The excitation of high‐amplitude Langmuir waves in a warm plasma is investigated in the framework of the hydrodynamic model. An integral expression is found describing the maximum possible amplitude in the limit of stationary waves. The steep increase of compression energy in the density peaks is shown to be the reason for this limitation of amplitude. If electron trapping in the potential of the Langmuir wave is taken into consideration, two consequences occur: The limit for the density peaks in the stationary wave is lowered, and the dispersion relation is modified. Because of the trapped electrons, allowed frequency bands appear below the original plasma frequency. Thus plasma waves containing trapped electrons are able to propagate in the overdense region.