Ogilvie 



and the functions fp are to be deternnined in such a way that the 

 approximation is uniformly valid for all t. In treating this parti- 

 cular problem, Cole immediately assumes that t « T and further 

 that t/r = 1 + 0(e2). These extra assumptions speed the solution 

 considerably, but it is not clear how one would know to make them 

 if the exact solution were not available. The exact solution takes 

 the form: 



y(t;€) = e' [ cos t + (t /t) sin t] , 



in terms of the new variables. (The factor (t/r) does not depend 

 on t.) Here it is clear how the two time scales enter into the 

 solution as well as the problem. One may expect the relationship 

 betwee n t and t to be equivalent to the expansion of the quantity 

 V(l-€^). The reader is referred to Cole's book for further discus- 

 sion of the solution of such problems. 



One problem that will be discussed later is a close relative of 

 the classical problems mentioned above. The solution by Salvesen 

 [ 1969] of the higher-order problem of the wave resistance of a sub- 

 merged body leads to a situation in which the first approximation is 

 periodic downstream and that period is modified in the third-order 

 approximation. (Otherwise the waves downstream in the higher 

 approximation would grow larger and larger, without limit.) A 

 similar problem involves the oscillation of a body on the free sur- 

 face, in which the wave length of the radiated waves must be modified 

 in the third approximation. For example, see Lee [ 1968] . 



A quite different application of this method is the problem of 

 very low speed motion of a body under or on a free surface. The 

 simplest such case has been discussed by Ogilvie [ 1968] . For a 

 translating submerged body, there are two kinds of length scales: 

 length scales associated with body dimensions and submergence, 

 and the length scale U /g, which is associated with the presence 

 of the free surface. Presumably, the latter has effects primarily 

 near the free surface, in a "boundary layer" with thickness which 

 varies with U /g as that variable approaches zero. But the 

 effects of the body dimensions are also important near the free 

 surface (or at least near a part of it). Thus the effects of the two 

 length scales cannot be separated into distinct regions. A brief 

 discussion of this problem appears in Section 5.42 of the present 

 paper. 



There may be many other problems of ship hydro dynannics in 

 which this approach would be valuable. For example, many authors 

 have obtained approxiniate solutions of problems involving submerged 

 bodies by alternately satisfying a body boundary condition, then the 

 free-surface condition, then again the body condition, etc. At each 

 stage, when one condition is being satisfied, the other is being 

 violated, but it is assumed that the errors become smaller and 



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