Underwater Explosion Bubbles I the Effect of Compressibility of the Water
Underwater Explosion Bubbles I the Effect of Compressibility of the Water
Joseph B Keller
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Wc first note 8. That this fiels has just cv;o singular points--a saddle point at a = 0, V = _-, and a spiral point at a - 1, v = 0. . We next observe that v'(a) bocoiiies infinite on the lines a = 0, v = and V = c and it becordes zero on the line h(v, a) = 0, It also changes sign on crossing those lines. This information sxiXfices to determine the topological character of the direction field. In order to present a more precise sketch of this field, vre vjill first cxaninc the curve h(v, a) = 0.... The equation h(v, a) = mas'- bo rewritten as (28) a^Y = 2(3Y-l)(v-:r-^)[(v-c)^-(2+3c^)(v-c) -iic-2c^]"^ . The denominator has only one real root, and thus a becomes infinite for only one value of v, provided that -(|+c^)^+ (2c+c^)^>0. This is satisfied if c >7t(/T-4) = . 103, which we assume to be the case. The root ^ ' 8 is then easily seen to lie in the interval 3c integrating both forwards and backwards, in -time from this point. •J The values y ~ 1»25> P = 1 g/cm-^, C = l, if85 n. /sGC wore vised.
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