Abstract :
paper is concerned with a general asymptotic analysis of the dispersion relation
associated with waves propagating in a pre-stressed, incompressible elastic plate. In the high wave
number limit it is well-known that, whenever a real surface wave speed exists, the fundamental
modes of both symmetric and anti-symmetric motions tend to this surface wave speed, with all
harmonics tending to a single shear wave speed limit. The character of the two dispersion curves in
the moderate and high wave number regimes falls into one of two distinct cases, these being
dependent on pre-stress. In the first case all the harmonics are monotonic decreasing functions and
as such the asymptotic analysis in this case offers a modest generalisation of an earlier study, see
Rogerson and Fu (Rogerson, G. A. and Fu, Y. B. (1995) An asymptotic analysis of the dispersion
relation of a pre-stressed incompressible elastic plate. Acta Mechanica 111, 59-77). In contrast, the
second case is quite different in character with the passage to the high wave number limit
accompanied by sinusoidal behaviour. This behaviour is fully elucidated by obtaining asymptotic
expansions which give phase speed as a function of wave number, pre-stress and harmonic number,
sinusoidal terms being found to occur at third order. Both these asymptotic expansions and ones
obtained for high harmonic number are found to provide excellent agreement with numerical
solutions for Varga materials in the appropriate regimes. It is envisaged that the expansions derived
in this paper may well find important potential applications in the numerical inversion of the
transform solution sometimes used in impact problems. f(?‘ 1997 Elsevier Science Ltd
