208 
49 APPENDIX II 
3. PRESSURE AND IMPULSE 
[7]. At large r, the Bernouilli cerm(1/2)pu? can be neglected. Hence, integrating, 
we obtain 
J(w— n)dt = prdw = 2 A(riu,) [23] 
where u denotes the velocity of the liquid at the point in question, and A the total 
change during the time covered by the integration. The value of r is held constant 
here whereas 7, varies with the time. 
The total positive impulse during an oscillation can then be found, provided 
we know the maximum value of 7,7u,. From[14] 
1 
2 Ww 2 p 4\2 [24] 
rus = (Cr, =F 2Q7p Ty a i) 
This is a maximum for such a value of r, that 
Ww iS fy, 8 
ee EE SE Eos 0 
27 p 3: up" 4 
In the case of large oscillations, i.e., large r2/7m, this last equation can be solved 
approximately. For then we can write it, using [20], 
2 Do sels 8 Dy Bg W 
es Dee fe 2p 
The value, r, = sa Tra, is too large, for it makes the left side of this equation zero; 
but for such values of r,, as we have seen, W/2mp is small as compared with C or (2p,/ 
3p) r3. Hence a small decrease in r, will satisfy the equation. Neglecting this de- 
crease, we have, therefore, approximately, 
sage 
at 
Inserting this value of r, in [24], dropping the term in W, and using [21a,b], we find 
1 eat Ba 
2 2\2 ~6 / 3W, \3 
(7% Mp) max = (=) Po 4) 
Inserting twice this value for A(7?u,) in [23], we obtain finally for the total positive 
impulse, during the part of the oscillation in which p> Dy at a large distance rv from 
the center, 
SIN oe a ase 
— 0.432 2 6 3 
[(e—apat = Pr W, (25] 
r 
