Bounds and approximations for elastodynamic wave speeds in tetragonal media

This paper presents an analytical study of elastodynamic waves propagating along an arbitrary direction in anisotropic materials with tetragonal symmetry. Upper and lower bounds are developed on the wave speeds through an additive decomposition of the acoustic tensor into an associated hexagonal counterpart and a rank-one modification, the eigenproperties of which can be determined analytically. The bounds are obtained by applying the minimax property of eigenvalues. Linear approximations of the wave speeds are obtained from a first-order expansion of the eigenvalues of the acoustic tensor, with respect to a tetragonality index, about the value of that index for which tetragonal symmetry degenerates to hexagonal symmetry. A numerical example shows that the linear approximations agree remarkably well with the exact values of the wave speeds.