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Photo source: the sound of "sawing wood" from upstairs Unsplash is always strange in tone, but persistent and determined.

If there is a neighbor upstairs who suddenly starts to learn musical instruments, I think most people hope that he does not learn erhu and violin. After all, the sounds made by these two instruments at the beginning stage are not wonderful, not to mention that some people always take a long time. Both the erhu and the violin make sound by pulling the bow to rub the strings and make it vibrate, which is very similar to the process of pulling and rubbing with a hacksaw to cut through wood. Therefore, we often use "sawing wood" to describe the ungraceful sounds of the two instruments.

Photo: Pixabay, but in fact, the hacksaw itself is also a musical instrument, with a bow, this piece of steel that gives us countless nightmares will be transformed into a singing saw.

At first glance, the sawing piano plays in a way similar to that of the erhu, requiring the bottom of the instrument to be placed on the leg. The sawing player can hold the handle of the saw with his knees, then control the pitch with one hand and pull the bow with the other to make a sound. Unlike the sound made by a hacksaw when rubbing against other objects, the sound of the sawing piano is long and ethereal, and it sounds a bit like a female voice. However, what makes the sawing piano special is not just the material and timbre.

Sound is produced by the vibration of an object, and the pitch depends on the frequency of the vibration. And almost all the musical instruments we are familiar with control the pitch by adjusting the length of the sound-producing device so as to adjust the vibration frequency. For example, stringed instruments such as zither, violin and guitar control pitch by adjusting the length of strings; wind instruments such as flute and clarinet adjust the length of the air column by pressing keys; and some percussion instruments with fixed pitch, such as chimes, determine the pitch by the size of the bell.

However, the sawing piano has neither strings nor keys, let alone a change in volume. So how do you control its tone? This thin piece of metal adjusts its tone by bending the body of the piano to change its shape. If you look at the saw piano from the side, it is obvious that it is a straight line. When the line is geometrically deformed and bends into an S shape, there is an inflection point, which is called "sweet spot" by musicians. Striking at a sweet spot or playing with a bow will make the sawing piano sound clear, lasting for a long time and with a specific pitch. By adjusting the shape of S, the tone of the sawing instrument can be changed. This peculiar acoustic property has aroused the intense curiosity of scientists. In a study published this year in the Proceedings of the National Academy of Sciences (PNAS), researchers from the Department of Physics at Harvard University revealed the unique mathematical physics behind the sound of the sawing piano.

How to make a wonderful sound? When we strike an elastic continuous medium, such as a piece of metal, the vibration starts from the hitting point, quickly spreads to the whole medium and vibrates together, and then dissipates quickly. This process is very much like dropping a stone in the water. When the stone touches the surface of the water, the vibration of the water will spread into the distance with the ripples until it disappears. So when the hacksaw is laid flat or bent into a J shape, striking it or playing with a bow will cause the whole piece of steel to vibrate together, with a dull and short sound.

Only when the sawing piano is bent into an S shape will there be a sweet spot (sweet spot). The amazing thing is that when the saw is bent into an S shape, everything is different, and the vibration at the sweet spot is long-lasting and stable. This means that the vibration does not spread to the whole hacksaw as before, but only occurs in the sweet spot. The vibration pattern of the saw piano at this time is like, after throwing the stone into the water, the whole surface of the water only has ripples near the point of entry of the stone, and the ripple does not spread around, so strange.

Through experiments, the researchers found that if you play or knock away from the sweet spot, or change the shape to J-shape while playing, the sound of the sawing piano will disappear quickly. No matter what the geometric shape of the saw piano is, it doesn't matter whether the top curvature is larger or smaller, as long as it is S-shaped and can have sweet spots with curvature reversal, a stable and lasting local vibration mode can occur.

We know that when the curvature of a noodle increases, so does its structural strength. for example, if we want to elegantly eat the tip of a pizza, we need to bend it along the central axis so that the pizza tip does not fall softly. On the other hand, for the sawing piano bent into S, the curvature of the upper and lower bends will play a similar structural enhancement role. Due to the curvature reversal, the curvature near the sweet spot is basically zero, which means that it is flat and soft, and it is more likely to vibrate. Once away from the sweet spot, it will enter the part that is affected by curvature and is not easy to vibrate. As a result, the vibration occurs only in the part where the curvature is zero and does not spread to the curved part, so as to avoid dissipation and attenuation along the edge, lose most of the energy, and therefore the vibration can be more lasting.

The local vibration mode of a high-quality resonator saw looks very much like a state protected by the surrounding bending structure. If described mathematically, the local vibration depends on the space of varying curvature, and the topological protection is obtained at the inflection point of the curvature of the space, that is, the edge of curvature. The fact that the vibration is not sensitive to the specific shape of S also verifies that the local vibration is protected by topology. All this reminds researchers of a special quantum material with internal insulation and conductive edges, topological insulators (topological insulators). No matter how the topology insulator is divided, or there are defects, impurities or even external stimuli, it will not affect its internal insulation and edge conductivity, which is the result of topology protection.

The sweet spot with zero curvature in the saw piano is essentially similar to the topologically protected edge state in the topological insulator, which acts as the edge of the upper and lower bending parts of the S-shaped steel sheet, except that this strange edge appears on the inside of the steel sheet. This reveals the geometric characteristics of the local vibration mode of the saw piano, which has nothing to do with the vibration source, the size of the vibration body and the material. Therefore, it can also be applied to any other elastic continuum material, such as graphene with atomic thickness.

Image source: due to the topology protection of Unsplash, the local vibration mode of the saw can continue to vibrate for a long time when there is no continuous energy source, and the energy will almost only be dissipated into the surrounding environment. Therefore, this geometric feature can also be applied to the design of high-quality oscillators. Some micromachined oscillators that oscillate hundreds of millions of times per second are often used as timers. Usually, we do not want it to consume too much energy, but also want its oscillation frequency to remain as constant as possible, so it is particularly important to isolate and protect the oscillation. This study shows that instead of constantly experimenting with various materials, it is better to achieve higher-quality oscillations by adjusting geometric shapes.

Reference link:

Https://www.pnas.org/doi/10.1073/pnas.2117241119

Https://physicstoday.scitation.org/do/10.1063/PT.6.1.20220505a/full/

Https://www.pnas.org/post/podcast/saw-sings

Https://www.world-today-news.com/unique-mathematical-physics-for-singing-saws/

This article comes from the official account of Wechat: global Science (ID:huanqiukexue), written by: bad, revised: chestnut, Erqi

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