Cymatics Nature
Paul
From ancient Greek: κῦμα, meaning "wave", Cymatics is a subset of modal vibrational phenomena. The study of visible sound and vibration. The term was coined by Hans Jenny (1904-1972) a physician and natural scientist. In 'Cymatics: The Study of Wave Phenomena' he concluded, "This is not an unregulated chaos; it is a dynamic but ordered pattern."
Hans Jenny studied visual sound intensively. A physician, fine artist, pianist, philosopher, historian, and empirical researcher. He is nicknamed the "father of cymatics," Between 1958 and his death in 1972, Jenny conducted a wide range of experiments documenting the effects of sound and energy on various media.
Mary D. Waller became Professor of Physics at the Royal Free Hospital Medical School in London. She became fascinated by Chladni's work and recreated all the forms he discovered, taking his work to a higher level.
She approached the subject of Chladni Figures with scientific rigor and her work represents a rich resource for students of this branch of acoustics, including some of the mathematical equations that describe the phenomena.
When the vibration of a transducer (speaker) for example vibrates a circular dish of water, it is transforming into a type of transverse wave. As the surface waves on the water are moving perpendicular (right angled) to the direction of energy transfer (the propagation of the original sound wave).
As the wave radiates from the centre of the waters surface as a surface wave, it encounters a boundary. (The walls of the circular dish) the energy is then reflected back on itself. This creates nodes and antinodes (behaviour of the waves at the points of minimum and maximum vibrations, in this case, points of amplified peaks and troughs)
Here below is an example of Chladni figures, and the resonant modes of a violin body, when vibrated to a range of frequencies. The violin body as the medium is held constant, and the sweeping range of frequencies (one by one, from low to high) vibrate the body. There are certain frequencies that will naturally excite it more than others, where standing waves, and thereby chladni figures arise.
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Based on rigorous theoretical findings, we present a proof-of-concept design for a structured square cloak enclosing a void in an elastic lattice. We implement high-precision fabrication and experimental testing of an elastic invisibility cloak for flexural waves in a mechanical lattice. This is accompanied by verifications and numerical modelling performed through finite element simulations. The primary advantage of our square lattice cloak, over other designs, is the straightforward implementation and the ease of construction. The elastic lattice cloak, implemented experimentally, shows high efficiency.
In this paper we present a novel practical design and an experimental implementation of an approximate cloak in a structured flexural plate. It is based on the rigorous theoretical findings of Colquitt et al.1,2. In particular, for the lattice implementation, the cloaking transformation is not singular and hence does not require unrealistic infinite wave speed at the interior contour of the cloak. Since the first experimental demonstration of the microwave invisibility cloak3, there has been an explosion of theoretical and practical advances in the design and analysis of electromagnetic metamaterials4. In contrast, the significantly more challenging problem of creating invisibility cloaks and metamaterials for elastodynamics has been much less studied. Notable recent advances in the theoretical analysis of cloaks for elastic waves have been made by Milton et al.5,6, Norris & Shuvalov7, Brun et al.8,9, Farhat et al.10, Jones et al.11, Colquitt et al.1,2, Guenneau et al.12 and Parnell et al.13,14,15,16. These theoretical developments have been complemented with a series of experimental implementations of multi-scale mechanical cloaks performed by the group led by Wegener17,18,19.
The design of the cloak presented here is distinct from the cloaks used by Wegener et al.17 and Chen et al.26, in so far as the cloak designed and constructed in the present paper is matched with a multi-scale mechanical structure, as opposed to a homogeneous continuous ambient matrix. Moreover, the cloak which we use is not singular, i.e. it does not require infinite wave speeds on the interior boundary; it is based on the regularised cloaking transformation and theoretical design by Colquitt et al.1 for membrane waves and later for flexural waves in plates2. We emphasise that the analysis and theoretical design presented in2 is two-dimensional; in contrast, the cloak design and numerical implementation described here is fully three-dimensional and corresponds precisely with the experimental parameters used, including the presence of non-negligible dissipative effects.
