course, that the velocity distribution U(y) of the secondary flow is 

 known. In actual cases it is hard to predict the character of such 

 secondary motions because in general they depend upon local conditions. 

 It is, however, possible to deduce reasonable estimates of their behavior 

 by statistical methods based on long time records. Since the wave char- 

 acteristics themselves are usually evaluated by similar methods it be- 

 comes evident that the error in the estimated values of the parameters 

 entering our problem as independent variables has a bivariate distribution. 

 This, of course, reduces the accuracy of the results. There is at least 

 one case, though, in which the steady mean motion U(y) in a body of water 

 is induced by the surface wave itself. This particular case will be 

 described in the following section. 



7. A PARTICULAR CASE OF THE SECONDARY DRIFT 



This type of a second-order drift which is generated by the surface 

 waves in a direction parallel to the wave propagation has been originally 

 studied by Stokes (1851) and more recently by numerous investigators. The 

 works of Bagnold (1947) and Longuet-Higgins(1953) wil be singled out 

 because in addition to their scientific merits offer a rather convenient 

 application to the problem at hand. Associated with this secondary motion 

 is a steady mean water particle velocity which usually is called "mass- 

 transport velocity". Stokes' expression for this velocity, which we will 

 denote by U, is of the form 



a a u)k cosh2k(yi +d) a 3 0) coth kd 



— ± (71) 



u - 2 sinh^kd 2d 



where yi is measured from the mean free surface negative downwards. The 

 only necessary and sufficient condition to be satisfied for the derivation 

 of this expression is that the flow is irrotational . Longuet-Higgins 

 pointed out that the requirement of small amplitude surface wave, as was 

 suggested by Stokes, is not necessary. According to Stokes' theory, the 

 velocity near the bottom is negative (in opposite direction to the wave 

 propagation) and for typical values of the product kd it increases with 

 distance from the bottom attaining its maximum positive value at the free 

 surface. Bagnold (1947) on the other hand has shown experimentally that 

 the actual behavior is quite different. He observed a strong forward 

 velocity near the bottom and a weaker backward velocity at higher levels. 

 This confirmed the belief that Stokes' theory was not accurate, as it has 

 been evidenced from older experiments in which. forward velocities were 

 observed both near the bottom and the free surface and backward motion in 

 the interior. A more reliable theory which confirms the experimental re- 

 sults has been developed by Longuet-Higgins (1953). The two fundamental 

 assumptions are that the mean motion is periodic in time and that it can 

 be expressed as a perturbation of a state of rest. The theory recognizes 

 the existence of three distinct regions, the first near the free surface, 

 the second in the interior and the third near the fixed boundary. Associ- 

 ated with each of these regions is a particular mode of secondary motion. 



23 



