## Journal Highlights Math Research by Professor and Student

By Rebecca Goldfine*Physical Review Research*in its April issue tagged a paper by Associate Professor of Mathematics Chris Chong and Evelyn Wallace ’22 as an "editor's suggestion" and featured it on its homepage.

The , "Modulation instability and wavenumber bandgap breathers in a time layered phononic lattice," is the result of a yearslong collaboration of Chong, Wallace, and two researchers at CalTech.

Their project focused on how pressure waves can be manipulated by materials. It began in 2021 with a theoretical component—which Wallace worked on for her honors theses, ".” She is now wrapping up a master's degree in applied mathematics at Columbia University.

Wallace said she reached out to Chong in 2021 to inquire whether she could assist him with a research project. She had excelled in his ordinary differential equations and partial differential equations classes, Chong said, and he was just launching a new research program. So he invited her to work as his research assistant in the summer, a partnership that extended through her senior year.

The work Wallace completed in the first stage of the research program was followed by numerical and experimental simulations, conducted by Chong and his Caltech teammates, which was then backed up with rigorous mathematical proofs in a follow-up paper. “It is a nice spectrum that goes from Evy's analytical computations to computer simulations, the laboratory, and to mathematical proof,” Chong said.

## New Insights

Their April 11 paper is garnering attention because it opens new ground in the field of physical energy. Their project specifically focused on understanding how pressure waves are affected by media that varies over time. (Pressure waves, of which the most commonly known is a sound wave, are set off when a mass or object is compressed or decompressed by an external force, such as an explosion.)

Traditionally, researchers have experimented with pressure waves using media that vary in space while they remain constant in time.

For those who are not familiar with his field, Chong came up with a helpful analogy to explain his research, comparing the time-varying media he and Wallace studied to a floor, and the waves passing over the media to a person walking across that floor.

“A traditional floor might have a pattern of alternating spots of rough and smooth areas,” he began. “As you’re walking across the floor, how your feet touch the ground will affect how you walk along it.”

In contrast, a medium that changes over time resembles a wacky floor in an amusement park fun house. Rather than having both smooth and rough patches, it is entirely smooth one moment before shifting to a rough surface, switching back and forth between these two patterns. “How you walk through that medium will be different than walking across a traditional floor,” Chong said.

Wallace's original calculations showed exactly where this system would be unstable, which is desirable. When a system is stable, “it means the amplitude of the wave doesn't grow,” Chong said, whereas an unstable system results in a wave that amplifies quite a bit.

“What Evy did was to tell us exactly what material parameters would be needed for that to happen, an exact formula for when and under what circumstances you get interesting instability,” he said. “Having a formula that exact is quite rare. Usually you have to approximate it or do some numerical simulations.”

By the way, the materials they studied aren't linoleum or wood panels or anything most of us would recognize as typical flooring materials.

“Our experiment consists of magnets,” Chong said. They looked at variables such as the distance and strength of magnets, as well as the effect of dampening from air friction. “Each of those properties has a parameter that you can tune, so having a formula saying this combination of parameters leads to amplification is very helpful.”

What is of particular interest to researchers working in this area is how to maintain the force of a wave or signal, because over time, all waves degrade. Each electrical circuit, for example, needs an amplifier to work, Chong said.

“So when talking about the applications of our work, our system would act as an amplifier,” he added. “Based on what Evy did, you would know exactly how to make the material so that the wave would amplify.”

## Possible applications

Though Chong and his team are working on a basic research level, in the long term, understanding the systems they are studying could lead to innovations in “acoustic logic,” Chong explained, which is like the logic underlying electrical circuits, but which don't use electricity but rather pressure waves. They could even help create a situation where “acoustic circuits” could operate with the use of ambient vibrations only, and thus avoid detection from sensors.

“We’re still doing work that is more fundamental,” Chong said. “But understanding how these systems behave could contribute to that the much-longer project of having a goal like that.”

Wallace's undergraduate work with Chong led to a research collaboration with a Columbia faculty member. With him, Wallace expanded Chong's one-dimensional study to two dimensions.

Starting this type of high-level research at 50業子, Wallace said, “synthesized everything I had learned at 50業子 and learned about math in general.”

She continued, “Doing research allowed me to take what I learned out of the classroom and apply it to something I could see in the the real world, and see how it would be very important and relevant.”

The journal, which is open access, and 50業子 College provided support to cover the publication fees via the Fletcher Family Research Grant. Wallace's research was supported by a National Science Foundation grant and the 50業子 Kaufman Family Fellowship.