Radiation and Dispersion of Internal Waves 



dF/dt = on F = , (7) 



p is continuous across F= . (8) 



In (8) the surface tension effect, if any, is not included. 

 The following special cases are of particular interest. 



Incompressible Stratified Flows 



When the effect of compressibility of the fluid is negligible in flows of con- 

 cern, S is not considered, andEq. (3) is replaced by the equation of incompres- 

 sibility, which is, in general, 



dp/dt = , (9) 



stating that p remains constant along a material line. Then Eq. (1) reduces to 



div q = Q(x,t) . (10) 



The momentum Eq. (2) remains the same in form, with X = -ge^. The dual role 

 of both hydrodynamic and thermodynamic nature played by p in the general case 

 now reduces to one of only the hydrodynamic origin in incompressible flows. 

 The basic flow is now characterized by a shear and a density stratification 



q = (U(z), V(z), 0) , p = Po(0 - (11) 



in which p^iz) is assumed to be 50 for static stability. A sufficient condition 

 for stability is that -^^Pq/Po > (U')^ + (V')^ (see Howard (8). 



Steady Stratified Flows 



When the flow is steady the time derivatives of all flow quantities drop out 

 from the above set of equations. In this case there exists a useful transforma- 

 tion, due to Yih (1,9,10), by which flows of nonhomogeneous fluids can be reduced 

 to flows of homogeneous ones (called the related flows) under certain conditions. 

 However, the related flow involves, in general, a nonuniform force field and an 

 additional shear in the free stream. 



For steady, two-dimensional incompressible flows, with q = (u,0,w), a 

 stream function ^(x, z) exists such that 



u = 3i///3z , w = -a^/Bx . (12) 



Then density p and the total head H remain constant along each streamline, 



p = p(^) , H(>//) - p + — /0(u2 + w2) + gpz . (13) 



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