It is known that head seas cannot travel without deformation along a cylinder of full constant cross-section, and recent calculations have indicated that the wave amplitude near the cylinder ultimately decreases as the waves travel along the cylinder, i.e. that the waves are refracted away from the axis of the cylinder. It was assumed in these calculations that the cross-section was a half-immersed circle of radius a of the same order as the wavelength 2π/K, but the method can probably be adapted to arbitrary full constant cross-sections. (There is however another calculation which indicates that for a thin ship the wave amplitude ultimately increases.) In the present paper these calculations are extended. The circular section is again studied but it is now supposed that the wavenumber Ka may be small. Uniformly valid expressions for the wave potential are obtained which show that for small Ka the refraction becomes significant only when Kx (the dimensionless distance along the cylinder) is so large that the product (Kx)½ v0(Ka) is also large; here the function v0(Ka) ∼ 2Ka arises in the solution of a certain eigenvalue problem. (The uniformly valid expressions also suggest an interpretation of the thin-ship calculation which resolves the apparent inconsistency.) The same method is applied to the waves generated by a pulsating source on an infinite cylinder, and similar results are obtained.