The propagation of acoustic waves along a duct of elliptic cross-section, the eccentricity of which varies slowly along the axis, is considered as a model of the unsteady #ow inside a realistic aeroengine nacelle. This is a development of recent work on a circular duct by Rienstra. The acoustic "eld at each axial location is expanded in terms of the local even and odd Mathieu modes, and the slow variation of the axial wavenumber and the modal amplitude along the duct are determined as part of the solution. The duct eccentricity is seen to have a very signi"cant e!ect right across the range of practical azimuthal orders, and has di!erent e!ects on the even and odd modes. For instance, the point at which a given mode changes from being cut-on to cut-o! depends on the eccentricity; for even modes, and for the odd modes apart from at a very low frequency, the eccentricity makes modes more cut-on than in a circular duct. Even for cases in which a given mode is cut-on all along the duct according to standard circular duct theory, the modal amplitude is seen to depend strongly on the cross-sectional eccentricity. In a representative case the amplitude of a high order mode propagating upstream along a slowly varying elliptical duct is seen to be signi"cantly lower than that of the equivalent mode propagating in a circular duct of the same cross-sectional area.