We present approximate analytical solutions of the Helmholtz equation for a slightly bent metallic rectangular waveguide with infinite aperture cylindrical mirrors. The low‐order, overmoded, low‐losses eigenmodes of the ‘‘cold resonator’’ are naturally expressed as products of cylindrical Gaussian–Hermite and trigonometric functions. In first order in perturbation theory, the correction to the attenuation constant is proportional to the straight waveguide attenuation constant. We show that &pgr; modes propagate with negligible losses in the far‐infrared region. These results are compatible with preliminary experimental data from the University of California, Santa Barbara Free Electron Laser experiment.