• The physics of a singing saw

    From ScienceDaily@1:317/3 to All on Friday, April 22, 2022 22:30:48
    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

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