singular Perturbation Problems in Ship Hydrodynamics 



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Flg. (3-4). Contour of Integration Defining the 

 Velocity Potential of a Line of Pul- 

 sating Sources: Forward-Speed Case 



[-=^r-[^r-. 



and the contour is indented as shown at the poles on the real axis 

 in the i plane. The contour C extends from -co to +00. The 

 poles in the i plane all fall on the imaginary axis if k, < k < k2, 

 and then C is the entire real axis, with no special interpretations 

 being necessary. 



The above expression for i|j(x,y,z,t) is a one-term outer 

 expansion, but it is not a consistent one-term expansion. It is 

 shown by Ogilvie and Tuck that a much simpler expression is pos- 

 sible if r = (y^ + z^)'^^ is 0(i) as e -^ 0; emphasis should be 

 placed here on the restriction that r is not extraordinarily large. 

 If (T (k) is restricted in a rather reasonable way, it follows that: 



f*CD 2 



, / j.\ ^ "^* \ ji 'kx */i \ v(z-ilyl)(l + Uk/a)) ,, ^n^ 



4i(x,y,z,t) ~ •^— e \ dk e 0- (k) e . (3-39) 



^■f J-00 



We can take this as our one-term outer expansion of ij;(x,y ,z ,t). 



The inner expansion of this expression is obtained by letting 

 r = 0(e). Then we find that: 



4;(x,y,z,t)~ie"'^''^"*'^'^ [ o-(x) - 2i((^U/g)(z - i | y | )o-'(x)] . (3-40) 



Since vr = 0(i), It Is not appropriate to expand the exponential 

 function further. This is a two-term inner expansion of the outer 

 expansion of ijj; the first term represents an outgoing, two-dimen- 

 sional, gravity wave, just as in the zero- speed problem (see (3-25)), 

 but the second term represents a wave motion in which the amplitude 



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