Reflection of Wave Trains in Semi-Infinite Plates

A method for using the well known modes of vibration for the infinite plate as elements of an expansion applicable to semi-infinite plates is given. The method is employed to study the reflection of an infinite train of waves from the free edge of an elastic plate. The incident train of waves induces other, nonpropagating, modes. As has previously been discovered, a resonance may occur in such a mode that will increase the portion of the plate for which the evanescent modes affect the stress distribution. Above a critical frequency, a transfer of energy from one propagating mode to another occurs during the reflection. Generally, energy brought to a free edge by waves of one wavelength will depart in wave trains of several different wavelengths. The presence of a mode which, for a limited range of frequency, carries energy in a direction opposite to the motion of the waves is confirmed. Some observations are made about necessary restrictions in an extension of St. Venant's principle to dynamic loadings.