54 



HYDRODYNAMICAL RELATIONS 



opposite sign in order that the net pressure be zero at the boundary, 

 and that the velocity of the surface is twice the normal component of 

 velocity in the incident wave, Avhich may be expressed as 



P' = -P, 



2u cos a 



where a is the angle of incidence. Initially, therefore, the surface of 

 water above an explosion will be thrown upward and the pressure can- 

 celled. As the wave progresses, however, the pressure in the incident 

 wave at a point near the surface decreases and the head of the rare- 

 faction wave moves down and away from the boundary and the source, 

 as sketched in Fig. 2.9. It is evident that the negative change in pres- 

 sure at this front will encounter regions in which there is a smaller value 



Reflected Wave 



Direct Wave 



(o) Direct Wove and Surface Reflection (b) Resultant Pressure -Time Curve 



Fig. 2.9 Effect of surface reflection on observed pressure below a free surface. 



of excess pressure. The net pressure behind the rarefaction will there- 

 fore become negative, and if this regime persists the water will be 

 required to develop a state of negative absolute pressure. (It is to be 

 remembered that the water at the surface is initially under a hj^dro- 

 static pressure of one atmosphere and this pressure increases at the rate 

 of one atmosphere for each thirty-three feet of depth.) The pressure- 

 time curve at a point C in Fig. 2.9(a) would, therefore, look somewhat 

 as sketched in Fig. 2.9(b), the time of arrival of the reflected wave being 

 readily calculable from the geometry. It is well known that water will 

 in favorable circumstances withstand considerable tensions, values as 

 high as 600 Ib./in.^ having been reported, but under ordinary conditions 

 much lower values are to be expected. Hence for sufficient negative 

 absolute pressure we must expect that the water mass is pulled apart 

 from itself and "holes," so to speak, are formed which prevent an in- 

 crease in tension, a region of cavitation being formed instead. The 

 pressure after this has occurred is indicated qualitatively by tlie solid 

 line of Fig. 2.9(b), the dashed line being the unrealized state of tension. 

 This "surface cutoff" effect plays a considerable role in the surface 



