182 



EXPLOSIONS AS SOURCES OF SOUND 



8.4.3 



The Law of Similarity 



According to the theory outlined in Sections 8.4.1 

 and 8.4.2 the disturbance created in the water by an 

 explosion is uniquely determined by : 



1. The forces which the explosive gases exert on 

 the water near them. 



2. The Hugoniot equations (16), (17), and (18), 

 which hold across the advancing shock front. 



3. The equation of continuity [equation (2) of 

 Section 2.1.1]. 



4. The equations of motion (2), (3), and (4), 

 which hold true to a very good approximation at all 

 points of the water except points in the shock front. 



5. The thermodynamic properties of the water, 

 that is, the relations, such as the equation of state, be- 

 tween pressure, density, and energy. Moreover, from 

 what has been said previously concerning the nature 

 of detonation waves, it is likely that the course of 

 events within the explosive material itself is deter- 

 mined by a similar set of equations, so that factor 1 

 can be derived from laws of the same type as 2, 

 3, 4, and 5. Now suppose a disturbance to be 

 given which satisfies all these equations, and is 

 described by 



V = vix,y,z,t) 

 P = p{^,y,z,t) 



Ur = u^{x,y,z,t) 

 Uy = Uy{x,y,z,t) 

 Uz = Uz{x,y,z,t) 



Then it can easily be verified by substitution that all 

 the equations mentioned in 2, 3, and 4 outlined 

 previously and the laws 5 as well, will be satisfied 

 at all points except those in a thin layer at the shock 

 front, by a disturbance described by p' ,p' ,u!„Uy,ul 

 where 



p'(x,y,z,t) = pipx,fiy,Pz,m 



p'{x,y,z,t) = pil3x,Py,Pz,pt) 



u'^{x,y,z,t) = u^{^xfiy,pzfit) 



etc., that is, by a disturbance identical with the first 

 except that the distance and time scales have been 

 changed by a factor fi. A scale relationship of this 

 sort may be expected to hold both in the explosive 

 material and in the water, and if the linear dimension 

 D' of the explosive in the second case is made equal 

 to /3 times the corresponding dimension D in the first 

 case, the disturbances in both the explosive and the 

 water can be scaled together. 



This law of similarity relating to disturbances pro- 



duced by different quantities of the same explosive 

 has been fairly accurately verified experimentally at 

 ranges for which the peak pressure in the shock wave, 

 is of the order of an atmosphere and above.' Provided 

 the shape of the explosive charge and the position at 

 which the detonation is initiated are the same on the 

 two scales, an appreciable departure from the simi- 

 larity law could be caused only by failure of the equa- 

 tions of motion (2), (3), and (4) to hold behind the 

 shock front in the water, or by a failure of either the 

 water or the explosive to have thermodynamic prop- 

 erties independent of the scale of times involved. A 

 phenomenon of the latter type might conceivably 

 occur, for example, in an aluminized explosive, if the 

 reaction of the grains of aluminum with the hot gases 

 were so slow that the reaction occupied an appreciable 

 part of the volume behind the detonation front. 

 However, the fact that significant departures from 

 the scaling law have not been observed at the ranges 

 mentioned indicates that such phenomena are not 

 serious. As for the possibility of failure of the basic 

 assumptions to be fulfilled in the water, such a failure 

 could be caused only by bubbles or other extraordi- 

 nary dissipative mechanisms; as far as is known, the 

 effect of these only becomes appreciable at very long 

 ranges (see Section 9.2.1). Ordinary viscosity and 

 heat conduction can be shown to have a negligible 

 effect on the scale used in experimental work. It 

 should be remembered, of course, that the derivation 

 we have given of the similarity rule does not apply 

 in the very thin region of the shock front in which the 

 abrupt rise of pressure takes place; as will be shown 

 in the next section, the thickness of this region does 

 not ordinarily scale proportionally to the factor /3 

 used previously. 



8.4.4 Theoretical Thickness of a 

 Shock Front 



We have seen in the preceding sections that it is 

 for many purposes unnecessary to consider the de- 

 tails of phenomena occurring in a shock front, and 

 that many useful conclusions can be drawn by think- 

 ing of a shock front merely as a surface in the fluid 

 across which the pressure and other quantities change 

 discontinuously. However, these conclusions will be 

 valid only if the thickness of the region in which the 

 pressure rise takes place is very small; and to make 

 our theoretical discussion complete we should verify 

 that this is the case. Moreover, a study of the factors 



