique mathematical physics of the singing saw
The mysterious and ethereal sound of the singing saw is part of folk musical traditions around the world.
Made by bending a metal handsaw and tilting it like a cello, the instrument reached its peak in the vaudeville scenes of the early 20and century and has experienced a resurgence thanks, in part, to social media.
It turns out that the unique mathematical physics of the singing saw may hold the key to designing high quality resonators for a range of applications.
In a new paper, a team of researchers from the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) and the Department of Physics used the singing saw to demonstrate how the geometry of a curved sheet, like curved metal , could be engineered to create high-quality, long-lasting oscillations for applications in sensing, nanoelectronics, photonics and more.
“The common handsaw can be converted into a bowed musical instrument capable of producing extremely sustained notes when its blade is properly bent. Acoustic modes localized to an inflection point are known to underlie the sound quality of the saw, but the origin of the localization has remained mysterious. Study Quotes.
“Our research offers a robust principle for designing high-quality resonators independent of scale and material, from macroscopic musical instruments to nanoscale devices, simply through a combination of geometry and topology,” said L Mahadevan, Valpine Lola England Professor of Applied Mathematics, Organism and Evolutionary Biology and Physics and lead author of the study.
The research is published in The Proceedings of the National Academy of Sciences (PNAS).
“The ability to sustain notes or vibrations underlies the design of most acoustic devices, ranging from musical instruments to nanomechanical resonators. Inspired by the singing saw that gains its musical quality from the unusual curvature of its blade, we wonder how geometry can be used to trap and isolate acoustic modes of dissipative decay in a continuous elastic medium. Study Quotes.
“How the singing saw sings is based on a startling effect,” said Petur Bryde, a graduate student at SEAS and the paper’s co-first author. “When you hit a flat elastic sheet, such as sheet metal, the whole structure vibrates. Energy is quickly lost through the boundary where it is held, resulting in a dull sound that quickly dissipates. The same result is seen if you bend it into a J-shape. But, if you bend the sheet into an S-shape, you can vibrate it in a very small area, resulting in a clear, long-lasting sound.
The curved saw geometry creates what musicians call the sweet spot and what physicists call localized vibrational modes – a confined area on the sheet that resonates without losing energy around the edges.
Importantly, the specific geometry of the S-curve doesn’t matter. It could be an S with a large curve at the top and a small curve at the bottom or vice versa.
“Musicians and researchers have known about this robust effect of geometry for some time, but the underlying mechanisms have remained a mystery,” said Suraj Shankar, a Harvard Junior Fellow in physics and SEAS and co-first author of the study. “We have found a mathematical argument that explains how and why this robust effect exists with any shape of this class, so that the details of the shape are unimportant, and the only fact that matters is that there is a reversal of curvature along the saw.”
Shankar, Bryde and Mahadevan found this explanation via an analogy with a very different class of physical systems – topological insulators. Most often associated with quantum physics, topological insulators are materials that conduct electricity on their surface or edge but not in the middle, and no matter how you cut these materials, they will still conduct on their edges.
“In this work, we have established a mathematical analogy between the acoustics of bent sheets and these quantum and electronic systems”, Shankar said.
Using the mathematics of topological systems, the researchers found that the localized vibrational modes in the singing saw sweet spot were governed by a topological parameter which can be calculated and which relies on nothing more than the existence of two curves opposites in the material. . The sweet spot then behaves like an internal “edge” in the saw.
“Using experiments, theoretical and numerical analyses, we have shown that S-curvature in a thin shell can locate topologically protected modes at the ‘sweet point’ or inflection line, similar to exotic edge states in topological insulators”, Bride said. “This phenomenon is material-independent, meaning it will show up in steel, glass, or even graphene.”
The researchers also found that they could tune the mode localization by changing the shape of the S-curve, which is important in applications like sensing, where you need a resonator tuned to very specific frequencies. .
Futuristic vision of the study:
“Our work suggests an alternative strategy inspired by the singing saw, which relies solely on the scale separation intrinsic to any curved thin sheet; by manipulating the curvature in space, the topological modes localized within therefore remain vibrationally isolated and decay extremely slowly, allowing very high quality oscillations, perhaps even in the ultimate limit of atomically thin graphene. Study Quotes.
- Suraj Shankar, Petur Bryde and L. Mahadevan, Geometric Control of Topological Dynamics in a Singing Saw. Proceedings of the National Academy of Sciences. 119 (17) e2117241119 DOI: 10.1073/pnas.2117241119