Elementary water waves

New elementary farfield and nearfield time-harmonic water waves, observed from a Galilean reference frame, are given. These two related classes of waves are modifications of classical elementary plane waves that are obtained in a simple way, using only elementary fundamental considerations, by analyzing waves that slowly grow from rest at time T = −∞ and are bounded in the farfield. This approach circumvents the need for a radiation condition, which may indeed be regarded as (indirectly) satisfied ab initio by the elementary farfield and nearfield waves given here. The farfield waves are the product of classical elementary waves by a function, called radiation function, that is defined explicitly in terms of the dispersion function. Thus, this radiation function is valid not only for water waves, but more generally for a broad class of linear dispersive waves. For illustration and verification purposes, elementary stationary-phase considerations are used to determine the main properties (phase and group velocities, wave patterns, asymptote and cusp lines) of farfield waves, and two particular classes of water waves—steady ship waves and time-harmonic waves generated by an offshore structure—in uniform finite water depth are considered. The elementary nearfield waves can readily be used to construct free-surface Green functions or in a spectral representation of nearfield flows.