271-272] WAVES IN HETEROGENEOUS MEDIUM. 493 



where H denotes as in Art. 256 the height of a 'homogeneous 

 atmosphere' at the given temperature. Hence 



P<xe-*! (3). 



Substituting in Art. 271 (7), and putting s = 0, we find, in 

 the case of no disturbing forces, 



For plane waves travelling in a vertical direction, s will be a function of z 

 only, and therefore 



d 2 s /cPs I ds 



If we assume a time-factor e tcrt , this is satisfied by 



provided ra 2 -m/H + <r 2 /c 2 =0 



or w 



The lower sign gives the case of waves propagated upwards. Expressed in 

 real form the solution for this case is 



The wave-velocity (o/#) varies with the frequency, but so long as a- is large 

 compared with c/2H it is approximately constant, differing from c by a small 

 quantity of the second order. The main effect of the variation of density is 

 on the amplitude, which increases as the waves ascend upwards into the rarer 

 regions, according to the law indicated by the exponential factor. This 

 increase might have been foreseen without calculation ; for when the variation 

 of density within the limits of a wave-length is small, there is no sensible 

 reflection, and the energy per wave-length, which varies as a 2 p (a being the 

 amplitude), must therefore remain unaltered as the waves proceed. Since 

 Po oc e~ z/sl , this shews that a oc e z/2H . 



When <r<c/2H, the form of the solution is changed, viz. we have 



s=(A l e m * z +Atf> m *)coa<rt (vii), 



where m u m 2 (the two roots of (iii)) are real and positive. This represents a 

 standing oscillation, with one nodal and one * loop ' plane. For example, if 

 the nodal plane be that of 2 = 0, we have m 1 A 1 -\-m 2 A 2 =O t and the position of 

 the loop (s=0) is given by 



1 i w^i i ">\ 



z= log 1 (vm). 



