Eigenvalues of the Steklov problem in an infinite cylinder

The Steklov problem is considered in cylindrical domains; the coefficient in the boundary condition has a compact support and is an even function of a coordinate varying along the generators. We study the dependence of eigenvalues on the spacing between two symmetric parts of the coefficient's support. It is proved that the antisymmetric (symmetric) eigenvalues are monotonically decreasing (increasing) functions of the spacing and formulae for their derivatives are obtained. Application to the sloshing problem in a channel covered by a dock with two equal rectangular gaps is given.