Familias de campos ondulatorios fundamentales de la ecuación de Helmholtz en sistemas de coordenadas curvilíneas ortogonales
JOSE ADAN HERNANDEZ NOLASCO
The Scalar Helmholtz Equation is separable in eleven orthogonal coordinate systems. Despite being one of the most studied equations in Mathematical Physics, this fact is not mentioned and the whole story is hard to find in the classical textbooks or even specialized literature. Moreover, even when mentioned, the geometry of some of those coordinate systems is not properly illustrated in the literature. In order to transform the differential operator of the Helmholtz equation the corresponding metric for the curvilinear coordinates is used. For some of these, after performing the transformation, the resulting equation can be very cumbersome and the method of separation of variables does not apply in a simple and straightforward way. This leads to look for an alternative method also barely known that is the Stackel determinant, this allows to get the three separated differential equations for each of the curvilinear coordinates. The method can be used for the eleven coordinate systems. From which emerge fifteen different differential equations to solve. To represent the whole families of scalar wave fields it is necessary to solve each of the three equations for each one of the curvilinear coordinates. In most of the cases the solutions are given in terms of not widely known special functions and in some cases their numerical evaluation is not an easy task. We will present in an understandable and visual way the eleven coordinate systems in which the Helmholtz equation is separable. We will show the surfaces associated with them, whose intersections determine univocally a point in a three dimensional space. We will present the normalization in a canonical form, the corresponding sets of ordinary differential equations resulting from the separation of the Helmholtz equation, providing a graphical picture of the fundamental solutions that enable the representation of radiating electromagnetic wave fields for each of the curvilinear coordinate systems.
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