In a domain method of solution of exterior scalar wave equation, the radiation condition needs to be imposed on a truncation boundary of the modelling domain. The Bayliss, Gunzberger and Turkel (BGT) boundary dampers of first- and second-orders, which require a circular cylindrical truncation boundary in the diffraction-radiation problem of water waves, have been particularly successful in this task. However, for an elongated body, an elliptic cylindrical truncation boundary has the potential to reduce the modelling domain and hence the computational effort. Grote and Keller [On non-reflecting boundary conditions. Journal of Computational Physics 1995; 122: 231-243] proposed extension of the first- and second-order BGT dampers for the elliptic radiation boundary and used these conditions to the acoustic scattering by an elliptic scatterer using the finite difference method. In this paper, these conditions are implemented for the problem of diffraction of water waves using the finite element method. Also, it is shown that the proposed extension works well only for head-on wave incidence. To remedy this, two new elliptic dampers are proposed, one for beam-on incidence and the other for general wave incidence. The performance of all the three dampers is studied using a numerical example of diffraction by an elliptic cylinder. Copyright © 2001 John Wiley & Sons, Ltd.