176 



EXPLOSIONS AS SOURCES OF SOUND 



explosions which depend on other factors beside the 

 pressure jump will be treated in Section 8.5. 



It is a familiar fact that the laws of acoustics are a 

 limiting case of the laws of hydrodynamics for a com- 

 pressible fluid and are valid only in the limit of very 

 small amplitudes of pressure and velocity. According 

 to these laws of acoustics, all pressure disturbances 

 are transmitted as waves with velocity c = (dp /dp)', 

 the derivative being understood, in the case of all 

 ordinary fluids, to relate to the change of pressure p 

 with density p in an adiabatic change. For the special 

 case of a plane wave traveling in the positive x direc- 

 tion, the pressure in the acoustic approximation is 

 simply a function of (x — ct), where t is the time; any 

 such wave is thus propagated forward with velocity c 

 without change of shape. For a disturbance of large 

 amplitude this is no longer true. The shape of the 

 wave will, in general, change as it progresses. The 

 way in which the changes of shape take place can be 

 calculated by a method of reasoning due to Riemann, 

 the mathematical form of which will be given later in 

 Section 8.4.1. Here we shall be content to give a 

 simple qualitative explanation of Riemann's ideas. 



-VELOCITY (e,1-u,) 



FiGUEE 3. Development of a shock wave in one dimen- 

 sion. 



Suppose we have a plane wave of the form shown 

 in Figure 3A advancing in the positive x direction. 

 Let the particle velocity at x = Xi be mi, and let that 

 at 2 = a;2 be 1^2. If we use acoustic theory as a first 



approximation, we have Mi « pi/c, Ui ~ pi/c, where 

 Pi and p2 are the pressures at Xi and Xi respectively.* 

 Now imagine an observer moving with the velocity Mi, 

 so that to this observer the fluid at the point xi is 

 instantaneously at rest. Any small additional dis- 

 turbance, such as the bump A, will seem to this ob- 

 server to be propagated forward with the velocity 

 C\ = {dp/dp)\^p^. Relative to the original system of 

 reference, therefore, this bump will advance with 

 velocity (ci -|- Mi). Similarly the bump B will ad- 

 vance with velocity (c2 + Ui). Now the fact that 

 Pi > P2 implies, as shown above for the acoustic 

 approximation, that Mi •> Wi\ moreover, the equa- 

 tion of state of all ordinary fluids is such that this 

 fact also implies C\ > (h- Thus (ci -f- Ui) > (ci + U2) 

 and bump A advances faster than B, so that after a 

 short interval of time the pressure pulse will have 

 somewhat the form shown in Figure 3B. This il- 

 lustrates Riemann's result, that in an advancing 

 wave the parts of higher amplitude move faster than 

 those of lower amplitude. If continued long enough, 

 this difference in velocity would cause the high pres- 

 sure point A to overtake the low pressure B; how- 

 ever, before this occurs, the curve of p against x will 

 acquire a vertical, or nearly vertical, tangent at some 

 intermediate point C, as shown in Figure 3C. In the 

 neighborhood of this point the pressure gradient and 

 velocity gradient will be very large, and it will no 

 longer be permissible to neglect the effects of viscos- 

 ity and heat conduction, which have been omitted 

 from Riemann's equations. It turns out that viscosity 

 and heat conduction, by converting mechanical 

 energy into heat, slow up the rate of advance of the 

 high pressure regions, the amount of this slowing up 

 becoming greater the larger the gradient of pressure 

 and velocity. Thus a stage will eventually be reached, 

 as in Figure 3D, where the steepness of the rise from 

 A to E is just sufficient to keep A from overtaking E. 

 This state of affairs constitutes a shock wave. Since 

 in practice this limiting value of the time of rise is 

 extremely short, the shock wave begins with an almost 

 instantaneous rise of pressure. 



The importance of this phenomenon of Riemann's 

 in the understanding of explosive soimd is not merely 

 that it explains the origin of shock waves, which 

 could simply be taken for granted, but that it also 

 explains how the characteristics of the disturbance 

 behind a shock front change with the time. This vari- 

 ation will be discussed in Section 8.5. 



' Throughout this chapter the symbol 

 denote "is approximately equal to." 



will be used to 



