Wave propagation in transversely isotropic elastic media - III. The special case a 5 = 0 and the inextensible limit

Abstract
It has been noticed in part I that the simplified form of transversely isotropic elasticity obtained by taking the moduli to satisfy the relation c 13 + c 44 = 0 (<=> a 5 = 0) has favourable theoretical features, one of which is the possibility of approaching analytically the limiting case of inextensibility along the axis of symmetry. This advantage is exploited in the present paper. On the basis of the restricted theory the surface-wave function F(v) is calculated in closed form for any orientation of the axis relative to the surface-wave basis. The exceptional configurations classified in part I are then considered. Case 2 is unaffected by the vanishing of a 5 while, as shown in part II, case 3 does not arise. The continuous transition between subsonic and supersonic surface-wave propagation encountered in part II persists under the conditions of case 1 and is displayed here with particular simplicity. The passage to the inextensible limit of F( is immediate and the secular equation for surface waves, derived previously by a purely algebraic method, is retrieved directly. No surface wave exists in the α configurations when the inextensibility constraint applies and, although an exceptional plane wave appears, it can no longer be interpreted as a degenerate form of the subsonic wave which propagates in neighbouring configurations. All the other configurations admit a unique surface wave and this wave is subsonic except, perhaps, in the β = ½π configuration where the possibility of supersonic transmission is retained when a 4 < a 2 . The extension of the existence theorem of Barnett and Lothe to the constrained medium is not straightforward. Part 1 of the theorem fails to hold in the α = ½π configuration and part 2 is not fully effective in the remaining α configurations. The theorem holds good for the other configurations, barring β = ½π, if the degenerate part B e of the slowness surface is ignored in the calculation of the limiting speed v ˆ . In the β =½π configuration the theorem remains valid provided that B e is taken into account in the determination of the transonic state.

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