178 



EXPLOSIONS AS SOURCES OF SOUND 



have been treated briefly in the preceding sections. 

 Although this material is essential to a complete 

 understanding of explosive phenomena, the con- 

 tinuity of the chapter will not be impaired by omis- 

 sion of these sections provided the reader is willing 

 to accept on faith those results from them which 

 have already been quoted. 



A more complete account of the theory of waves of 

 finite amplitude and shock waves is given in a report 

 issued by the Applied Mathematics Panel/ and in 

 textbooks on hydrodynamics.* 



8.4.1 Riemann's Theory of Waves 

 of Finite Amplitude 



In Sections 2.1.2 and 2.1.3 the equations of motion 

 of a perfect fluid were derived on the assumption that 

 the amplitude of the disturbance was small, so that 

 the acceleration of a particle of the fluid could be ap- 

 proximated by the partial derivative of the velocity 

 with respect to time. Since we wish in this section to 

 treat disturbances for which this approximation will 

 not be valid, we must start from a more exact form 

 of the equations of motion. As before, let x,y,z be 

 three rectangular coordinates in space, t the time, 

 and Ui,Uy,Us the three components of particle velocity 

 in the fluid. As shown in Section 2.1.2, the x com- 

 ponent of the acceleration of a particle of the fluid is 



3Mi dU:c dUi dUi dUi 



— = h Mx h Uy h Mj — , 



dt dt dx dy dz 



(1) 



and this, when multiplied by the density p, must 

 equal fx, the x component of force per unit volume. 

 According to Section 2.1.3, fx is related to the distri- 

 bution of pressure p by 



dx 



Thus the exact equation of motion for the x com- 

 ponent of velocity is 



dUx . dUx 1 dp 



p dx ' 



and applying the same reasoning to the y and z com- 

 ponents, we get the remaining equations of motion 



dUx , dUx ^ - -. , 

 dt dx dy dz 



(2) 



dUy 



dUy 



dUy 



dUy 



dt dx dy dz 



du 



du. 



Idp 

 pdy' 



Idp 



p dz 



Let us now consider a disturbance in which the pres- 

 sure and velocity are functions only of x and t, inde- 



dUz du^ 



dt dx dy dz 



(3) 

 (4) 



pendent of y and z. For such a disturbance the equa- 

 tion of continuity [equation (2) of Section 2.1.1] 

 becomes 



dp d{pUx) 

 dt dx 



and equation (2) becomes 



= 0, 



(5) 



(6) 



dUx dUx 1 dp 



dt dx p dx 



Riemann discovered that the two equations (5) and 

 (6) could be put into a symmetrical and useful form 

 by introducing the variable 



dp 

 Kdp/ p 



where po is a reference value of the density which is 

 most conveniently chosen equal to the density of the 

 imdisturbed fluid. By using this equation and the 

 abbreviation c = {dp/dp)% equation (5) becomes 



r/dp\h 



J pa \dp f 



(7) 



dp 9^ dp d\f/ dUx 



= h Ux- h p 



d<p dt d-if/ dx dx 



or 



a^ d^ 



h Ux — = 



dt dx 



and equation (6) becomes 

 dUx dUx 



1- Ux — = 



dt dx 



dij/ dUx 



p- = — c- 



dp dx 



1 dp dp dTp 



p dp d}// dx 



d^p 



-c 



dx 



dUx 

 dx 



(8) 



(9) 



Adding equations (8) and (9) gives 



+ (Ux + c) 



dt 



dx 



= 0, (10) 



and subtracting equation (9) from equation (8) gives 

 similarly 



djyp — Ux) 

 dt 



+ (Ux — c) 



d{^ - Ux) 

 dx 



0. (11) 



Equation (10) states that the quantity {\p -\- Ux) is 

 propagated in the x direction with velocity {Ux + c) 

 while equation (11) states that the quantity (^ — Ux) 

 is propagated with velocity {Ux — c), that is, since 

 ordinarily c > Ux, is propagated in the negative 

 X direction with velocity (c — Ux). These are Rie- 

 mann's results. 



The significance of these equations can be seen by 

 considering a disturbance which is initially confined 

 within the range a < a; < 6. For such a disturbance 

 both (i// + Ux) and (tp — Ux) are initially zero below 

 X = a and above x = b. The region in which (^ + Ux) 



