Exploring the Origin of Maximum Entropy States Relevant to Resonant Modes in Modern Chladni Plates

Abstract

The resonant modes generated from the modern Chladni experiment are systematically confirmed to intimately correspond to the maximum entropy states obtained from the inhomogeneous Helmholtz equation for the square and equilateral triangle plates. To investigate the origin of maximum entropy states, the inhomogeneous Helmholtz equation is modified to consider the point interaction coming from the driving oscillator. The coupling strength associated with the point interaction is characterized by a dimensionless factor α. The δ potential of the point interaction is numerically modelled by a truncated basis with an upper index N. The asymptotic behavior for the upper index N is thoroughly explored to verify that the coupling strength of α = 1.0 can make the theoretical resonant modes agree excellently with the maximum entropy states as N→∞. It is further authenticated that nearly the same resonant modes can be obtained by using a larger coupling strength α when a smaller upper index N is exploited in the calculation.

Keywords

• maximum entropy
• pattern formation
• point interaction
• modern Chladni plates