Plane Waves of Sound. 23 



v? + j/2 = I so that we may put cos0 = \i, sinQ = v ; after a 

 little reduction we obtain 



^ = cosmx.ei^^nz [Acosntco^iymz - /3) + A' ?,m7itcos{vi?iz - /3') } 

 + smmxei^^'^^{Aco^ntcos{vmz - /3 - 0) + A' sm?itcos{vinz -ft'- 0)} 

 when A, A', /3, /3' are absolute constants. 



Now suppose that in the compartment at the origin we 

 have the motion given by 



\ = llcos(7if - mx). 

 Applying this condition to determine the constants, we 

 obtain for the motion in a compartment distant z from the 

 origin 



\ = H^^''''|cos j'W^cos [7it - mx) - ^^^^^in {nt - mx) 



+ cot0sini^;;/5cos(;// + mx) 

 But from the equations 



we get 



so that 



P = KD 





V D 



— "R /Do f / ^ \ sinvmz 



Thus finally we obtain 



- A / -^°\ cosymzcos int - mx) — ^^^^^^^sin(;// - mx) 



+ cot0sin)'wscos(;2/+ mx) v . . . v. 



This solution will allow us to obtain a very good idea of 

 the changes which a sound wave undergoes when travelling 

 through the continuously varying atmosphere, and will 

 serve us to show how extremely small is the error in the 

 ordinary approximation where we suppose there is no 

 reflected wave. 



Examining the solution v. we notice that it consists partly 



