RADIATION AND DISPERSION 

 OF INTERNAL WAVES 



T. Yao-tsu Wu 



California Institute of Technology 



Pasadena, California 



ABSTRACT 



For small perturbations of a basic flow having a prescribed stratifica- 

 tion of entropy and a vertical shear of horizontal free stream in a grav- 

 ity field, a linearized equation for the vertical component of velocity is 

 derived. The dispersion function is examined in detail for the special 

 case of incompressible stratified flows with a uniform specific-density 

 gradient and without shear. This dispersion relationship is applied to 

 consider the radiation of internal gravity waves due to (a) an oscillating 

 dipole and (b) a steadily moving singularity. Analyses of these two 

 problems are also carried out in order to present a self-contained dis- 

 cussion of the results. 



INTRODUCTION 



Consideration of the propagation of gravity and infrasonic waves in the at- 

 mosphere or ocean generally requires full- wave methods. Previous treatments 

 of this class of flow problems have shown that the integral representations of 

 the solutions of steady and quasi- steady flows involve, in general, some branch 

 points on the path of integration. The indeterminancy of the branch cuts had 

 been a subject of discussion for some time (see, for example, a review by Yih 

 (1), p. 65 ff .), until a clarification by the approach of an initial value problem 

 was given by Crapper (2). It is of interest to examine if the solution can also 

 be made determinate by the consideration of the propagation of wave energy 

 alone. One of the purposes of the present investigation is to discuss and com- 

 pare these two approaches. 



For the general purpose, a linearized equation is derived for the vertical 

 velocity component of a perturbed flow while the basic flow is entropy- stratified 

 and has a vertical shear. The gravity-acoustic mode and the effect of shear are 

 not considered further here; discussions of some phases of these problems can 

 be found in the recent literature, e.g., see Bjerknes (3), Hines (4) on isothermal 

 atmosphere without winds, Weston (5), Pierce (6) on isothermal atmosphere with 

 constant winds. The dispersion relation is investigated in detail for an incom- 

 pressible flow with a constant specific-density gradient in the absence of shear. 



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