# Localized Faraday patterns under heterogeneous parametric excitation

## TL;DR

This paper investigates how localized and heterogeneous parametric excitation influences Faraday wave patterns, revealing the formation of localized subharmonic patterns and providing a theoretical model that aligns well with experimental results.

## Contribution

It introduces a combined experimental and theoretical study of Faraday waves under localized forcing, extending understanding beyond uniform excitation scenarios.

## Key findings

- Localized forcing induces subharmonic wave patterns.
- Theoretical model predicts pattern evolution accurately.
- Onset of instability is altered by heterogeneity.

## Abstract

Faraday waves are a classic example of a system in which an extended pattern emerges under spatially uniform forcing. Motivated by systems in which uniform excitation is not plausible, we study both experimentally and theoretically the effect of heterogeneous forcing on Faraday waves. Our experiments show that vibrations restricted to finite regions lead to the formation of localized subharmonic wave patterns and change the onset of the instability. The prototype model used for the theoretical calculations is the parametrically driven and damped nonlinear Schr\"odinger equation, which is known to describe well Faraday-instability regimes. For an energy injection with a Gaussian spatial profile, we show that the evolution of the envelope of the wave pattern can be reduced to a Weber-equation eigenvalue problem. Our theoretical results provide very good predictions of our experimental observations provided that the decay length scale of the Gaussian profile is much larger than the pattern wavelength.