WAVES AND MODES IN FINITE BEAMS - APPLICATION OF THE PHASE-CLOSURE PRINCIPLE

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 f irst analyzed. Boundary conditions are identified which cause no evane scent wave to be reflected when a propagating wave is incident, and vi ce versa. The phase differences between the incident and reflected wav es are analyzed for the case in which two waves (one propagating and o ne evanescent) are incident. The phase-closure principle is formally p roved for single span uniform beams and is used to set up the exact fr equency equation for a fully fixed beam. Whereas the principle has tra ditionally 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 ident ical roots. A physical interpretation becomes obvious for the conventi onal frequency equation for the fully fixed beam.