This work concerns the numerical finite element computation, in the frequency domain, of the diffracted wave produced by a defect (crack, inclusion, perturbation of the boundaries etc..) located in an infinite elastic waveguide. The objective is to use modal representations to build transparent conditions on the artificial boundaries of the computational domain. This cannot be achieved in a classical way, due to non standard properties of elastic modes. In particular, the derivation of a ``Dirichlet-to-Neumann'' operator (relating the normal stress to the displacement) is not tractable. However, a biorthogonality relation allows to build an operator, relating hybrids displacement/stress vectors. An original mixed formulation is then derived and implemented, whose unknowns are the displacement field in the bounded domain and the normal component of the normal stresses on the artificial boundaries. Numerical validations are presented in the two-dimensional case.