542 XL HYDRODYNAMICS. 



By introducing the complete expressions for the accelerations with 

 respect to revolving axes, given in 104, and applying the principles 

 of forced oscillations, we obtain the more complete theory given by 

 Laplace. 



2O1. Sound -Waves. Let us now consider the motion of a 

 compressible fluid which takes place in the propagation of sound. 

 In the production of all ordinary sounds, except those violent ones 

 produced by explosions, the motion of each particle of air is extremely 

 minute. We shall therefore suppose that the velocity components 

 u, v, w and their space derivatives are so small that their squares 

 and products may be neglected. Let us put 



190) p = p (l-M), 



where Q O is a constant and s is a small quantity, of the same order 

 as the velocities, called the compression. From the equation of 

 continuity we have 



1Q1 x _ du cv dw _ 1 d$ _ 1 ds 



-fa^d^^fa- ~~Q~di~ ~1+10*' 

 or neglecting the product of s and its derivative, 



-t f\c\\ ds 



192) "=-** 



In order to calculate P, we have, since the changes in Q are small 



193) dp = a 2 dQ = a 2 Q Q ds, 



where a 2 is a constant representing the value of the derivative J- 



for Q = Q O , the density of the air at atmospheric pressure. We 

 therefore have 



194) P _ =- log (1 +.) = ,, 



to the same degree of approximation. 



Neglecting small quantities the equations of hydrodynamics 6) 

 become, when there are no applied forces, 



o o 



CU 9 08 



-or = a 3 7 



dt dx 



dv 9 ds 



195) - = a?2-> 



dt dy 



dw o ds 



- = a 2 -^-) 

 dt cz 



with 



192) =-f, 



Differentiating the equations respectively by x, y, z, adding and 

 observing the definition of 6, we obtain 



