initial conditions for the subsequent underwater motion of the missile. The most impor- 

 tant of these conditions is the change in angular velocity, or "whip," experienced by a 

 missile on water entry; design engineers have already at their disposal a variety of head- 

 shape devices which provide pitching moments of varying magnitude and sign. These 

 devices, most of which have been arrived at by empirical methods, give the designer 

 considerable control over the subsequent trajectory. 



a. Normal Entry. A theoretical study of the flow forming stage is difficult, even 

 with normal entry, as it involves three independent variables: two space coordinates 

 and time. An additional difficulty is that the exact boundary conditions on the free 

 surface are non-linear. For these reasons no exact mathematical solution for a water 

 entry problem, even with the simplest headshape, can be expected. The best we can 

 hope for are either approximate solutions or solutions of the exact equations obtained by 

 numerical methods. 



The problem is simplified for wedges and cones entering at a constant speed 

 since the flow pattern produced by such bodies may be pseudo-stationary; under these 

 conditions the pattern of the disturbed flow increases in size uniformly with the depth 

 of penetration. The vertical impact of a wedge on a water surface was investigated 

 many years ago by Wagner (1933); this problem was further developed by Pierson 

 (1950) who calculated in detail the conditions on wedges of several angles. Likewise, 

 Schiffman and Spencer (1951) considered the similar problem of the vertical entry 

 of a cone and dealt in detail with the case of a cone of 60 degs. semi-angle. Solutions 

 of problems of this type can be obtained in a number of ways; for example Schiffman 



WEDGE - PIERSON 

 CONE - SHIFFMAN, SPENCER 

 t KILLMAH. 



10 20 50 4-0 



Figure 2. Pressure and velocity distributions on 60° semi-angle wedge and cone during water entry. 



217 



