Wave propagation in slowly varying guiding geometries is of fundamental importance across many disciplines, and can be described in terms of mutually uncoupled quasi‐modes. These are a generalization of the concept of normal modes in a uniform waveguide to the weakly non‐uniform case. Quasi‐modal propagation is dependent upon the wavelength and two geometrical length‐scales, those of the longitudinal variations and the guide thickness. By changing these length‐scales one enters different asymptotic regimes. Our aim is to present an asymptotic theory for quasi‐modal propagation, in the various regimes, and for a canonical geometry, that is, an arbitrarily curved waveguide of constant thickness. Potential applications of our results are very wide and include acoustic or electromagnetic propagation in curved waveguides, shear horizontal wave propagation in curved elastic plates, and the determination of electronic states in quantum wires.
We derive practically useful asymptotic expressions of the quasi‐modes of a weakly curved waveguide; these are particularly important since the adiabatic approximation for this problem simply coincides with the normal modes of a straight waveguide of the same thickness. The basic ideas and methodology should carry across to other geometries, boundary conditions and governing equations.