In the subfigures the dashed/black lines show the positions of the nodal lines of the vibrating plate. In the vicinity of the shaking clamp there cannot be any nodal lines because the edge is vibrating. Any powder adjacent to the shaking clamp does not represent a physical nodal line, and it is due to the clamp restricting the movement of the powder.
It is emphasised that here we consider the full elastodynamical problem of wave propagation in a discrete metamaterial lattice cloak embedded within a multi-scale ambient medium; this should be distinguished from the earlier work of Wegener et al. on mechanical lattice cloaks18,19 wherein the static cloaking problem is considered and no waves propagate within the system.
The structure of the paper is as follows. We first advocate the idea of the Hooke-Chladni-Faraday visualisation, which is extremely efficient for the case of elastic structures such as flexural plates. We then present the proof of concept for the square cloak. This includes computational and experimental implementations, followed by a discussion of the experimental design and main results. We also discuss the concept of the regularised cloaking transformations, and the lattice approximation. Finally, we draw together important concluding remarks.
For the purpose of illustration, we include in Fig. 2 examples of the Chladni patterns for eigenmodes of a square elastic plate with a free boundary, which are accurate and fully consistent with analytical findings. The patterns depend on the boundary conditions and on any inhomogeneities that may occur in the plate, such as voids or inclusions. In particular, Chladni patterns were never constructed for a plate with a hole surrounded by a structured cloak.
In the present paper, we visualise Hooke-Chladni-Faraday patterns for flexural waves around an obstacle surrounded by a multi-scale structured cloak and elegantly illustrate the efficacy of the cloak. An ABAQUS simulation for the mechanical configuration, identical to the one used in the experiment, provides accurate numerical data on the displacement amplitudes and stress distribution. We present the experimental visualisation to confirm the predicted wavefront profiles.
Compared to the classical settings for the frequency response problem for a rectangular Kirchhoff-Love plate mentioned above, we go further and consider a structured plate with a cloaked hole. Figure 1 includes Hooke-Chladni-Faraday patterns used for visualisation of the cloaking effect for flexural waves. These observations are new and demonstrate scattering patterns for three configurations, which include (a) a rectangular lattice-type plate, (b) a plate with a square hole, and (c) a plate with a structured cloak enclosing the hole. As in the original experiments by Hooke, the powder collects along the nodal lines of the vibrating plate thus indicating the position of the wavefronts, i.e. the locus of points on the wave with the same phase and zero displacement. The Hooke-Chladni-Faraday patterns allow us to conveniently visualise the wave front in the structured plate and, in particular, the cloaking effect for the case shown in Fig. 1(c). A detailed discussion of the experiment is given in the text below. Results of numerical simulations for several cases, corresponding to different frequencies of the incident wave, are reported in Fig. 3.
We have performed both three-dimensional finite element (FE) computations, in ABAQUS, and also experiments to verify the efficacy of the structured square cloak in reducing the scattering of flexural waves by a void. The structured cloak has been designed and implemented in SOLIDWORKS. Each elastic ligament has a specified variable cross section that provides the required rigidities D1 and D2 (see equation (5)), first derived in the paper2 which addressed the corresponding two-dimensional model. The physical three-dimensional lattice cloak was created by milling holes into a polycarbonate plate; both ABAQUS and the milling machine were programmed using the same SOLIDWORKS original code. In so doing, experiments and simulations have identical three-dimensional geometry, material parameters, constraints and applied out-of-plane vibrations. Figure 5 illustrates both the lattice geometries implemented in ABAQUS and in the experiment. In the cloaked configuration (see Figs 1 and 3), we observe a significant reduction in the scattered field and, in particular, the reduction in the shadow region behind the scatterer and the restoration of the incident field represented by a plane wave.
We have compared the wave field for three cases: the first for a homogeneous lattice in the absence of any void, the second in the presence of a void, and the third in the presence of a void surrounded by our specially designed invisibility cloak. The simulations have been performed using a parametric python script for ABAQUS, run by means of MATLAB.
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