388 SURFACE WAVES. [CHAP. IX 



there is now no slipping at the common boundary of the two fluids. The 

 second root of (v) is 



and for this the ratio (vi) assumes the value 



' (ix)- 



If in (viii) and (ix) we put kh f = oo , we fall back on a former case ; whilst if we 

 make kk' small, we find 



and the ratio of the amplitudes is 



.(xi). 



These problems were first investigated by Stokes*. The case of any 

 number of superposed strata of different densities has been treated by Webbf 

 and GreenhillJ. For investigations of the possible rotational oscillations in a 

 heterogeneous liquid the reader may consult the papers cited below . 



224. As a further example of the method of Art. 222, let us 

 suppose that two fluids of densities p, p', one beneath the other, 

 are moving parallel to x with velocities U, U', respectively, the 

 common surface (when undisturbed) being of course plane and 

 horizontal. This is virtually a problem of small oscillations about 

 a state of steady motion. 



The fluids being supposed unlimited vertically, we assume, for 

 the lower fluid 



^r = -U{y-pe k ycoskx] (1), 



and for the upper fluid 



(2), 



* " On the Theory of Oscillatory Waves," Camb. Trans, t. viii. (1847) ; Math, 

 and Phys. Papers, t. i., pp. 212219. 



t Math. Tripos Papers, 1884. 



(i \Vave Motion in Hydrodynamics," Amer. Journ. Math., t. ix. (1887). 



Lord Kayleigh, "Investigation of the Character of the Equilibrium of an 

 Incompressible Heavy Fluid of Variable Density." Proc. Lond. Math. Soc. t t. xiv., 

 p. 170 (1883). 



Burnside, " On the small Wave-Motions of a Heterogeneous Fluid under Gravity." 

 Proc. Lond. Math. Soc., t. xx., p. 392 (1889). 



Love, "Wave-Motion in a Heterogeneous Heavy Liquid." Proc. Lond. Math. 

 Soc., t. xxii., p. 307 (1891). 



