A semi-infinite plate of homogeneous isotropic, linearly elastic material occupies the region x >/0, lYl < 1, -~o < z < oo; the faces y = + 1 are free of tractions, the end x = 0 may be either fixed or traction free, and there are no body forces. A plane strain, time-harmonic, symmetric Rayleigh-Lamb wave propagates in the plate and is normally incident upon the end x = 0. The problem of determining the resulting reflected wave field is solved by the "method of projection", a method developed by the authors for solving corresponding problems in elastostatics. The solutions obtained for the dynamic problem fully satisfy the equations and boundary conditions of the linear theory, and (in the fixed-end case) proper account is taken of the singularities of the stress field at the corners x = 0, 3. = + 1. In each case the division of energy between the various reflected modes is found, and the dynamical stress intensity factors at the corners are determined in the fixed-end case. The existence of an "edge-mode" for the free-end case at a single isolated value of the frequency is confirmed, but a careful search revealed no similar phenomenon for the fixed-end case.
* The material of the plate is supposed to be homogeneous, isotropic and linearly elastic; body forces are supposed absent. ** The small-displacement linearised theory is assumed throughout, and we shall restrict attention to motions which are symmetrical about the middle-plane y = 0. * More strictly, modes propagating energy to the right; the corresponding values of a are real but not necessarily positive (see Appendix 1). ** Here k is the compressional wave number of the elastic material at angular frequency ~; recall also that the plate has thickness 2. ** From now on, the factor e -i'Β°r will be suppressed.
* This completeness has never been proved, but will be assumed; see Gregory [4] for a proof of completeness of the corresponding static stress functions.