Expressions are derived for the amplitudes of the second-harmonic waves generated when a uniform wave train is normally incident upon a two-dimensional body, submerged in water of illfinite depth. These amplitudes are given in terms of integrals over the free surface of products of first-order quantities. For a submerged, circular cylinder, it is shown analytically that there is no second-order reflected wave at any frequency. This extends the classical result that there is no reflection a t first-order for this body.