value of the friction angle, <J), it is of extreme importance that the 

 reflection coefficient be accurately determined from the experimental 

 data. 



According to linear wave theory the wave motion in the constant 

 depth region in front of the slope is given by equation (85). 

 Introducing the expression: 



I 1 i5 

 a e 

 I -pi 



(117) 



for the reflected wave amplitude in equation (85), where 6 is an 

 arbitrary phase angle, the resulting wave amplitude, \t,\, may be 

 expressed as 



a(x) = Ul = (a.^ + la^I^ + 2a.|a^| cos(2k^x + 6))^^^ , (118) 



which shows the wave amplitude to vary with distance along the constant 

 depth part of the flume in a periodic manner. For values of 

 2k X + 6 = 0, + 2tt, etc. (i.e., at the antinodes) the resulting 

 amplitude is a maximum, 



a = a. + la I = a. (1 + R) , (119) 



max 1 ' r ' i 



and for values of 2koX + 6 = +_ tj , etc. (i.e., at the nodes) the resulting 

 amplitude is a minimum, 



a . = a. - la I = a.(l - R) . (120) 



mm 1 ' r' i 



Since the wave height, H, according to linear wave theory is twice 

 the amplitude the preceding formulas show it, in principle, to be 

 possible to determine the reflection coefficient, R, and the incident 

 wave height, Hj^ = 2aj^, by merely seeking out a node and an antinode along 

 the flume. Thus, 



H - H . 

 max mm 



H + H . 

 max min 



(121) 



and 



H. = 2a. = ^ (H + H . ) . (122) 



1 1 2 max mm 



As discussed by Ursell, et al . , (1960) the above formulas are valid 

 also when the wave motion is weakly nonlinear, i.e., consists of a small 

 second harmonic motion in addition to the primary first harmonic motion 



64 



