[ 669 J 



LXXVI. On the Integration of the Differential Equation 

 applicable to a Plane Progressive Wave. By J. Rose- 

 Innes, M.A., B.Sc* 



LORD RAYLEIGH in his < Theory o£ Sound' has given 

 the differential equation applicable to a plane pro- 

 gressive wave as 



He quotes Earnshaw's trial solution 



and also Earnshaw's final conclusion that the integral of 

 the equation is to be found by eliminating a between the 

 equations 



y=«* + F («)* + £ (a), 1 



O=a+F(a)* + 0'(a)-J 



This form o£ integral, however, does not lend itself readily 

 io the introduction of the initial conditions, and I therefore 

 think it of some interest to present another solution of the 

 equation which enables us to impose the initial conditions 

 with more facility. 



From Earnshaw's trial solution we readily obtain 



dt 2 



dy =¥ (dy 

 dt 



V \dJj da* 



In order that this may coincide with the original differential 

 equation, we have to choose F so as to fulfil the condition 



I Wj w " do' 



and since 



dy_ _Po 

 dx p' 



we may write the condition 



Hf)YCf)-t- 



# Communicated by the Author. 

 Phil Mag. S. 6. Vol. 27. No. 160. April 1914. 2 Y 



