The propagation of electron Bernstein waves (EBWs) close to the tokamak mid-plane is considered. It has been shown that in the case of a concave electron cyclotron resonance (ECR) surface, typical for an ECR located between the upper hybrid resonance layer and the discharge axis, there is a plasma waveguide for EBWs, inhomogeneous in the poloidal direction and slowly varying in the radial direction, where the waves propagate in the form of discrete eigenmodes. Since the WKB theory is inapplicable to small mode number eigenfunctions, this means that the ray method is inapplicable to a part of the waves. To investigate this effect, a solution to the wave equation is found using a generalization of the parabolic equation method. For the Gaussian initial condition used, the solution is often written explicitly. Close to the ECR it has the form of an adiabatic eigenmode superposition. For illustration, the eigenmode spectrum for the case of the Gaussian beam strongly elongated in toroidal direction is investigated as a function of the initial poloidal beam width. For the usual experimental poloidal beam width of several centimetres, the eigenmode spectrum is rather wide; nonetheless, about half of the launched RF power is deposited into non-WKB modes. With the initial beam width diminishing, this share increases rapidly. The damping of the non-WKB modes is analysed in the relativistic approximation. In contrast with 'relativistic' rays, these modes lose their energy before reaching the relativistic ECR layer, in the region where damping is weak, so that the power deposition profile is rather smooth. In the case of a convex ECR surface there is a 'potential hump' in the mid-plane region and the EBW behaviour is adequately described by the ray method.