36. LONGUET-HIGGINS, M.S., and STEWART, R.W., "The Changes in Amplitude 



of Short Gravity Waves on Steady Non-Uniform Currents," Journal of 



Fluid Mechanics, Cambridge, England, Vol. 10, No. 4, June 1961, pp. 

 529-549. 



Keywords. Conservation Equations; Current Refraction; Currents, Non- 

 uniform; Currents, Opposing; Currents, Shearing; Currents, Slowly 

 Varying; Radiation Stress; Theory; Theory, Ray; Wave Energy. 



Discussion . A formal perturbation analysis gives the second-order 

 interaction between gravity waves and a current which has a small linear 

 variation in magnitude. The result is used to resolve an ambiguity in a 

 wave-energy conservation equation deduced directly from the expression 

 for energy flux. This leads to the correct, unsteady two-dimensional 

 energy conservation equation. 



As well as extensive physical discussion, the following particular 

 examples are examined: waves on currents varying (a) in the direction 

 of flow, (b) across the direction of flow, (c) sinusoidally, as in tidal 

 currents, and (d) along the "centerline" of a flow convergence. 



Coastal Engineering Significance. This important paper gives the first 

 major theoretical results for the interaction of surface gravity waves 

 with large-scale currents. The examples given are the primary simple 

 examples to which these equations can be applied. The results are of 

 significance to all wave-current interactions. 



37. LONGUET-HIGGINS, M.S., and STEWART, R.W., "Radiation Stress and 

 Mass Transport in Gravity Waves, with Applications to 'Surf- 

 Beats'," Journal of Fluid Mechanics, Cambridge, England, Vol. 13, 

 No. 4, Aug. 1962, pp. 481-504. 



Keywords . Currents, Wave-Induced; Mass Transport; Radiation Stress; 

 Setdown; Setup; Surf Beat; Theory; Wave Breaking; Wave Groups. 



Discussion . The concept of radiation stress is used to determine the 

 mass transport associated with groups of surface gravity waves. Three 

 methods of derivation are given (a) a direct perturbation approach, (b) 

 an indirect perturbation approach, and (c) a consideration of 

 conservation of mass and momentum following WHITHAM (1962) and 

 indicating the limitation of Whitham's results to wave groups which are 

 much longer than the depth. Two simple examples are discussed in 

 detail; wave groups long compared with the depth and groups formed by 

 only two sinusoidal components. It is found, contrary to all expecta- 

 tion, that high waves are associated with a negative mass flux. 



Steady wave trains in water of variable depth are considered and 

 the mean water level is shown to decrease as the depth decreases and 

 wave height increases, that is "setdown." It is indicated that the 



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