388 SURFACE WAVES. [CHAP. IX 



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

 second root of (v) is 



c 2 - p ~ p&amp;gt; - $- (viii) 



~ *&quot; ...ivrnj, 



and for this the ratio (vi) assumes the value 



............... (ix). 



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

 make kh small, we find 



-(l-0fV (x), 



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 Greenhill|. 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 



Tjr = - U [y - $e k y cos kx\ (1), 



and for the upper fluid 



(2), 



* &quot; On the Theory of Oscillatory Waves,&quot; Carnb. Trans, t. viii. (1847) ; Math, 

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



t Math. Tripos Papers, 1884. 



$ &quot;Wave Motion in Hydrodynamics,&quot; Amer. Journ. Math., t. ix. (1887). 



Lord Kayleigh, &quot;Investigation of the Character of the Equilibrium of an 

 Incompressible Heavy Fluid of Variable Density.&quot; Proc. Lond. Math. Soc., t. xiv., 

 p. 170 (1883). 



Burnside, &quot;On the small Wave-Motions of a Heterogeneous Fluid under Gravity.&quot; 

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



Love, &quot;Wave-Motion in a Heterogeneous Heavy Liquid.&quot; Proc. Lond. Math. 

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



