Tuck and Taylor 



^"^1Tf2 ^2-^5) 



where F, d, s, h are again defined by (2.6) - (2,9) respectively, 

 although all these quantities may now in principle vary with station 

 coordinate x. However, if they do not, (2,15) can easily be shown 

 to be the result of direct approximation of (2,5) for small s. 



The most interesting feature of (2, 15) is of course the singu- 

 larity at the critical Froude number F = 1 , which is to be expected 

 from the fact that the former transcritical region F, < F < F- has 

 shrunk down to an isolated "forbidden" Froude number at F = 1, We 

 shall make use of linearized results like (2.15) throughout the re- 

 mainder of this paper; however, it is well to bear in mind in each 

 case that we may expect singularities at the critical Froude number 

 and that, should these be of concern, they may be explained, studied 

 or removed by non-linear considerations similar to those of the 

 present section. 



III. TWO-DIMENSIONAL THEORY OF SQUAT IN WIDE, SHALLOW 

 WATER 



The theories of the previous section are useful only in widths 

 of water comparable with the beam of the ship. Since the important 

 blockage parameter is the ratio of the maximum ship cross -section 

 area to the cross-section area of the channel, naive use of these 

 theories for very wide channels leads to the conclusion that the squat 

 effect tends to zero for a given ship as the channel width tends to 

 infinity. But of course the basic Bernoulli effect must still be present, 

 even in an infinite expanse of water, so that there will still be squat, 

 and indeed substantial squat in this case. 



Analysis of shallow-water flow past ship-like bodies in in- 

 finitely-wide water was first attempted by Mlchell [ 1898] In his 

 famous wave-resistance paper. The relatively greater Importance 

 of Mlchell's infinite depth formula, the derivation of which consti- 

 tutes the first part of his paper, has perhaps led to little Interest 

 being taken In the second part of the paper, where he treats a shallow 

 water problem. This Is unfortunate, since Mlchell's approach Is 

 what we might now call an aerodynamic analogy, even though his 

 paper ante-dates aerodynamics! 



The problem treated by Mlchell concerns steady flow at speed 

 U In the x-dlrectlon past an obstacle of thin cylindrical form, with 

 equation 



y = ± ib(x), |x| < i, (3.1) 



634 



