A Variational Principle for Periodic Waves of Infinite Depth (Classic Reprint)

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Excerpt from A Variational Principle for Periodic Waves of Infinite Depth This paper deals with periodic waves on an ocean of infinite depth. The flow is assumed to be two-dimensional, incompressible, steady, and irrotational. The impossibility of the existence of an asymmetric wave is proved. This is accomplished through an application of Steiner symmetrization. Also discussed is the shape of possible periodic waves. Using the calculus of variations, we set up an extremal problem involving the kinetic energy, an area integral, and the potential energy. For waves of small amplitude the kinetic energy is shown to be a minimum if we fix the area and the potential energy. This is accomplished by showing the first variation to be zero and the second variation to be positive. Since the kinetic energy is closely related to the Dirichlet integral, this is a generalization of the Dirichlet principle. This result is applicable in showing the existence of periodic surface waves. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.