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A two parameter asymptotic analysis is employed to investigate some unusual long wave dispersion phenomena in respect of symmetric motion in a nearly incompressible elastic plate.The plate not subject to the usual classical traction free boundary conditions, but rather has its faces fixed, therefore precluding any displacement on the boundary. The abnormal long wave behaviour results in the derivation of some non-local approximations, giving frequency as a function of wave number, for symmetric motion. Motivated by these approximations, long wave asymptotic integration is carried out and the asymptotic forms of displacement components established.