Sec. 613 



PREDICTED BEHAVIOR IN CONFINED WATERS 



391 



Range. The quantitative eiTect of shallow water 

 of unlimited extent on the resistance and speed 

 of ships is expressed in at least three different 

 ways, when the basis of comparison is a combina- 

 tion of resistance and speed in unlimited deep 

 water: 



(1) The effect of limited depth on ship speed at 

 constant (or at a given) resistance 



(2) The effect of limited depth upon resistance 

 for a speed equal to a given deep-water speed 



(3) Depths of water of unlimited extent beyond 

 which there is no shallow-water effect on either 

 resistance or speed. 



The discussion in this section is limited to ship 

 speeds less than the critical speed Cc of a wave of 

 translation in water of the given depth h, where 

 Cc = {9h)°-\ Table 72.a of Sec. 72.3 lists values 

 of Cc for a range of depths h from 2 through 40 ft. 



The problem of the wavemaking resistance 

 only in shallow water, and in confined waters as 

 well, has been tackled on a purely analytical or 

 theoretical basis by Sir Thomas H. Havelock, 

 J. K. Lunde, and others. One of Lunde's contri- 

 butions is his paper "On the Linearized Theory 

 of Wave Resistance for Displacement Ships in 

 Steady and Accelerated Motion" [SNAME, 1951], 

 in which Part 2, on pages 50 through 60, applies 

 directly to resistance due to wavemaking in 

 shallow water of imlimited extent as well as in a 

 canal. A more recent contribution is that of A. A. 

 Kostyukov entitled "Resistance of Bodies in a 

 Fluid to Motion near a Vertical Wall" (in Russian) 

 [Dokladi Akad. Nauk, SSSR (N.S.) 99, 1954, 

 pp. 349-352]. This paper is abstracted briefly in 

 Applied Mechanics Reviews, December 1955, 

 page 534, number 3846. 



None of the existing (1955) analyses in the 

 foregoing category is in a form to be readily 

 useful to the marine architect. There have been 

 a number of much more practical solutions pro- 

 posed for this problem but many of them do not 

 embody parameters which appear logical or 

 scientific. 



What appears to be the most satisfactory 

 method of deahng quantitatively with these 

 matters was developed some years ago in Ger- 

 many by Otto Schhchting ["Schiffswiderstand 

 auf beschrankter Wassertiefe; Widerstand von 

 Seeschiffen auf flachem Wasser (Resistance of 

 Seagoing Vessels in Shallow Water)," STG, 1934, 

 Vol. 35, pp. 127-148; EngUsh version in EMB 

 Transl. 56, Jan 1940; also van Lammeren, 



W. P. A., RPSS, 1948, p. 56]. This method has a 

 partly theoretical and partly experimental basis. 

 Some of its assumptions are open to question 

 but it has the merit that it works, as an engineer- 

 ing solution to the shallow- and confined-water 

 problems. It will undoubtedly give way in time 

 to a more rigorous treatment but in the meantime 

 it produces results in fair agreement with observed 

 model and ship data. 



The basis of 0. Schlichting's method is illus- 

 trated graphically by Fig. 61.B, adapted from 

 one of Schlichting's illustrations (Fig. 6 in the 

 reference cited). The point Ai represents the 

 relationship between the total resistance Rt^ in 

 unrestricted deep water and the corresponding 

 ship speed V„ achieved with a given power. This 

 may be assumed as the customary design point 

 for a normal deep-water ship. Unless otherwise 

 indicated, the subscript <» (infinity) in this 

 chapter applies to the value of a designated 

 quantity in water of infinite depth and width. 

 The total resistance Rt«, is composed of the usual 

 friction resistance R^^ and a pressure resistance 

 Rwo, , which is assumed by Schlichting to be 

 due entirely to wavemaking. These components 

 are indicated at the right of the diagram. 



The wavemaking resistance Rwo is associated 

 with a train of deep-water waves, belonging to 

 the Velox system, whose speed is the same as the 

 ship speed, so that the crests and the troughs 

 occupy certain fixed positions relative to the ship. 

 The fixed relationship between the wave length 

 L,ra. and the speed c„ of these waves, assuming 

 they are of trochoidal character and form, is 

 given under (2) of Sec. 48.4. Squaring the equality 

 given there, 



2-K 



cl=Vl^ 



In shallow water the speed of a trochoidal 

 wave of the same length Lwa, is less than V„ . If 

 Ca is this speed for a water depth h, the ratio 

 between c^ in shallow water and Ca, or F„ in 

 deep water is, from Sec. 18.10 and Fig. 48.N 

 of Sec. 48.15, expressed by 





= <;tanh (^1^2 



(61. i) 



When the ship passes from deep water into 

 shallow water of depth h, it can not make the 



