An oscillating shower head lets you discover a new mode of vibration in nature

During the hot summer of 2020, confined to his Pasadena home during the COVID-19 pandemic, National Medal of Science-winning applied physicist Amnon Yariv took frequent, long showers to cool off. One surprising result, adding to his record-breaking water bill, was a proposal and theoretical model for a new class of vibrations that could convert a constant force, such as wind or water, into a mechanical oscillation.

It wasn’t exactly a eureka moment. The idea didn’t come to him in the bathtub, Archimedes-style. But it was in the shower that Yariv first noticed something unusual about the way the jet showerhead behaved when left hanging by its hose. To Yariv, a scientist who has studied waves and their properties for more than 70 years, the showerhead was more than just a fixture attached to a flexible cable that sprayed water onto the wall. It was part of an oscillating system.

Oscillations are rhythmic or periodic variations in the world around us. The ebb and flow of the sea are oscillations. The vibrations of a plucked guitar string are oscillations. Even light is an oscillation, according to quantum theory.

Yariv observed that as he increased the flow rate in the shower, the system began to behave in unexpected ways. In fact, he saw a joint bimodal oscillation—two different oscillations synchronized with each other. As the showerhead swung back and forth like a pendulum, it also rotated in synchrony in one direction and then the other. It was clear that these two oscillatory modes were moving toward each other, because damping one would immediately cause the other mode to stop oscillating.

Moreover, Yariv found that the joint’s oscillation was predictably unstable. Once a certain threshold of water pressure was reached, the amplitude of the oscillation continued to increase even when the water flow remained constant.

“This bimodal oscillation is like an Argentine tango, where each dancer has to stay perfectly synchronized with the other or they crash into each other,” says Yariv, the Martin and Eileen Summerfield Professor of Applied Physics and Electrical Engineering. “The idea that a constant force could be used to excite this kind of entangled bimodal oscillation has never been proposed or demonstrated.”

Yariv spent the next few years developing the mathematical model that explained what he had observed during those hot summer months. The mathematics underlying Yariv’s two-mode model, which he dubbed “the Yariv groove,” is a radical two-mode extension of a single-mode oscillation model proposed by physicists Michael Faraday and Lord Rayleigh a century and a half ago.

Yariv’s new paradigm could have implications for fields ranging from civil engineering to quantum electronics. For example, one possibility is harnessing the essentially limitless energy of wind. However, Yariv cautions that because the bimodal oscillations described in the new work become increasingly intense once the driving force (in this case, wind) reaches a threshold, some way of controlling this instability would need to be found.

This instability also highlights factors that should be taken into account when constructing structures such as buildings and bridges, to prevent this type of oscillation from becoming uncontrollable and damaging or completely destroying them.

Oscillations, an introduction

To understand Yariv’s new paradigm, it is helpful to first consider a classic example of a swing mode, such as a child on a swing. Described in physics terminology, the child and the swing are a system. If the child is young, another person must push him or her periodically, usually once per swing or once every few swings. Otherwise, the child will eventually stop swinging because of friction.

“It turns out that almost all oscillations in nature are driven by a periodic force,” Yariv says.

As the child grows, he can learn to maintain the motion without the parent pushing him. He does this by pumping his legs forward and back or, if he is standing, by pulling his weight up and down on the swing chains. In both cases, the child periodically modulates or varies a parameter of the system (the moment of inertia in the first case and the child’s weight as felt by the swing seat in the second). The modulation rate in both cases is twice the swing frequency.

Such “parametric” pumping of oscillations was observed by Michael Faraday and mathematically explained some 50 years later, in 1883, by Lord Rayleigh.

“Lord Rayleigh’s work laid the foundations of parametric physics, which, in the field of nonlinear optics, has become one of the most exciting branches of modern physics and a major activity in the departments of applied physics and physics at Caltech over the last 50 years,” Yariv says.

Back to the shower

Yariv hypothesized that this parametric oscillation was behind the bimodal oscillation he had observed in the behavior of the hanging showerhead. But here, rather than an external agent modulating a parameter (mass, gravitational constant, or moment of inertia) at twice the resonant frequency, Yariv saw that it was the intricate collaboration of two modes of oscillation due to a nonlinearity in the system that powered the oscillation with a constant, unchanging force.

“We have a cycle of excitement here,” Yariv says.

Once the shower head begins its twist, or torsional oscillation, the constant force of the water pushing perpendicularly toward the face of the shower head generates the periodic force that drives the pendulum to excursion in the conventional way, once per cycle. This pendulum motion in turn nonlinearly modulates the torsion spring constant (here, the parameter, twice per cycle), thereby internally generating what is called the second harmonic, necessary according to Lord Rayleigh, to drive the torsional oscillation, thus completing the cycle.

“My study only follows the system from the beginning of the bimodal oscillation to the initial stage of the unstable oscillation, and stops before the heavy showerhead crashes the wall,” Yariv says. “But the new entangled bimodal oscillation is unstable. It doesn’t reach a steady state. It keeps getting bigger.”

Implications far beyond the shower

The oscillation that Yariv first noticed in the shower serves as a model for a whole class of oscillations, and his mathematical analysis should apply to all members of that class, he says. Yariv cites other examples that most likely belong in that class. A stop sign fluttering in windy weather is one example. A more infamous example, he says, is most likely the collapse in 1940 of a suspension bridge that spanned the Tacoma Sound in Washington.

The bridge, nicknamed Galloping Gertie because of the way it bucked and swayed in the wind, eventually collapsed during a storm in 1940. Videos of the bridge collapse show the roadway undergoing vertical and transverse oscillations leading to failure. Some of the key characteristics of showerhead oscillation exist in both cases: bimodality, stable force threshold, and instability.

“The abundance of stable forces in nature will provide rich areas of investigation to identify bimodal oscillations that can be excited by these forces,” Yariv says.

He notes that the mechanical motion generated by such a bimodal oscillation could also be converted into rotational motion and, therefore, into electricity generation. This discovery could also have applications in optics, electronics and cosmology, where Yariv says bimodal oscillations could be related to the dance before the merger between colliding black holes.

The paper describing Yariv’s theory, “On a Class of Bimodal Oscillations Driven by a Constant Force at Zero Frequency – Implications for Energy Conversion and Structural Stability,” appeared in the September 11, 2023, issue of the journal Proceedings of the National Academy of SciencesA commentary on the work by Demetrios Christodoulides of USC appeared in the October 4, 2023, issue of the same journal.