312 MOTION OF THE GAS SPHERE 



compared with a during the expansion of the bubble, but increases 

 rapidly as the bubble contracts, becoming so large near the minimum 

 (greater than a) that the whole calculation is invalid. This result, 

 which is similar to that found by Penney and Price, thus again shows 

 the inadequacy of assuming a spherical bubble near its minimum size, 

 and indicates also the difficulty of obtaining better methods of ap- 

 proximation. The failure of theory is closely related to the existence 

 of a large upward velocity at these times, and so should be least when 

 the vertical motion is least: at great depths or close enough to surfaces 

 that the upward effect of gravity is compensated. 



8.8. Effects of Boundary Surfaces: The Method of Images 



The mathematical developments which have so far been made were 

 all based on the assumption that the gaseous explosion products moved 

 in a medium of infinite extent, and the flow of water around the gas 

 sphere was therefore restricted only by the condition that the velocity 

 vanish sufficiently rapidly at infinity. In actual practice, however, the 

 fluid medium is always limited in extent by the natural boundaries of 

 the surface and bottom, and may also be limited by artificial boundaries 

 such as the wall of a tank or the hull of a ship. The simplest cases ap- 

 proximating actual conditions are of an infinite rigid boundary (e.g., 

 the sea bottom), to which the flow must be parallel, and a free surface 

 (surface of the water), at all points on which the pressure is the same, 

 it being supposed that any pressure differences which might otherwise 

 develop are instantly relieved by displacement of the surface. Neither 

 of these boundary conditions is satisfied by the results for radial and 

 axial gravity flow around a spherical cavity in a fluid, and the motion of 

 the cavity and surrounding water must therefore be appropriately modi- 

 fied to permit their being satisfied. The differences in the resulting 

 motion will evidently be greater as the region of large flow velocities 

 near the bubble surface approaches either the surface or the bottom. 

 These boundaries will thus appear to exert appreciable forces on the 

 bubble if they are near the bubble during its motion. The evaluation 

 of such effects is most easily made by the use of the standard method of 

 sources and images to be described.^ 



A. Hydrodynamical sources. The motion of the gas sphere bound- 

 ary can be analyzed as the resultant of radial pulsation of the sphere 

 superimposed on a translational velocity of its center. Considering 

 first the radial motion, it is convenient to think of an outward radial 

 flow of the surrounding water as being produced by a simple source of 

 fluid at the center of the sphere, the strength M of this source being 

 defined as the fraction (l/47r) of the volume of fluid emitted in unit time. 



' These methods are given in more detail in standard texts on hydrodynamics, 

 for example the book by Milno-Thomson (74). 



