''Added mass'' theory is usually employed in a momentum analysis. In connection with the vertical impact of a wedge, Pierson1 noted an anomaly when it is employed in an energy analysis, because the two approaches yielded different results. This led the writer to conjecture that the ''difference'' might be due to the energy dissipated in the spray sheets. The analysis which follows appears to support that conjecture, in that the size of the spray sheets so calculated is the same as those of Pierson's [(1950), The penetration of a fluid surface by a wedge. Stevens Institute of Technology Report No. 381 (July) (IAS paper FF-3)] numerical calculations. The theory predicts, for both vertical wedge impact at constant vertical velocity and for the steady high speed planing of a slender prismatic hull, that half the pressure drag energy (Du(o)) appears as kinetic energy in the spray sheets, the other half being absorbed by the downward deflection of the main body of water. This does not agree with the conclusions drawn from the experimental work of Latorre and Tamiya [(1975), An experimental technique for studying the planing boat spray. Proceedings of the Fourteenth International Towing Tank Conference, Ottawa, Canada, Vol. 4, pp. 562-5711 and Latorre [(1983), Study of prismatic planing model spray and resistance components. J. Ship Res. 27, 187-196]. Nor does it agree with two-dimensional planing theory [summarized in ch. 3 of Payne [(1988), Design of High-speed Boats, Volume I: Planing. Fishergate, Inc., Annapolis, Maryland]] where all the energy appears in the spray sheet at very high Froude numbers. But the equal division of energy into spray and water deflection for slender surfaces was predicted for a high speed slender planing surface by both Wagner [(1932), Uber Stoss-und Gleitvorgange an der Oberflache von Flussig-keiten. Zeitschrift fur Angewandte Mathematic und Mechanik, Band 12, Heft 4 (August)] and Tulin [(1957), The theory of slender surfaces planing at high speeds. Schiffstechnik 4, 125-133]. Having calculated the volume of spray, we then use this to compute the spray sheet momentum flux and, hence, the additional vertical force that can be obtained by deflecting the spray sheets horizontally (as with a spray rail) and straight down, as with a spray reverser. It is found that the latter enables a wedge to develop more vertical force than a flat plate. The effect of a spray rail, in contrast, is much less, by comparison, for the deadrise angles conventionally employed in high speed boats. When applied to the steady planning of a two-dimensional flat plate, the calculation leads to a jet thickness which is identical with that calculated by Wagner [(1932), Uber Stoss-und Gleitvorgange an der Oberflache von Flussig-keiten. Zeitschrift fur Angewandte Mathematic und Mechanik, Band 12, Heft 4 (August)] with the aid of a conformal transformation. For a given trim angle, the jet thickness is proportional to the submerged length (or chord) of the two-dimensional plate. Using the same analysis, it is found that the jet thickness of a slender planing plate, again at a given trim angle, is a function only of its beam. The variation of thickness with trim angle is the same in both extremes. Because of its historical significance, and because the writer is not aware of any available English translation, Wagner's [(1932), Uber Stoss-und Gleitvorgange an der Oberflache von Flussig-keiten. Zeitschrift fur Angewandte Mathematic und Mechanik, Band 12, Heft 4 (August)] analysis of the slender planing problem is appended to this paper.
