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There are numerous reports on the numerical analysis of variational expressions for a waveguide with an arbitrary cross section and structure by means of a finite element method. In conventional methods, cumbersome procedures or special programming techniques are required to discriminate the true solution of the guided mode from the spurious solutions. In this paper, a new variational expression for the β propagation constant is derived in which the transverse electromagnetic field components e.t and ht are used as the testing functions. A method is presented for numerical analysis of this new variational expression by means of the finite element method. It is pointed out that the guided mode and the spurious mode can easily and completely be discriminated by only the realistic condition that the propagation constant of the guided mode is real, without resorting to cumbersome procedures or special programming techniques. The new variational expression is well suited for a finite element analysis and resolves difficulties and ambiguities associated with the finite element methods based on the conventional variational expressions.