  This paper is devoted to Maxwell modes in three-dimensional bounded
electromagnetic cavities that have the form of a product of lower dimensional
domains in some system of coordinates. The boundary conditions are those of the
perfectly conducting or perfectly insulating body. The main case of interest is
products in Cartesian variables. Cylindrical and spherical variables are also
addressed. We exhibit common structures of polarization type for eigenmodes. In
the Cartesian case, the cavity eigenvalues can be obtained as sums of
Dirich-let or Neumann eigenvalues of positive Laplace operators and the
corresponding eigenvectors have a tensor product form. We compare these
descriptions with the spherical wave function Ansatz for a ball and show why
the cavity eigenvalue of the ball are also Dirichlet or Neumann eigenvalues of
some scalar operators. As application of our general formulas, we find explicit
eigenpairs in a cuboid, in a circular cylinder, and in a cylinder with a
coaxial circular hole. This latter example exhibit interesting '' TEM ''
eigenmodes that have a one-dimensional vibrating string structure, and
contribute to the least energy modes if the cylinder is long enough.
