in which J is the Bessel function of the first kind of order zero. The 

 general solution includes also the Bessel function of the second kind, 

 Y . However, this solution blows up at the origin, x = 0, and for the 

 solution to remain finite at x = this part of the general solution is 

 omitted. For x ^ 0, Jq approaches unity so that the arbitrary constant 

 A in equation (90) is the complex vertical amplitude of the wave motion 

 at the intersection of the Stillwater level and the slope, i.e., |a| 

 may be interpreted as a measure of the runup on the slope. It should 

 be realized that the linearized solution, as discussed in Appendix A, 

 is based on the assumption that |ri,| << h. Thus, since h = at x = 

 the solution given by equation (90) cannot be considered valid near 

 x = 0. However, Meyer and Taylor (1972) have shown that a linear 

 solution gives essentially the same value of the runup as does the more 

 realistic solution based on the nonlinear shallow-water wave equations. 

 Thus, some physical significance may be attached to the magnitude of A, 

 |A|, as being an approximate value of the runup on the slope. 



With ? given by equation (90) the horizontal velocity is evaluated 

 from equation (87) 



CO (1-if, )x 



,, .r: ,^ , — ^ J, (2 7 — ;: — ^ — ); < X < I , (91) 

 (1-if, )x tanB IV g tan6 — — s 



in which J is the Bessel function of the first kind of order one. 



With the general solution given by equations (85), (90), and (91) 

 the complex amplitude of the reflected wave, aj,j and the complex 

 runup amplitude, A, are determined by matching the solutions for c and 

 u at their common boundary, x = Jl . Thus, at x = Jl 



a. e 



ik £ -ik I 



°^ + ae °^ = AJ(2kJ2, /T^IfT) (92) 



i r o s b 



and 



ik £ ik £ . 



°^-ae °^ = A — i J, (2k I /1-if^) (93) 



r n r-j— 1 ^ O S b-' ^ ' 



b 



in which h = £ tang and equation (86) have been introduced. 



OSS ^ 



These equations are readily solved to give the complex amplitude 

 of the reflected wave. 



