Subrahmanyan Chandrasekhar

“ And it appears not to have been recognized that there exists a special class of shock pulses for which the problem of decay admits of an explicit solution. The case in question arises when shock waves of moderate intensities (i.e., shocks with Mach numbers less than 1.5) are considered. For, under these circumstances (when the ratio of pressures on either side of the shock front is less than 2.5) the increase in entropy of an element of gas as it crosses the shock front can be ignored. ”

“ For while a reduction in the effective target area requires a greater surface density of fragments this can be achieved by even a slight reduction in since by so doing we bring into range the smaller fragments which increase in number very rapidly with decreasing mass. ”

“ Under these circumstances a new shock wave (the Mach wave) is formed and moreover in the region included between the reflected and the Mach wave there is a surface of discontinuity . Across the normal components of the velocities remain continuous while there is a discontinuity in the tangential components: in other words, is a vortex sheet. Strictly speaking, the idealization of the Mach effect in terms of plane shock waves and a plane of discontinuity cannot be valid. ”

“ We allow the micro waves of a given frequency emitted by a transmitter to be reflected from a projectile in motion. On account of the Doppler effect these reflected waves have frequencies slightly different from that of the incident waves (see equation [1] below) . Consequently, if we now superpose the reflected wave trains on a wave train having the same frequency as the incident waves we shall obtain a beat phenomenon. It is evident that quantitative measures based on the existence of this beat phenomenon should enable us to determine the dynamical characteristics of the trajectory. ”

“ Essentially the existence of an optimum height for burst depends on the extent to which the area on the ground where we are super-efficient (region I of §4) can be reduced. It cannot however be concluded that the optimum height corresponds to reducing this area to zero. For the fragments which are most numerous are those with small masses and these have ranges of the order of itself. Thus, when , the fragments which are effective (i.e., with velocities greater than [Cf. eq. 1] ) and which arrive on the ground directly below the point of burst must have masses greater than 0.0021 lb. ”

“ Fragments with in this neighborhood are the most numerous, but their effect is confined to a very small area. Accordingly we must go to somewhat larger masses (with correspondingly longer ranges) to be really efficient. In other words, at the optimum height we must necessarily be super efficient in the regions directly below the point of burst. In other words the area over which we are super efficient must be reduced as much as is compatible with the circumstances of the problem. ”

“ In equations (3) , θ denotes the temperature which the element of gas under consideration would have when it is reduced to a standard pressure adiabatically. It is evident that θ remains constant during the motion of any element of gas (except when it crosses a discontinuity) . ”

“ Before proceeding further with the interpretation of when is a function of time we should perhaps remark that in order that the procedure may at all have an unambiguous meaning it is necessary that two conditions be satisfied. ”

“ It is possible that the laws
are more general than their derivations from equation (31) and (32) would suggest.
Table II
⁠3. On the General Solution of Shock-Pulses with Q = Constant. In §2 we have discussed the special case of linear shock pulses which are characterized by Q = Constant throughout. In this section we shall briefly indicate how the most general shock pulses under the circumstances Q = constant can be constructed. For this purpose we require the general integral of equation (16) . ”

“ The method described in the preceding paragraph is suitable for the purpose of isolating numerically configurations of three shocks in equilibrium, particularly if tables of solutions of the appropriate Rankine-Hugoniot equations are available. ”

“ Again, it it clear that since a change in the half angle of the spray can be formally incorporated into the calculations as a change in target area, the optimum height is relatively insensitive also to changes in . For a given total number of fragments, the surface density varies inversely as . Hence, other things being equal a change in alters the surface densities by a constant factor and the final efficiency will be clearly the same if were kept unaltered but the target area increased by the same factor. ”