231 



past a breaking-front into the region ahead of it. The production of the 

 cavitated region is thus a consequence solely of processes occurring in the 

 unbroken water or on its boundaries, as a result of which the pressure in 

 successive portions is lowered to the breaking-pressure; the front is merely 

 a particular surface of constant pressure advancing in accordance with the 

 ordinary equations of wave propagation. Its speed of advance is found to 

 be (2) 



dvi dvy dv^ 

 ir _ 2 dx dy dz roi 



^^-P' ai f^^ 



dn 



where dp/dn denotes the normal pressure gradient or the rate of Increase of 

 the pressure along a normal to the front drawn into the unbroken water, and 

 v^, Vy, Vj are components of the particle velocity taken In the directions 

 of cartesian axes. In order that the pressure may sink as the front ap- 

 proaches, the numerator in Equation [2] must be positive. 



If p^ is less than p^, there is a discontinuity of pressure at the 

 breaking-front, so that the pressure is pj ahead of it and p^ behind it. 

 Thus, while the front is traversing an element of water, the element is 

 kicked forward by the excess of pressure acting on its rear face. If v^ is 

 the particle velocity just ahead of the front, and if v ^^^ is the component of 

 this velocity in a direction perpendicular to the front or to the boundary of 

 the cavitated region, taken positive toward the unbroken water, and if Vc and 

 v^^ denote corresponding quantities in the cavitated region just behind the 

 front, then the analysis (2) indicates that 



Components of velocity parallel to the boundary are, however, left unaltered. 

 Thus, if p^^ = p^, the particle velocity Is left entirely unaltered by the 

 passage of the breaking-front, but if p Is less than p^ there is a discon- 

 tinuity in Its component perpendicular to the front. 



THE CAVITATED REGION 



Conditions within the region of cavitation must be comparatively 

 simple. Since there is no pressure gradient, and the pressure is uniformly 

 equal to p^, the particle velocity must be constant in time, retaining the 

 value at which It was left by the passage of the breaking-front. 



If p^ = p^, the particle velocity, being unaltered by the passage 

 of the breaking-front, retains its expanding character. In this case, ac- 

 cording to our assumptions, the fraction 77 of the space that is occupied by 



