A boundary integral equation (BIE) method is presented for the calculation of the motion of an elastic plate in a two-dimensional setting. A wave maker is placed next to the plate and the model is therefore representative of recent wave tank experiments. The model includes the plate’s draught and surge motion, the latter of which is restrained by a spring and damping system. However, the corners created by the non-zero draught produce firstorder singularities in the normal derivative of the Green’s function, in addition to the logarithmic singularities in the Green’s function itself. Both of these singularities are separated using Kummer transforms, and the associated boundary integrals are evaluated analytically. The numerical computation of the modified BIEs explicitly decomposes the plate’s displacement into its heave, pitch and flexural motions.