Mr. stokes, on THE THEORY OF OSCILLATORY WAVES 453 



15. The preceding investigation applies to two incompressible fluids, but the results are 

 applicable to the case of the waves propagated along the surface of a liquid exposed to the air, 

 provided that in considering the effect of the air we neglect terms which, in comparison with those 

 retained, are of the order of the ratio of the length of the waves considered to the length of a wave 

 of sound of the same period in air. Taking then p for the density of the liquid, p for that of the 

 air at the time, and supposing A^ = os , we have 



c^= - 



■^^-m'-i'^m--"- 



g (p- p,) D gP 

 m pS 



If we had considered the buoyancy only of the air, we should have had to replace g in the 



formula (14) by " — '-< g. We should have obtained in this manner 

 ■ P, 



^. g ip-p}D ^ffl^fi _P, 

 m pS mS \ p 



Hence, in order to allow for the inertia of the air, the correction for buoyancy must be increased 



in the ratio of 1 to 1 + — . The whole correction therefore increases as the ratio of the length of a 



wave to the depth of the fluid decreases. For very long waves the correction is that due to 

 buoyancy alone, while in the case of very short waves the correction for buoyancy is doubled. 



Even in this case the velocity of propagation is altered by only the fractional part - of the whole ■ 



P 

 and as this quantity is much less than the unavoidable errors of observation, the effect of the air in 

 altering the velocity of propagation may be neglected. 



16. There is a discontinuity in the density of the fluid mass considered in Art. 14, in passing 

 from one fluid into the other ; and it is easy to show that there is a corresponding discontinuity in 

 the velocity. If we consider two fluid particles in contact with each other, and situated on opposite 

 sides of the surface of junction of the two fluids, we see that the velocities of these particles resolved 

 in a direction normal to that surface are the same ; but their velocities resolved in a direction tan- 

 gential to the surface are different. These velocities are, to the order of approximation employed 



„ dd) d(b 



in the investigation, the values of -i and —£-' when y = Q. We have then from (4.3) and (44), for 



a CO dx 



the velocity with which the upper fluid slides along the under, 



iS S 

 mac (--- + - 



17. When the upper surface of the upper fluid is free, the equations by which the problem 

 is to be solved are the same as those of Art. 14, except that the condition (."31) is replaced by 



dd) , d'd) , , , , 



^ -^ -'^ T-^' = "' "''"^" y= -'k'- (*-^); 



dy d.v' 



and to determine the ordinate of the upper surface, we have 



C,+gp^y + cp,-^ = 0, 



where y is to l)e replaced by - A^ in the last term. Let us consider the motion corresponding to 

 the value of <^ given by (35). We must evidently have 



^ ^ (Ae"-^ + B^e"^) sin mx. 



