We derive the modes inside a cylindrical waveguide of finite surface impedances, assuming the waveguide transverse dimensions are large compared to the wavelength λ. This paper restricts its consideration to the modes with , where β is the propagation constant and k = 2π/λ. For these modes we show that asymptotically, for large values of k, the field ψ becomes infinitesimal (of the same order of 1/k) at the boundary. Taking this into account, we obtain simple expressions for the asymptotic properties of ψ for large k. The theory applies to a variety of waveguides: corrugated waveguides, optical fibers, waveguides with smooth walls of lossy metal, and so on. An important property of the modes considered here is that their attenuation constant is very low, i.e., asymptotic to 1/k2 for large k. Thus, these modes are useful for long‐distance communication. Another consequence of ψ → 0 at the boundary is that for large k the distribution of ψ inside the boundary is essentially independent of the boundary parameters, i.e., independent of the surface impedances in the longitudinal and transverse directions. This consequence implies that the same radiation characteristics of the corrugated feed can be obtained using other structures and, therefore, construction can be simplified in many cases, with little sacrifice in performance. We also derive general expressions for ψ and the propagation constant β.