Earlier work has indicated that kinematic advection associated with a linear superposition of standing sinusoidal Lagrangian gravity waves generates, at sufficiently high amplitudes, nonsinusoidal Eulerian perturbations and broadened vertical wavenumber spectra. We find that those same Lagrangian waves also produce singularities in the Eulerian fields. Even at the lowest wave amplitudes consistent with observations of saturated atmospheric waves, we find that kinematic parcel compressions are still far too large compared to adiabatic constraints, while the spectral broadening is far weaker than that indicated by atmospheric measurements. These results demonstrate that kinematic advection by linear Lagrangian waves cannot provide a suitable basis for understanding at least some saturated wave spectra.