Wu 



It is found that the group velocity and phase velocity of these internal gravity 

 waves can differ drastically from each other in both magnitude and direction. 

 Based on this dispersion relation, the radiation of internal waves is traced for 

 two specific problems: (a) an oscillating dipole and (b) the three-dimensional 

 fundamental solution of a steady flow. From the comparison with the corre- 

 sponding analytical solutions, it is found that the usual dynamic concept of the 

 group velocity associated with the propagation of wave energy remains valid 

 here. 



BASIC EQUATIONS AND BOUNDARY CONDITIONS 



When the effects of viscosity, heat conduction, and mass diffusion can be 

 neglected, the basic equations of motion for a compressible fluid are; 



continuity 77 ^ ^ ^^^ ^ " pO(x,t) , (1) 



momentum dt ~ ~ p" g'"^^ p + X(x,t) , (2) 



dS /«» 



energy dF " ° • ^^' 



state f(p,/o,T) = , (4) 



in which t denotes the time, x = (x,y,z) the Cartesian coordinates fixed in an 

 inertial frame, q the flow velocity, p the density, p the pressure, T the temper- 

 ature, S the specific entropy of the fluid, Q the fluid source strength, X the ex- 

 ternal force acting on a unit mass of the fluid, and d/dt = B/Bt + q • v. Equation 

 (3) states that the process is isentropic along a material line, which is a valid 

 assumption for not too rapid change of state in the absence of heat conduction. 



The class of flows of our present interest is characterized by a uniform 

 gravity field acting in the direction of z decreasing, or X= (O,0,-g),ona basic 

 reference flow which has a specified vertical shear and entropy stratification 



q = (U(z), V(z), 0), S = So(z). (5) 



It is assumed that So(z) = dSg/dz > o to ensure thermodynamic stability, e.g., 

 see Landau and Ltfshitz (7) (but we shall not go here into the condition relating 

 U'(z), v'(z), and So(z) for dynamic stability). 



At the surface of a solid boundary the condition, as usual, is that the veloc- 

 ity component normal to the surface vanishes, 



q • n = (6) 



on solid surface. At a free surface, say described by F(x, t) = 0, across which 

 the tangential velocity, density, temperature, and entropy may have a jump, 

 there are two conditions: 



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