431 



A SIMPLIFIED THEORY OF THE EFFECT OF SURFACES 

 ON THE MOTION OF AN EXPLOSION BUBBLE 



A. R. Bryant 



October 1944 



Summary . 



A simple physical explanation of the effect of surfaces on the displacement of an explosion 

 bubble is suggested. This explanation leads to the same quantitative expression as the treatments 

 given by Conyers Herring and by Taylor. It is shewn that in order to calculate the effect of any 

 given surface it is only necessary to determine the space gradient of the velocity potential due 

 to the 'image' sources 'induced" by a unit source at the explosion centre. A table of this 

 factor for a number of comnon surfaces is added for convenience. 



Introduction. 



The purpose of this note is to suggest a simple physical explanation of the influence of 

 surfaces on an explosion bubble. It will be shown that this explanation leads, with comparatively 

 simple mathematics, to the same quantitative expressions as the treatment given by Conyers Herring (l) 

 and Taylor (2). 



For purposes of explanation it will be convenient to consider a simple case first, viz. the 

 influence of an infinite rigid plane on an explosion bubble. The generalization to any arbitrary 

 Surface follows at once. For the moment the effect of gravity will be ignored. 



Figure 1. 



In Figure I, is the centre of an explosion bubble of radius a, assumed small compared to 

 its distance d from the rigid plane AB. The motion of the water outside the bubble due to its 

 pulsation Is the same ^s that produced by a point 'source' at of strength 



177 a J 



(1) 



The effect of the plane surface AB is then the same as that produced by a source e at the mirror 

 image point o . 



The image source e at will produce a velocity in the water everywhere directed radially 

 away from or to according as the sign of e is positive or negative, and this velocity is 

 superimposed on the radial flow due to the explosion bubble at 0. Hence the water In the 

 neighbourhood of will have a net velocity U, towards given by 



*1 



2j} 

 3 



(2) 



where <^ is the velocity potential due to the image source at the origin of the x 

 co-ordinate being taken at 0^, 



