In the present case our interest is concentrated on the boundary layer 

 near the bottom only. Omitting the lengthy procedure of Longuet-Higgins * 

 rigorous derivation it will suffice to present the final results for the 

 case of progressive waves in water of uniform depth. The first-order 

 motion of the fluid is described by means of the irrotational theory so 

 that the horizontal component of the velocity at the boundary may be ex- 

 pressed, as we have seen in section 3, as 



u = acosinujt (7-2) 



o 



The "mass-transport velocity" can then be obtained from the second-order 

 approximation. 



_ a 8 (JUk 



U = : 2—- f (n) (7-3) 



4 sinh^kd 



where f (mO = 5-8 e^cosu, + 3 e' 2 ^ (7-4) 



These are Longuet-Higgins' equations (254) and (253) respectively. 



Consistent with our notation 



u> = by 



So that the expression for U may be written as 



- = a^ r 5 _ 8 e -ey cos3y + 3e -2@yi (7 _ 5) 



4 sinh 8 kd L J 



where y^ measures the distance upwards from the effective bed. It is 

 evident therefore that the rate of sediment transport of a certain size 

 can be determined by combining the "sediment-transport equation" (6-2) 

 and the "mass-transport velocity equation" (7-5). A given set of surface 

 wave characteristics, temperature and depth of water and grain size of 

 uniform bed material is sufficient for the calculation of this rate. 



8. SUMMARY OF RESULTS 



The independent variables entering the problem of the sediment trans- 

 port by wave action are the following: 



The amplitude H, the length \ and the angular velocity u) of 



the surface wave. 

 The depth d and the temperature T° of the water (and 



consequently its kinematic viscosity \>) . 

 The grain diameter D and the density p s of the granular 



material forming the bed which is assumed to be uniform. 



A method has been described in the preceding sections of this study which 

 can be used to predict the rate of sediment transport associated with any 



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