  A boundary integral based method for the stable reconstruction of missing
boundary data is presented for the governing hyperbolic equation of
elastodynamics in annular planar domains. Cauchy data in the form of the
solution and traction is reconstructed on the inner boundary curve from the
similar data given on the outer boundary. The ill-posed data reconstruction
problem is reformulated as a sequence of boundary integral equations using the
Laguerre transform with respect to time and employing a single-layer approach
for the stationary problem. Singularities of the involved kernels in the
integrals are analysed and made explicit, and standard quadrature rules are
used for discretisation. Tikhonov regularization is employed for the stable
solution of the obtained linear system. Numerical results are included showing
that the outlined approach can be turned into a practical working method for
finding the missing data.
