The physics of a singing saw
Date:
April 22, 2022
Source:
Harvard John A. Paulson School of Engineering and Applied Sciences
Summary:
Researchers have used the singing saw to demonstrate how the
geometry of a curved sheet, like curved metal, could be tuned to
create high-quality, long-lasting oscillations for applications
in sensing, nanoelectronics, photonics and more.
FULL STORY ==========================================================================
The eerie, ethereal sound of the singing saw has been a part of folk
music traditions around the globe, from China to Appalachia, since the proliferation of cheap, flexible steel in the early 19th century. Made
from bending a metal hand saw and bowing it like a cello, the instrument reached its heyday on the vaudeville stages of the early 20th century
and has seen a resurgence thanks, in part, to social media.
==========================================================================
As it turns out, 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 tuned to create high-quality, long-lasting oscillations for applications in sensing, nanoelectronics, photonics and more.
"Our research offers a robust principle to design 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, the Lola England de Valpine Professor of
Applied Mathematics, of 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 acoustic resonators of a kind, none
work quite like the singing saw.
==========================================================================
"How the singing saw sings is based on a surprising effect," said Petur
Bryde, a graduate student at SEAS and co-first author of the paper. "When
you strike a flat elastic sheet, such as a sheet of metal, the entire
structure vibrates.
The energy is quickly lost through the boundary where it is held,
resulting in a dull sound that dissipates quickly. The same result is
observed if you curve it into a J-shape. But, if you bend the sheet into
an S-shape, you can make it vibrate in a very small area, which produces
a clear, long-lasting tone." The geometry of the curved saw creates
what musicians call the sweet spot and what physicists call localized vibrational modes -- a confined area on the sheet which resonates without losing energy at the edges.
Importantly, the specific geometry of the S-curve doesn't matter. It could
be an S with a big curve at the top and a small curve at the bottom or
visa 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 found a mathematical argument that
explains how and why this robust effect exists with any shape within
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 that explanation via an
analogy to very different class of physical systems -- topological
insulators. Most often associated with quantum physics, topological
insulators are materials that conduct electricity in their surface or
edge but not in the middle and no matter how you cut these materials,
they will always conduct on their edges.
==========================================================================
"In this work, we drew a mathematical analogy between the acoustics of
bent sheets and these quantum and electronic systems," said Shankar.
By using the mathematics of topological systems, the researchers found
that the localized vibrational modes in the sweet spot of singing saw
were governed by a topological parameter that can be computed and which
relies on nothing more than the existence of two opposite curves in the material. The sweet spot then behaves like an internal "edge" in the saw.
"By using experiments, theoretical and numerical analysis, we showed that
the S-curvature in a thin shell can localize topologically-protected modes
at the 'sweet spot' or inflection line, similar to exotic edge states
in topological insulators," said Bryde. "This phenomenon is material independent, meaning it will appear in steel, glass or even graphene."
The researchers also found that they could tune the localization of
the mode by changing the shape of the S-curve, which is important in applications such as sensing, where you need a resonator that is tuned
to very specific frequencies.
Next, the researchers aim to explore localized modes in doubly curved structures, such as bells and other shapes.
The research was supported in part by National Science Foundation under
Grant No. NSF PHY-1748958, DMR 2011754 and DMR 1922321.
========================================================================== Story Source: Materials provided by Harvard_John_A._Paulson_School_of_Engineering_and_Applied
Sciences. Original written by Leah Burrows. Note: Content may be edited
for style and length.
========================================================================== Related Multimedia:
* Saw_clamped_in_two_configurations ========================================================================== Journal Reference:
1. Suraj Shankar, Petur Bryde, L. Mahadevan. Geometric control of
topological dynamics in a singing saw. Proceedings of the National
Academy of Sciences, 2022; 119 (17) DOI: 10.1073/pnas.2117241119 ==========================================================================
Link to news story:
https://www.sciencedaily.com/releases/2022/04/220422114732.htm
--- up 7 weeks, 4 days, 10 hours, 51 minutes
* Origin: -=> Castle Rock BBS <=- Now Husky HPT Powered! (1:317/3)