III-72 



for periodic surfaces and could therefore be used as well for the more or less 

 cnoidal surfaces* to be expected from the theory of propagation of surface waves . 

 Marsh, however, obtains an approximate solution in terms of the correlation func- 

 tion (or equivalently the spectrum) of the surface. 



1 . Periodic Surfaces 



When the surface S(x,y) is periodic, the scattered wave p can be 

 represented by the superposition of a countable number of plane reflected waves. 

 In other words, the function (k , [i) becomes discrete. This may be seen as 

 follows . 



For simplicity in the argument, let us restrict ourselves to a "cor- 

 rugated" periodic surface, generated by straight line generators parallel to the 

 y-axis and periodic with period % in the x direction. The geometry is illustrated 

 in Figure III-29. 



2 = S(X) 



u 



/ 



FIGURE III-29 SCHEMATIC DIAGRAM OF A PLANE WAVE INCIDENT 



ON A CORRUGATED SURFACE 



If the plane wave 



ik(ax+Yz)- iuJt 



p (x,z,t) = p e 

 inc inc 



is incident on this surface, it causes a scattered wave p (x,z)e 



•iUJt 



(III- 51) 



*J. J. Stoker, "Water Waves," Interscience Publishers 1957. 



Arthur 21.little.3nf. 



S-7001-Q307 



