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 [...]

Exploration of Resonant Modes for Circular and Polygonal Chladni Plates

“…In addition, the boundary conditions and restrictions will result in mode mixing, resulting in further deviations from the eigenmodes. The resonant modes have successfully been determined by solving the inhomogeneous Helmholtz equation to find the response function in terms of the driving wave number, where the wave number is theoretically determined from the maximal entropy states, as determined from the standard Shannon equation for entropy [ 21 , 24 , 25 , 26 , 27 , 28 ]. However, in the study presented here, we determine the maximal entropy states as the points of equilibrium, defined in terms of the wave number and the response function, where the response function is defined in terms of the geometry of the spatial boundaries.…”

“…The nodal points, where the sand collects, correspond to the points of zero displacement, also described as equilibrium points or maximum entropy states, e.g., see ref. [ 28 ]. For each mode, these points can be found by setting z r in Equation (3) to equal 0, i.e., we can then assume that, giving, and, thus, the time it takes for the wave to reach a given point is given as, If the wave is reflected and oscillates at the resonant frequency of the system, then there is no phase difference between the waves; therefore, ϕ = 0, and a standing wave is formed.…”

“…The nodal points, where the sand collects, correspond to the points of zero displacement, also described as equilibrium points or maximum entropy states, e.g., see ref. [28]. For each mode, these points can be found by setting z r in Equation (3) to equal 0, i.e.,…”

“…For example, utilising Green’s function, Tuan et al [ 25 ] solved the inhomogeneous Helmholtz equation and found the response function for a vibrating wave on a thin plate as a function of the driving wave number, where the wave numbers are determined from the maximum entropy states. Thus, by substituting the theoretically determined wave numbers into the derived response function, the resonant modes can be calculated, and the experimental nodal line patterns can be successfully reconstructed for both the square and equilateral triangle plates [ 21 , 24 , 25 , 26 , 27 , 28 ]. Other approaches have also been successful; for example, instead of using the usual numerical approaches, Amore [ 29 ] found solutions to the Helmholtz equation (both homogeneous and inhomogeneous) by using the little sinc method [ 29 , 30 , 31 ].…”

Visualization of Chladni Patterns at Low-Frequency Resonant and Non-Resonant Flexural Modes of Vibration

“…Chladni patterns are the geometrical figures obtained by sprinkling particles on the surface of a vibrating plate [1], [2]. Using violin bows [1], [3], shakers and speakers [4]- [7], laser beams [8], etc., a stationary flexural vibration mode is to be generated into the horizontally supported plate.…”