2. Hydrodynamical Relations 



The purpose of this chapter is to develop from first principles the 

 equations describing the motion of a fluid, whether liquid or gas, which 

 are the basis of the more detailed theories of the phenomena in under- 

 water explosions, to discuss the implications of these equations, and to 

 indicate the manner of development appropriate to problems later 

 treated more explicitly. 



2.1. The Differential Equations for Ideal Fluids 



As a first step in discussing the propagation of waves in fluids, it is 

 necessary to put the basic laws of mechanics into a suitable mathe- 

 matical form. It is assumed in what follows that the fluid is ideal in 

 the sense that viscous stresses and effects of heat conduction may be 

 neglected. The discussion is further restricted to regions of space and 

 instants of time for which there are no discontinuities of pressure, 

 velocity of the fluid, or internal energy. 



A. Conservation of mass. The simplest restriction on the motion of 

 the fluid is the conservation of mass. If we consider a small fixed 

 region of space in the interior of the fluid, it must be true that any 

 change in mass of fluid contained in the volume is equal to the net 

 quantity of fluid which flows through the boundary surface. If the 

 region is a small cube of volume dxdydz, the change in mass in a time dt 

 resulting from change in the density p at the point x,y,z is 



— dt dxdydz 



di 



If such a change occurs it must be as a result of motion in the fluid. 

 Let the velocity of a point moving with the fluid be described by its 

 three components u, v, w, in the x, y, z directions, these components 

 being functions of the space coordinates and time. The net transport 

 of fluid into the fixed volume in time dt resulting from motion in the x 

 direction is the difference in amounts flowing through the two faces of 

 area dydz and is given by 



\_{pu):c — [pu)x^dx^ dtdydz = (pu) dt dxdydz 



dx 



higher order terms in the expansion of {pu)x+dx disappearing in the 

 limit of small displacements. Similar terms are obtained for the other 



