Steep progressive waves in deep and shallow water

Analytical solutions for steady progressive waves in deep and shallow water are developed and explored. The solution presentations emphasise the significant overlap in their formulations. Each theory is numerically confirmed as correct to fifth order and details of the solutions are archived in a manner that anticipates application code. Finally, the limits of validity, separate and overlapping, of the respective analytical solutions are extensively examined. It is suggested that ω²h/g ≈ 0·65 separates the region of applicability of the shallow and deep water theories.