2 



-^ = cos^ e (1) 



where N is the Brunt-Vaisala frequency, a is the wave frequency, and 

 9 is the angle of the wave-number vector to the plane perpendicular 

 to the gravitational field. 



A beam of internal-wave energy represents a sum of waves of different 

 wave numbers but identical frequencies that tend to cancel each other 

 outside of a given region of space. Thus at one edge of the beam waves 

 appear to form and pass through the beam cancelling each other at the 

 opposite side of the beam; the energy flux is along the beam. 



The seaward-propagating internal-wave system is roughly as wide as 

 the depth of fluid above the continental shelf and propagates energy 

 seaward and downward from the region of fluid above the shelf break. 

 For the seaward-propagating intexmal-wave system, Fig. 1 shows the 

 lines of constant phase, and thus the direction of energy flux for the 

 internal-wave system. The phase of the wave increases in the direction 

 of decreasing depth. 



Such an internal-wave system when described by normal modes contains 

 a large amount of energy in the high mode numbers . A great deal of 

 vertical structure is found in the velocity and density fields and the 

 wave signal is correlated not in the vertical but along the lines of 

 constant phase. The slope of the cophasal lines, which depend on the 

 stability of the ocean, is small, and the beam of wave energy meets the 

 ocean floor some 30 to 50 km seaward of the continental shelf break. 



The inviscid theory predicts that the wave beam would reflect from 

 the ocean floor, then some distance further from the shelf reflect 

 downward again from the surface. A problem with which we are concerned 

 is the amount of energy that reaches the sea floor and is reflected 

 upward. Knowledge of this permits determination of the distance from 

 the shelf that the propagation of sound can be affected by baroclinic 

 tidal oscillations. 



Because each wave number is damped at a different rate, the viscous 

 dissipation of the seaward-propagating wave system is complex. Further- 

 more, because the total system consists of waves of many different group 

 velocities, a "Taylor hypothesis" based on a single group velocity may 

 not be used to translate from a temporal analysis of the dissipation to 

 a spatial analysis. We attempt here to discuss the problem of frictional 

 modification of the beam by an analysis of the rate of energy flux by the 

 wave system as a whole. Regions of space into which little energy is 

 propagated would be expected to be affected rapidly by energy loss. 



LeBlond (1966) considered a temporal analysis of the dissipation 

 of internal wave energy. He found that waves of long wavelength are 

 dissipated in time slower than waves of shorter wa\^elength, but under 

 no reasonable circumstances would any of the normal modes that comprise 



391 



