Lord Rayleigh on Waves. 261 



These results are due to Green, Kelland, and Airy*. The 

 same method may be even more easily applied to the sound- 

 wave moving in a pipe of gradually varying section. 



The theory of long waves may be applied in many cases to 

 ascertain the effect on a stream of a contraction or enlarge- 

 ment of its channel. If the section of the channel up to the 

 natural level of the stream be altered from A to A, the equa- 

 tion of continuity gives 



(A + bh)u = A u , 



where b y the breadth at the surface of the water, is supposed 

 not to vary with height. The condition of a free surface is 



or 



A* _, 2ffh 



= 1 



(A + bhf 



which shows that h can never exceed the height due to the 

 velocity u , as is indeed otherwise obvious. 



If the variations in A and b are small as well as gradual, 

 and if we put A=A -f 8 A, we find 



- b ' h% 



When the velocity u Q is less than that of a free wave, gA Q > bu\ , 

 and h has the same sign as 8 A ; viz. a contraction of the chan n 

 nel produces a depression of the surface, and an enlargement an 

 elevation. But if the velocity of the stream exceed that of a 

 free wave, these effects are reversed, and an enlargement and 

 contraction of the section entail respectively a depression and 

 an elevation of the surface. 



If the velocity of the stream is nearly the same as that of a 

 free wave, a state of things is approached in which a wave can 

 sustain itself in a stationary position without requiring a varia- 

 tion in the channel ; and then the effects of such a variation are 

 naturally much intensified. 



We must not forget that these calculations proceed on the 

 supposition that a steady motion is possible. It would appear 

 that the motion thus obtained is unstable in the case where the 

 velocity of the stream exceeds that of a free wave. If we sup- 

 pose the upper surface to consist of a movable envelope, it 

 would indeed be in equilibrium when disposed according to the 

 law above investigated; but if a displacement be made and 

 steady motion be conceived to be reestablished, the pressure of 

 * Stokes, Brit.-Assoc. Keport on Hydrodynamics, 1846. 



