Summary: The objective of the present work is to provide a theoretical analysis of the spectral properties of a selection of nonlinear finite difference schemes. The Nonlinear Spectral Analysis (NSA) of the authors [J. Comput. Phys 230, No. 12, 4488–4518 (2011; Zbl 1416.76174)] is applied in order to study the statistical behavior of the modified wavenumber of TVD schemes in combination with five common limiter functions and also the 5th-order WENO scheme, for a large set of synthetic scalar fields with prescribed energy spectrum and random phase. The theoretical observations are verified numerically for the linear propagation of a broadband wave.