Abstract: A new waveguide made of changing a wide wall of a rectangular waveguide to a uniform curved wall is called conformal waveguide.A kind of expression of approximate solution for domain mode of the new waveguide,including symmetric,unsymmetric arc,parabolic arch,hyperbolic arch,elliptic arch,etc.,is constructed by using boundary condition of mental waveguide,variational method and boundary element method.By using the equivalence principle,an approximate calculating formula for equivalent resonant conductance of a narrow longitudinal half wavelength slot cut in the curved wall of the unsymmetric arc rectangular waveguide is derived,and the result is shown.Although this solution is approximate,it can solve the problem in designing the slotted antenna of such conformal waveguide.