The physics of a singing saw

The researchers clamped the saw in two configurations: a J-shape (left) and an S-shape (right). The S-shape has an inflection point (the sweet spot) in its profile, while the J-shape does not. Credit: Mahadevan Lab/Harvard SEAS

The eerie, ethereal sound of the singing saw has been a part of folk music traditions around the world, from China to Appalachian Mountains, since the spread of cheap, flexible steel in the early 1800s. Made from bending a metal hand saw and bowing like a cello, the instrument reached its heyday on the vaudeville stages of the early 20th century and has enjoyed a resurgence thanks in part to social media.

As it turns out, the unique mathematical physics of the singing saw could be the key to designing high-quality resonators for a range of applications.

In a new publication, a team of researchers from Harvard’s 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 bent sheet of metal, similar to bent metal, can be could be tuned to produce high-quality, long-lasting oscillations for applications in sensing, nanoelectronics, photonics, and more.

“Our research provides a robust principle to design high quality resonators regardless of size and material, from macroscopic musical instruments to nanoscale devices, simply through a combination of geometry and topology,” said L Mahadevan, Lola England de Valpine Professor of Applied Sciences Mathematics , Organismic and Evolutionary Biology, and of Physics and senior author of the study.

The research is published in The Proceedings of the National Academy of Sciences (PNAS).

While all musical instruments are some form of acoustic resonator, none work quite like the musical saw.






“The way the musical saw sings is based on a surprising effect,” said Petur Bryde, a PhD student at SEAS and co-first author of the study. “If you hit a flat elastic plate, such as a sheet of metal, the entire structure vibrates. The energy is quickly dissipated by the boundary it is held at, resulting in a dull sound that quickly dissipates. The same result is observed if you curve it into a J shape. But if you bend the reed into an S-shape, you can make it vibrate in a very small area, producing a clear, long-lasting tone.”

The geometry of the curved saw creates what musicians refer to as the sweet spot and physicists refer to as localized modes of vibration – a confined area on the sheet metal that resonates without losing energy at the edges.

Importantly, the specific geometry of the S-curve is irrelevant. 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, Harvard Junior Fellow in Physics and SEAS and co-first author of the study. “We found a mathematical argument that explains how and why this robust effect exists for every shape within this class, so the details of the shape are unimportant and the only fact that matters is that there is a reversal of curvature along the saw there. ”

Shankar, Bryde, and Mahadevan found this explanation through an analogy with an entirely different class of physical systems—topological insulators. Most commonly associated with quantum physics, topological insulators are materials that conduct electricity at their surface or edge but not at the center, and no matter how you cut these materials, they will always conduct at their edges.






“In this work, we drew a mathematical analogy between the acoustics of bent sheets of metal and these quantum and electronic systems,” Shankar said.

By applying the mathematics of topological systems, the researchers found that the localized vibrational modes in the musical saw’s sweet spot are governed by a topological parameter that can be calculated and relies on nothing more than the existence of two opposing curves in the material. The sweet spot then behaves like an inner “edge” in the saw.

“Through experiment, theoretical and numerical analysis, we have shown that the S-curvature in a thin shell can localize topologically protected modes at the ‘sweet spot’ or the turning line, similar to exotic edge states in topological insulators,” said Bryde. “This phenomenon is material-independent, meaning it occurs in steel, glass or even graphene.”

The researchers also found they could tune the localization of the mode 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.

Next, the researchers want to study localized modes in doubly curved structures such as bells and other shapes.


The mathematical framework turns any sheet of material into any shape with kirigami cuts


More information:

Suraj Shankar et al, Geometric Control of Topological Dynamics in a Musical Saw, Proceedings of the National Academy of Sciences (2022). DOI: 10.1073/pnas.2117241119

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The physics of a singing saw (22.4.2022)
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