OgiZvie 



near field, it is still not possible to see the waves that exist far 

 away, but the latter have the effect of making the incident stream 

 appear to be rotated somewhat. It is like a downwash effect (although 

 the physical origin is quite different). 



If one were given a planing problem such as we formulated 

 early in this section, with the incident stream and all geometric 

 parameters prescribed, it would be necessary to solve for the 

 parameters a and b. One equation relating these parameters has 

 already been mentioned, namely, the equation relating the length of 

 the plate to these parameters. The other equation comes from the 

 expression (which was not written out here) for h as a function of 

 a and b . 



Rispin avoided much tedious algebra by solving the inverse 

 problem. He assumed that a, b, and c were given, then solved 

 to find h. He also had to treat the angle of attack as an unknown 

 quantity, and he found an asymptotic expansion for it. (Note that 

 only two of the basic parameters can be prescribed arbitrarily, 

 unless we are prepared also to let i be an unknown quantity.) 



One final comment on Rispin's work must be made. He finds 

 terms of six orders of magnitude: 0(1) , 0(p log p) , 0(p) , 0(p^ log^|3) , 

 0(p log P) , and 0(p ). But he finds also that they cannot be deter- 

 mined one at a time. Rather, they must be taken in groups: a) the 

 0(1) terms, b) the terms linear in p (the logarithm being ignored), 

 and c) the terms involving p , This is the same kind of matching 

 procedure that would have been used if he had adopted the working 

 rule that logarithms should be treated as if they were 0(1), (See 

 Section 1.2.) 



5.2. Flow Around Bluff Body in Free Surface 



A problem related to that of Rispin [ 1966] and Wu [ 1967] 

 has been studied by Dagan and Tulin [ 1969] . They have concerned 

 themselves with the flow at the bow of a blunt ship, where any kind 

 of linearization procedure must be completely wrong. In order to 

 handle such a situation, they have adopted essentially the same pro- 

 cedure that the previous authors used, namely, they set up inner- 

 and outer- expansion problems in which the nonlinearity is confined 

 initially to the near field and the effects of gravity are confined 

 Initially to the far field. Then, by limiting their study to a two 

 dimensional problem, the nonlinear near-field problem can be solved 

 by the hodograph method, and the far-field problem Is a simple vari- 

 ation of a well-studied problem In water-wave theory. The geometry 

 of their problem Is shown In Fig. (5- 5), which Is reproduced from 

 their paper. They argue that at very low speed there will be a 

 smooth flow up to and then down under the bow, with a stagnation 

 point at the location of highest free- surface rise, but that that flow 

 becomes unstable as speed Increases, until finally a jet forms, as 



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