Ogilvie 



<^(x,y,z)- — ^\ dke 0- (k) — — -e 



= ie'''-'''V(x) + ... . (3-24) 



With the time dependence reintroduced, we have: 



<t>(x,y,z,t)=- Re {io-(x)e'''e*^'^^-»'y^ } +... . (3-25) 



This approximation represents a travelling wave; for y > , in 

 particular, the wave is moving away from the line of sources. For 

 y < 0, we must start over, closing the contour for the i integration 

 on the lower side of the i plane. It turns out that the result is the 

 same if only we replace y by |y|. Thus, we have an outgoing wave 

 for y < also. In both cases, the outgoing wave has the form 

 appropriate for a gravity wave in two dimensions. 



In the approximations above, it is necessary to require that 

 r be not extraordinarily large; if one assumes that r = 0(1) and 

 CO = 0(c''/^), then the above results follow logically. Thus the very 

 simple approximation above is valid even in part of the far field. It 

 Is an example of the well-known physical principle that nearly 

 unidirectional waves can be generated if the wave generator is much 

 larger than a wave length. 



If we let r = 0(€), no change occurs in this approximation. 

 Since v = 0(e''), it is not permissible to expand the exponential 

 functions even when y and z are 0(e). The only effect of passing 

 from far field to near field now is to change the scale of the observed 

 wave motion. 



This far -field analysis has provided information that was 

 probably quite obvious intuitively: In the near -field, the condition 

 at infinity is that there should be outgoing, two-dimensioned , gravity 

 waves*. With this information in hand, we can move on to the for- 

 mulation and solution of the near -field problem. 



In the near-field, we make the usual slender-body assump- 

 tions: 



I cannot imagine that anyone would ever have doubted this fact, 

 even without the above analysis to show it. But in the forward- speed 

 problem, the condition at infinity in the near field is not at all ob- 

 vious , and such an analysis seems necessary. 



752 



