Water waves are a prime example of the kind of physical phenomena which are seen in nature every day. The scientific explanation of such phenomena is the core of applied mathematics, and indeed applied mathematicians have been studying water waves since their governing equations were written down two centuries ago by the English mathematician G. G. Stokes, following earlier pivotal work on fluid mechanics by L. Euler.
The equations governing water waves are rather intriguing from a mathematical point of view, and historically their study has relied upon so many different approaches from different branches of mathematics that they have become a paradigm for the subject. However, although engineers have achieved dramatic success by using simpler equations to model water waves in certain special situations, many aspects of the completely rigorous mathematical theory of water waves remain unexplored. Recently several groups of mathematicians have become interested in the development of rigorous mathematical theories for "three-dimensional" water waves, that is, wave patterns of the kind seen on the ocean surface as opposed to the "two-dimensional" patterns seen in channels and canals. In our paper we show how a mathematical method known as "spatial dynamics" can be employed to find such wave patterns, and in particular we develop a theory for waves of the kind shown in the sketch. The method has since been used to find several other kinds of "three-dimensional" water waves. Our result is published at a time when the mathematical theory of water waves is once again becoming the subject of intensive research. Other types of waves currently under scrutiny include hexagonal patterns on ocean surfaces, "standing waves" which oscillate in space and time, and "fully localized" waves which consist of an antisymmetric "trough" of water moving on an ocean surface. Engineers have known of the existence of such waves for many years, and strictly mathematical theories to describe them are now becoming available.