MOTION OF THE GAS SPHERE 321 



It is interesting to note the electrostatic analogy to the hydro- 

 dynamical situation. If sources are replaced by electrical charges, the 

 velocity potential has its counterpart in the electrostatic potential, 

 and streamlines of flow correspond to electric lines of force. The case 

 of a free surface in hydrodynamics is easily seen to be equivalent to a 

 perfectly conducting plane to which lines of force must be perpen- 

 dicular, and the image of a charge in a conducting plane is a charge of 

 opposite sign at the image point in the plane. A similar but less real- 

 istic analogy can be drawn for a rigid surface : the corresponding electro- 

 static surface is a boundary separating free space from a hypothetical 

 medium of dielectric constant zero, this condition being the one for 

 which lines of force are parallel to the surface, and the image is a charge 

 of the same sign. 



The boundary condition (jj? = at a free surface is clearly only an 

 approxim.ation to the true condition expressed by Eq. (8.42) and appli- 

 cation of the method of images to satisfy <^ = involves the further 

 difficulty that, strictly, the boundary surface cannot remain fixed or 

 even plane. This is evident if it is recalled that the streamlines ending 

 on the surface are lines along which fluid flows. Any motion of fluid 

 will thus displace its boundary, in general by unequal amounts at dif- 

 ferent points, and the result is formation of surface waves (this is briefly 

 considered in section 10.1). This effect will clearly be greater when any 

 sources required are relatively closer to the surface, and use of the con- 

 dition </5 = at the undisturbed surface is a poorer approximation. 

 Application of more exact boundary conditions would, however, be a 

 matter of considerable difficulty, and as a result most developments in- 

 cluding the effect of a free surface content themselves with the simpler 

 condition. If these developments were to be carried out to the same 

 degree as the treatment for a sphere and rigid boundary, a similar set 

 of images reflected in the sphere and plane (except of course for signs) 

 would be required, but the approximations inherent in satisfying (^ = 

 at a fixed boundary do not justify such elaboration. The development 

 for a rigid boundary does, however, permit inferring the errors resulting 

 from neglect of the higher order images required for a sphere of finite 

 rather than negligible radius, and this question is discussed in section 

 8.10. 



The representation of the free surface boundary condition by a 

 single image source is so simple that it is convenient to work out to the 

 same approximation the more general case of a bubble pulsating at a 

 distance h above the bottom, and distance d below the free surface, as 

 shown in Fig. 8.16. The images of the original source in these two 

 surfaces must in turn have images in the other surfaces, which again 

 must have images, and so on. As a result an infinite set of image sources 



