In this paper, the effects of wave–wave interactions of the lowest order, i.e., three-wave interactions, on parallel-propagating Alfvén wave spectra on a closed magnetic field line are considered. The spectra are then used to evaluate the transport parameters of energetic particles in a coronal loop. The wave spectral density is the main variable investigated, and it is modelled using a diffusionless numerical scheme. A model, where high-frequency Alfvén waves are emitted from the two footpoints of the loop and interact with each other as they pass by, is considered. The wave spectrum evolution shows the erosion of wave energy starting from higher frequencies so that the wave mode emitted from the closer footpoint of the loop dominates the wave energy density. Consistent with the cross-helicity state of the waves, the bulk velocity of energetic protons is from the loop footpoints towards the loop apex. Protons can be turbulently trapped in the loop, and Fermi acceleration is possible near the loop apex, as long as the partial pressure of the particles does not exceed that of the resonant waves. The erosion of the Alfvén wave energy density should also lead to the heating of the loop.