Similarities of the Chladni Patterns Observed at Low Frequencies on Square and Circular Plates: Experimental Evidence and Theoretical Justification

By using a combination of experimental and theoretical approaches, this study aims to provide new insights into the similarities of Chaldni patterns on square and circular plates, and to fill the gap in the existing body of knowledge by exploring the plate boundary shape on the formation of nodal lines. Patterns like parallel segments, hyperbolic and elliptical arcs, usually arising on the square plates, were unexpectedly visualized on the circular plate. Such similarities are explained by admitting solutions with unseparated variables to solve the wave equation. This approach is a departure from the standard textbook path, by challenging the traditional assumption that circular plates only produce concentric circles and diametral segments. By building a bridge between the classical Bessel function theory, the concept of modal density used in acoustics and structural dynamics, and the current experimental techniques, a more flexible framework to predict the plate response is provided, and this is particularly useful for the field of micro-particle manipulation where precise pattern control is required. Mathematical approach proposed here might be successfully used to solve the Helmholtz equation attached to many other phenomena in the field of physics, seismology, electromagnetics and acoustics.