A waveguide occupies a domain in an (n + 1)-dimensional Euclidean space which has several cylindrical outlets to infinity. Three classes of waveguides are considered: those of quantum theory, of electromagnetic theory, and of elasticity theory, described respectively by the Helmholtz operator, the Maxwell system, and the system of equations for an elastic medium. It is assumed that the coefficients of all problems stabilize exponentially at infinity, to functions that are independent of the axial variable in the corresponding cylindrical outlet. Each row of the scattering matrix is given approximately by minimizing a quadratic functional. This functional is constructed by use of an elliptic boundary value problem in a bounded domain obtained by cutting the cylindrical outlets of the waveguide at some distance R. The existence and uniqueness of a solution is proved for each of the three types of waveguides. The minimizers converge exponentially fast as functions of R, as R → ∞, to rows of the scattering matrix.