An investigation is made into water-wave propagation through an array of vertical cylinders extending to infinity and periodic in both horizontal directions. Methods are presented for the calculation of the frequency ranges for which wave propagation without change of amplitude is possible (‘passing bands’), and for which propagation without change of amplitude is not possible (‘stopping bands’). Some of the techniques may be used to determine the change of wave amplitude for frequencies within the stopping bands. Approximate and numerical techniques are used to show how this infinite-array problem is related to trapped modes, Rayleigh–Bloch waves, and the problem of wave diffraction by a grating made up of a finite number of cylinder rows.