472 WAVES OF EXPANSION. [CHAP. X 



this may be written 



dp dp .,. dp 



dt +u d^=-tf^d* .................. &amp;lt;&amp;gt;- 



In the same way we obtain 



The condition (4) is satisfied by 



Substituting in (5) and (6), we find 



\dpj j J dx 



Hence dP = 0, or P is constant, for a geometrical point moving 

 with the velocity 



*-$) + ........................ 



whilst Q is constant for a point whose velocity is 



.(10). 



Hence, any given value of P moves forward, and any value of Q 

 moves backward, with the velocity given by (9) or (10), as 

 the case may be. 



These results enable us to understand, in a general way, the 

 nature of the motion in any given case. Thus if the initial 

 disturbance be confined to the space between the two planes 

 x = a, x = b, we may suppose that P and Q both vanish for x &amp;gt; a 

 and for x &amp;lt; b. The region within which P is variable will 

 advance, and that within which Q is variable will recede, until 

 after a time these regions separate and leave between them a 

 space for which P = 0, Q = 0, and in which the fluid is therefore 

 at rest. The original disturbance has thus been split up into two 

 progressive waves travelling in opposite directions. In the 

 advancing wave we have Q = 0, and therefore 



