Wu 



It is noteworthy that the compressibility effect does not appear explicitly in the 

 above boundary conditions on free surface. 



Incompressible Flows of Small Amplitude 



The linearized differential equations for incompressible flows with density 

 stratification and shear can be deduced from the above case by passing to the 

 limit Co^-»co and dc' Vdz - o. In this limit, the order of Eq. (28) reduces by two 

 and (28) becomes 



(D2 + n2)V2^ + d (i-+ ^j d 



Bz ~o 



1-- 



'1 = 1-^(^0 DQ). (33) 



or, after some rearrangement, 



D^V^ + 



effect of: 



,:!!>' ^-sv= 



3z 



inertia gravity 



DD'-^iDD']}., = ^ A(p„Dg,, 



density gradient Shear 



Shear- stratification 

 interaction 



(34) 



in which 



D' 



,3 ,3 



3x 3y 



3 d 



and v^ is the Laplacian in x,y,z. The vertical displacement C(x,y,z,t), such 

 that Wj = D^, satisfies the equation 



D^V^ + 



Po 



Bz 



^ = ^^(^oDQ) 



(35) 



This equation may be called the Love Equation (Lamb (13)), although Love (1891) 

 first derived it for u' = v' = 0. For two-dimensional flows (in x-z plane), all the 

 y -derivatives in (33) and (34) of course drop out. 



The different physical nature represented by different terms in (34) is indi- 

 cated below (34). The effect of density gradient appears in two terms, one due 

 to inertial and the other due to gravitational origin. The quantity D^3w,/Bz gives 

 a measure of the acceleration of rotation of a fluid bulk, and (-p^/p'^) is related 

 to the radius of gyration of the heterogeneous bulk. The gravity term represents 

 the restoring force. The relative magnitude of these two terms gives rise to a 

 Froude number, 



Bw 



O^D^ -^ gV' w,j - UoVg^ = Fr . 



where Ug is a characteristic velocity and ^ a characteristic length, both appro- 

 priate to the particular problem in question. For small values of this Fr, g^2^*i 



556 



