On Isotropy

Isotropic and Orthotropic Materials

Some words you may not have heard of before but are definitely important in understanding cymatic phenomena are isotropic and orthotropic, and these terms relate to both liquid cymatics and Chladni plate cymatics. 

Isotropic materials “flex” or bend the same manner in all directions. An elastic balloon is isotropic - its material flexes the same way no matter the direction of applied force. A wooden 2x4, however, is orthotropic - it flexes differently in its three perpendicular directions. Flex a 2x4 along its face lengthwise and it can easily bend and bow and eventually snap. Wood easily bends when you flex it in the direction of the grain. Now lay the 2x4 flat and try to flex it to make a circle end to end. Nope! This is why headers above windows and doors are made with the 2x4s facing out from the wall rather than facing up or down in the wall. The wood is much stronger if the applied forces are in the direction perpendicular to (against) the grain.

Tonoscopes made from elastic rubber or latex are like drumheads and are examples of isotropic circular membranes. When a voice or other sound passes through the tonoscope, the elastic membrane flexes in various directions. But, because it is isotropic, the tensions and forces in the flexing are equal throughout the membrane except at the edge where the membrane is clamped like a drum.
Chladni plates are isotropic elastic materials that bend and flex the same in all directions but are not clamped at the edge. They are, however, designed to be larger than the decay length of the vibrations and thus should always produce modes of vibration without slinging too much sand off the plate.

The modes of vibration (that is, the ways a material resonates) are well-studied for fixed-edge membranes like drumheads and tonoscopes. If we know the tensile strength of the material, and the mass-density of the material, we can confidently calculate and predict not only whether a frequency will resonate with the membrane but also predict what shape it will induce. Similarly, the modes of vibration are well-studied for free-edge materials like brass plates, and we can also predict mode shapes with Chladni plate cymatics.

Liquid cymatics, however, is an entirely new creature in the world of physics, and represents another class of isotropic phenomena. The surface of the fluid is “held together” by surface tension, much like the tension applied to a rubber membrane when it is stretched across a frame, and is another example of an isotropic material. Though, unlike the rubber membrane, the liquid has no clamped edge. Sure, it experiences adhesion to the inside of its container, but with a hydrophobic liner we can induce a near-100% free edge like the Chladni plates. However, unlike the Chladni plates, the surface of the liquid experiences a boundary that reflects the energy back to the center of the fluid.

The research on the phenomenon of vibrational modes of liquid held in a circular container is fairly extensive, going back to the mid-80’s; however, one fact remains - after all of the searching and mathematical physics applied to this phenomenon, it is still a mystery to predict what will happen with a given frequency in a dish full of fluid with known properties. Perhaps instead of attempting to model the *entire* physics of the fluid in the dish, we should somehow focus more on the membrane at the surface. Sure, the viscosity and density of the fluid still matter, but the surface is where all of the geometry happens, so that is where we are focusing our research - to study the vibrational modes of an isotropic membrane held in a circular container with a near-100% free edge.

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