544 XI. HYDRODYNAMICS. 



r\ 



Integrating with respect to g, we find that C - J - is independent of q, 

 but may be an arbitrary function of p, say 



sf-*V(iO- 



Integrating again we find 



204) (p = F l (p) + -F 2 (g), 



where F 2 is an arbitrary function of q. Whatever the functions F 1 

 and F 2 , this value will satisfy the equation 203). Replacing p and q 

 by their values, we have the general solution 



205) (p^F^x- at) + F 2 (x + at). 



Let us first assume F 2 = and consider the solution 



9Qg\ Tfl / \ TP / i\ 



The value of (p depending only on p is unchanged when x has 

 increased by the amount at, that is to say, if (p be represented 

 graphically as a function of x at the time t = 0, it will be represented 

 at the time t by the same curve moved to the right a distance at. 

 Such a motion is termed a wave moving with the velocity a in the 

 positive X- direction. 



Similarly the solution 



represents a wave moving in the negative X- direction with the same 

 velocity. 



If the function F(p) is zero except for a certain small range 

 of values p 0> p lf the motion is sometimes called a pulse. A pulse is 

 none the less a wave. 



Thus the general solution of equation 201) represents two plane 

 waves propagated in opposite directions with the same velocity a. 



The velocity of sound a = y -JJT depends upon the elasticity of 

 the air and was calculated by Newton, assuming that the process 

 was isothermal, using Boyle's law. As this was found to give results 

 not agreeing with experiment Laplace suggested that the compression 

 was adiabatic, the vibrations being so rapid that the heat generated 

 did not have time to .flow from the heated to the cooled parts. Thus 

 the constant K, equation 17) 178 representing the ratio of the two 

 specific heats of the air is introduced. The velocity of sound gives 

 one of the most accurate ways of determining this ratio sc. 



The velocity of the particle of air is obtained by 



C)f\Q\ d<P 77T f / /\ 



^Uo ) u = 7p- = Jb i (x a t) 



in the wave going to the right. The compression by 



