Numerical solutions of wave problems are often influenced by uncertainties generated by a lack of knowledge of the input values related to the domain data and/or boundary conditions in the mathematical equations used in the modeling. Conventional methods for uncertainty quantification in modeling waves constitute severe challenges due to the high computational costs especially at high frequencies/wavenumbers. For a given accuracy and a high wavenumber it is necessary to perform a mesh convergence study by refining the discretization of the computational domain with an increased resolution, which leads to increasing the number of degrees of freedom at a much higher rate than the wavenumber. This effect also known as the pollution error often limits the computations to relatively small values of the wavenumber. To estimate the uncertainties, many model evaluations are required, but only a single surrogate model is created in the process. In the present work, we propose the use of a non-intrusive spectral projection applied to a finite element framework with enriched basis functions for the uncertainty quantification of waves at high frequencies. The method integrates (i) the partition of unity finite element method for effectively computing the solutions of waves at high frequencies; and (ii) a non-intrusive spectral projection for effectively propagating random wavenumbers that encode uncertainties in the wave problems. Compared to the conventional finite element methods, the proposed method is demonstrated to reduce the total cost of accurately computing uncertainties in waves with high values of the wavenumber. Numerical results are presented for two sets of numerical tests. First, the interference of plane waves in a squared domain and then a wave scattering by a circular cylinder are studied at high wavenumbers. Comparisons to the Monte Carlo simulations and the regression based polynomial chaos expansion confirm the computational effectiveness of the proposed approach.