Kinematics of an elliptical orbital motion resulting from the superposition of two concentric circular movements running in opposite directions

The change in the orbital movement that occurs with water waves (from approximately circular in deep water to increasingly elliptical with decreasing water depth) deviates from the conservative view of the shoaling process such that it is accompanied a priori by the phenomenon of reflection. Thereby the long principal axes of the ellipse grow at the expense of the short principal axes. For deep water conditions (d/L ≥ 0.5), according to the theory ERR used by the author [1], for example, the circular movement on that of the diameter D1 (= wave height H) on the water surface is superimposed on that of a smaller circular diameter D2. The familiar relationship applies D2 = D1 ·e-2πd/L
with d/L as the ratio of the water depth d to the wavelength L.
The phenomenon of exponentially reduced reflection (ERR) [2] is taken into account by the fact that the water depth d < L/2 is assumed for the reflection and accordingly a reversal of the direction of rotation of the orbital movement
occurs. At a given water depth (mirror depth) d2 < L/2, the circular orbital movement of the diameter D1 on the water surface would have to be superimposed by the orbital movement of diameter D2, which results from the above exponential law at twice the distance 2d2 from the water surface.
The theory at hand is checked on the one hand using the example of the calculation of the kinematics for water waves with regard to the dimensioning of offshore structures [3] and on the other hand the example of the earth's elliptical orbit around the sun is placed alongside the results of the well-known epicyclic theory.


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