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Nodal Lines

Mid-Term Report Summary - NODAL (Nodal Lines)

First observed by the physicist and musician Ernst Chladni in the 18th century, the nodal lines (also referred to as the Chladni Plates or Chladni Modes) appear in many problems in engineering, physics and natural sciences. Nodal lines describe sets that remain stationary during membrane vibrations, hence their importance in such diverse areas as musical instruments industry, mechanical structures, earthquake study and other fields. So far, the nodal structures have been mainly addressed in the physics literature, whose statement are lacking the mathematical precision; most of their results are based on numerical experiments and heuristic computations rather than analytic methods typical for mathematics. This project aims at understanding the nodal patterns and question arising from them with mathematical rigour.

In his seminal paper, Michael Berry (1977) suggested that the behaviour of the deterministic nodal patterns corresponding to the high frequency vibration on generic membranes is universal, and may be “miraculously” explained by a random ensemble of monochromatic waves. Extensive numerical experiments confirm Berry’s predictions, however no rigorous statement was known (or even formulated). We address questions like these with mathematical rigour, and discover certain universalities shared among all dynamical systems (with Chladni Patterns as a by-product), and other non-universalities depending on the peculiarities of the system. The questions we address or raise are of high importance in mathematical physics, probability theory, mathematical analysis, and number theory.