The analytic form of Green’s function in elliptic coordinates

Abstract

The purpose of this study is the derivation of a closed-form formula for Green's function in elliptic coordinates that could be used for achieving an analytic solution for the second-order diffraction problem by elliptical cylinders subjected to monochromatic incident waves. In fact, Green's function represents the solution of the so-called locked wave component of the second-order velocity potential. The mathematical analysis starts with a proper analytic formulation of the second-order diffraction potential that results in the inhomogeneous Helmholtz equation. The associated boundary-value problem is treated by applying Green's theorem to obtain a closed-form solution for Green's function. Green's function is initially expressed in polar coordinates while its final elliptic form is produced through the proper employment of addition theorems.

A second-order theory for an array of curved wave energy converters in open sea

“…Mathieu functions have the advantage of decreasing significantly the numerical tasks and reducing the algebraic manipulations with respect to methods based on integral and hypersingular integral equations. For this reason, their applications can be found in several surface wave diffraction phenomena of the first (Chen and Mei, 1973;Williams and Darwiche, 1988;Chatjigeorgiou and Mavrakos, 2010;Chatjigeorgiou, 2011) and second order (Chatjigeorgiou and Mavrakos, 2012) and other fields involving elliptic structures (Stamnes and Spjelkavik, 1995;Ruby, 1995;Gutiérrez-Vega, 2000;Gutiérrez-Vega et al, 2003). Hereafter we summarize the method of solution adopted in Michele et al (2016b).…”

“…This is because quadratic products of the first-order solutions generate a drift flow and several harmonics that are related to unexpected forces over the system components (Mei et al, 2005). Analytical models concerning cylindrical structures involved several authors in the past (Molin, 1979;Chau and Eatock Taylor, 1992;Kim and Yue, 1989;Newman, 1996), while the second-order diffraction theory for elliptical cylinders has more recently been developed by Chatjigeorgiou and Mavrakos (2012). Higher-order theories involving floating offshore structures have been developed in several contexts.…”

Explicit Equations to Transform from Cartesian to Elliptic Coordinates

“…The elliptical coordinate system ( , ) ξ η as a two-dimensional orthogonal coordinate system has many dynamical and engineering applications, such as Kirchhoff vortex [1], insect aerodynamics [2], hydrodynamic wave diffraction [3], and theoretical physics [4]. Its coordinate lines are confocal ellipses and hyperbolae and the transformation from elliptic to Cartesian coordinates is given by …”