# Complete modes of star-shaped oscillating drops

## TL;DR

This paper provides a comprehensive analysis of star-shaped oscillations in water drops, combining surface and azimuthal modes to improve frequency prediction accuracy.

## Contribution

It introduces a complete model coupling surface and azimuthal modes, advancing understanding of star-shaped drop oscillations beyond previous quasi-2D approaches.

## Key findings

- Coupling of surface and azimuthal modes explains star-shaped oscillations.
- New dispersion relation improves frequency prediction accuracy.
- Surface motion patterns are driven by parametric instability.

## Abstract

The star-shaped oscillation of water drops has been observed in various physical situations with different sources of vertical excitation. Previous studies apply a quasi-2D model to analyze the resonance frequency and only consider the azimuthal oscillation mode. In this paper, we find that the upper surface of water drops also develop different motion patterns due to parametric instability and it is the coupling of surface motion and azimuthal oscillation that leads to star-shaped oscillations. We will introduce the analysis of the surface mode, combining with the azimuthal mode to give a complete description of the motion of water drops. We propose a new dispersion relation based on the complete description, which provides a significant increase of accuracy in predicting the oscillating frequencies.