The relationship between the free wave motion and the natural flexural modes of beams is re-examined. The reflection of both propagating and evanescent waves, incident upon a linearly constrained boundary, is first analyzed. Boundary conditions are identified which cause no evanescent wave to be reflected when a propagating wave is incident, and vice versa. The phase differences between the incident and reflected waves are analyzed for the case in which two waves (one propagating and one evanescent) are incident. The phase-closure principle is formally proved for single span uniform beams and is used to set up the exact frequency equation for a fully fixed beam. Whereas the principle has traditionally been applied to the propagating wave motion, it is applied in this paper also to the evanescent wave motion. Frequency equations having quite different forms are thereby obtained, but they have identical roots. A physical interpretation becomes obvious for the conventional frequency equation for the fully fixed beam.
