Newman and Tuck 



this. In particular for u = 0, K^ j^ reduces to the zero speed kernel of Part II, 

 Eq. (2.12), i.e.. 



K(i)(x,co,0) = - -^ 



1 c^ 



+ Y„ 



2iJo 



(3.18) 



On the other hand, for zero frequency, K, ^. reduces (as it must) to the steady 

 kernel K^ J^) of Eq. (3.11), i.e.. 



K(,)(x,0,U) - -\j^ 



Ho(^).(2.sgnx)Y„(|-i^|) 



(3.19) 



For non-zero values of co and U, K^ has not yet been tabulated, and further 

 work is needed to investigate the properties of this function. We may note that 

 by a suitable non-dimensionalization K^ j^Cx.oj.U) may be expressed as a func- 

 tion of two dimensionless variables only, for instance 



K, , , (x, cd, U) = — , function ot , — 



(i> g \ g g 



It is easy to see that K( j^ has the familiar singularity when wu/g = 1/4, which 

 may complicate the task of numerical evaluation of the kernel. 



Pressure Calculation 



From Bernoulli's equation the hydrodynamic portion of the pressure field is 



-ia;0 + - 



1 I" ' I 2 _ 1 ^2 



(3.20) 



(here "-iw" may best be interpreted simply as the operator "B/3t"), which gives 

 on expanding with respect to a that 



1 I I 2 1 2 



2" ly(^(0)(x,y, z)| - 2" U - i^'?^( 1) + V(?5)^o^(x,y, z) 



• V0(i)(x,y,z,t) + 0(a2) 



that is, 



P = P(0)('''y'Z) + P(i)(x,y,z, t) + 

 where p^q^ is the steady pressure field 



(oy 



2.1,2 



(3.21) 



while p( 1) is the term of first order in a, namely 



148 



