ANALYTICAL BACKGROUND 



RANKINE OVOID 



A Rankine ovoid is the body formed by the stagnation streamline in a uniform flow 

 about a source-sink pair oriented parallel to the flow; see Figure 1. The flow is in the neg- 

 ative z-direction, the source of strength M/in is located at (c, 0), and a sink of strength 

 -M/in is located at (-c, 0). The equation of a Rankine ovoid may be written in the form 



y 2 + (cos 6> - cos 6.) = [1] 



where the coordinate system is shown in Figure 1. The half breadth h and the half length I 

 of a Rankine ovoid are obtained from the following set of equations: 7 



M 



a 2 - c 2 ) 2 = d — [2] 



2c 2nUh : 



[3] 



JlFTc 2 M 



A table of offsets for a 7 to 1 Rankine ovoid is also included in Figure 1. 



STATIONARY-PHASE WAVE HEIGHTS 



The steady-state wavemaking of a source moving beneath a free surface in the region 

 far downstream may be approximated by a stationary-phase expression. The accuracy of this 

 approximation is poor near the body but improves as the downstream distance increases. 

 From Equation [11] of Reference 5, the wave height £ on the centerline at any distance R 

 from the source is 



M 

 £ = 4 — (2jrg/RV 2 ) 1/2 exp(-gf/U 2 ) cos(gR/U 2 + ff /4) [4] 



where g is the acceleration of gravity, 



U is the velocity, 



M is the source strength, and 



/ is the submergence depth of the source. 



