246 -11- 



Appendlx TI« 



Page 



The Velocity Potential for a Pulsating Sphere Moving 

 Perpendicularly to a Wall. 



!• Statement of the problem 41 



2. Some theorems on images 42 



3. The construction of 4> 44 



' o 



4» The construction of <|>, 



49 



5» The kinetic energy of the water 51 



6» Approximate theory for a sphere pulsating between 54 

 a rigid wall and a free siirface 



Appendix IIIo 



Numerical Evaluation of Some Definite Integrals. 57 



TABLE O F GRAPHS 



Figure !• Scaling factor L 61 



Figure 2. Distance from bottom to produce maximum pressure 63 



Figure 3. Pressure factor for mine of 1500 lbs. of T.N.T. 65 

 in water 150 ft. deep 



Figure 4. Pressure factor at sea level for mine of 1500 67 

 lbs. of T. N. T. in water 150 ft. deep 



Figure 5. Dependence _of internal energy (u) on 69 

 momentum ( s ) 



Figure 6. Dependence_of pressure factor (q) on 71 

 momentum (s) 



Flgvo'e 7. The displacement of the bubble 73 



Figure 8. Radius of bubble; case 7 75 



Figure 9. Position of bubble; case 1 77 



Figvire 10. Position of bubble; case 3 79 



Figure 11. Position of bubble; case 4 81 



Summary of Formulas 83 



Bibliography 85 



