  This article aims to present a general study of the Helmholtz problem in
slowly varying waveguides. This work is of particular interest at locally
resonant frequencies, where a phenomenon close to the tunnel effect for
Schr\"odinger equation in quantum mechanics can be observed. In this situation,
locally resonant modes propagate in the waveguide under the form of Airy
functions. Using previous mathematical results on the Schr\"odinger equation,
we prove the existence of a unique solution to the Helmholtz source problem
with outgoing conditions in such waveguides. We provide an explicit modal
approximation of this solution, as well as a control of the approximation error
in H1loc. The main theorem is proved in the case of a waveguide with a
monotonously varying profile and then generalized using a matching strategy. We
finally validate the modal approximation by comparing it to numerical solutions
based on the finite element method.
