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Volume 12: Pages 775-780, 1999

The Surface Wave as a Set of Springs: An Application of One of Newton's Hypothetical Orbital Solutions

Kern E. Kenyon

4632 North Lane, Del Mar, California 92014‐4134 U.S.A.

Out of the four hypothetical orbital solutions obtained by Newton, excluding the Kepler problem, which so far have not been seriously associated with any practical situations, one is discussed here and then applied to surface gravity waves in order to increase the understanding of these geophysical waves. The orbit is the ellipse, and the attractive force on the moving particle always points to the center of the ellipse. In this case, Newton found that the central force necessary to keep the particle in its orbit varies directly as the distance, like a linear spring. The fluid particles of surface gravity waves move in closed ellipses for progressive waves of infinitesimal amplitude, according to potential theory, when the total mean depth of water is smaller than a wavelength. Since each particle feels an outward centrifugal force as it moves around its path, there must be a counteracting inward restoring force, assuming nonturbulent flow, and for surface waves this can only be a pressure force. Therefore, by applying Newton's result, we deduce that the restoring pressure force varies directly as the displacement of the particle from its equilibrium position at the orbit's center. This finding is not obvious a priori but is consistent with potential theory. It is as if each particle of a small‐amplitude surface wave is held in its orbit by a linear spring anchored to the orbit's center. Interactions between neighboring particles are taken to be negligible.

Keywords: Newton's orbits, spring force, surface waves, particle paths

Received: July 5, 1999; Published online: December 15, 2008