On the form of the highest progressive and standing waves in deep water

The form of a progressive gravity wave on deep water, which generally must be found by numerical integration (Michell 1893) is shown to be approximated with remarkable accuracy by a single term. Six consecutive waves are transformed conformally so as to surround the point corresponding to infinite depth. The free surface then corresponds closely to the boundary of a hexagon. In a similar way the profile of a standing wave is closely approximated to by transforming four consecutive waves conformally and taking the profile as the boundary of a square. The profile agrees closely with that calculated by Penney & Price (1952) and with the experiments of Taylor (1953).