Abstract
One of Newton’s mathematical solutions to a hypothetical orbital problem, recently verified by an independent physics model, is applied to the fluid particle motion in shallow water surface gravity waves. What is the functional form of the central force, with origin at the ellipse’s center, which will keep a body in the orbit? Newton found out it is the spring force, which is linear. All fluid particles in shallow water waves move in ellipses. By a superposition of solutions in a linear problem, the application of Newton’s result to shallow water waves is combined with a feature not noticed by Newton: the orbital period is independent of the semi-major and semi-minor axes. Two conclusions reached are that the wave period of shoaling waves should be constant and that there is no friction in these waves.
Highlights
IntroductionAccording to laboratory studies that have used small neutrally buoyant particles in a wave tank, it is inferred that the motion of fluid particles in propagating surface gravity waves are in circles when the mean depth of water greatly exceeds the wavelength [1, 2]
The present basically mathematical status of shallow water surface gravity waves, with respect to describing the paths of the fluid particles, is enhanced below by incorporating a physical model into the discussion.According to laboratory studies that have used small neutrally buoyant particles in a wave tank, it is inferred that the motion of fluid particles in propagating surface gravity waves are in circles when the mean depth of water greatly exceeds the wavelength [1, 2]
Digging deeper is required which turns out to mean going further back in time to the 1600s, Newton’s Principia in particular. It is better known [5] that Newton warmed up to solving the famous and practical Kepler problem by working out four hypothetical exercises involving different functional forms for a central force acting on an orbiting body
Summary
According to laboratory studies that have used small neutrally buoyant particles in a wave tank, it is inferred that the motion of fluid particles in propagating surface gravity waves are in circles when the mean depth of water greatly exceeds the wavelength [1, 2]. Shallow water waves occur when the waves travel in water with a mean depth comparable to or less than the wavelength. Whether or not the elliptical particle paths have been demonstrated in the lab is not known to me
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