  We derive and analyze a hybridizable discontinuous Galerkin (HDG) method for
approximating weak solutions to the equations of time-harmonic linear
elasticity on a bounded Lipschitz domain in three dimensions. The real symmetry
of the stress tensor is strongly enforced and its coefficients as well as those
of the displacement vector field are approximated simultaneously at optimal
convergence with respect to the choice of approximating spaces, wavenumber, and
mesh size. Sufficient conditions are given so that the system is indeed
transferable onto a global hybrid variable that, for larger polynomial degrees,
may be approximated via a smaller-dimensional space than the original
variables. We construct several variants of this method and discuss their
advantages and disadvantages, and give a systematic approach to the error
analysis for these methods. We touch briefly on the application of this error
analysis to the time-dependent problem, and finally, we examine two different
implementations of the method over various polynomial degrees and numerically
demonstrate the convergence properties proven herein.
