r^ = \ + \=Y 



cos 



2ttx 2iTt' 



+ cos 



ZtTX 2TTt 



which reduces to 



Ittx. 2-rrt 



n = H^ cos —r— COS -Y" 



(2-80) 



Equation 2-80 represents the water surface for a standing wave or alapotis 

 which is periodic in time and in x having a maximum height of 2Hj when 

 both cos(2Trx/L) and cos(2Tit/T) equal 1. The water surface profile as a func- 

 tion of 2Trx/L for several values of 2irt/T is shown in Figure 2-63. There are 

 some points (nodes) on the profile where the water surface remains at the SWL 

 for all values of t and other points (antinodes) where the water particle 

 excursion at the surface is 2YL or twice the incident wave height . The 

 equations describing the water particle motion show that the velocity is 

 always horizontal under the nodes and always vertical under the antinodes. At 

 intermediate points, the water particles move along diagonal lines as shown in 

 Figure 2-63. Since water motion at the antinodes is purely vertical, the 

 presence of a vertical wall at any antinode will not change the flow pattern 

 described since there is no flow across the vertical barrier and equivalently, 

 there is no flow across a vertical line passing through an antinode. (For the 

 linear theory discussion here, the water contained between any two antinodes 

 will remain between those two antinodes.) Consequently, the flow described 

 here is valid for a barrier at 2Tnc/L = (x = 0) since there is an antinode at 

 that location. 



3. Reflections in an Enclosed Basin. 



Some insight can be obtained about the phenomenon of the resonant behavior 

 of harbors and other enclosed bodies of water by examining the standing wave 

 system previously described. The possible resonant oscillations between two 

 vertical walls can be described by locating the two barriers so that they are 

 both at antinodes; e.g., barriers at x = and tt or x = and 2it , etc., 

 represent possible modes of oscillation. If the barriers are taken at x = 



and X = Ti , there is one-half of a wave in the basin, or if 



length, £g = L/2. 



'B 



is the basin 



Since the wavelength is given by equation (2-4) 



^ - f^ "- (¥) 



the period of this fundamental mode of oscillation is 



t1/2 

 4tt£. 

 T = 



B 



g tanh (nd/Ag) 



(2-81) 



The next possible resonant mode occurs when there is one complete wave in the 

 basin (barriers at x = and x = 2-n) and the next mode when there are 3/2 

 waves in the basin (barriers at x = and x = 3it/2, etc.). In general, i^ = 

 jL/2, where j = 1, 2, .... In reality, the length of a natural or manmade 

 basin Jig is fixed and the wavelength of the resonant wave contained in the 

 basin will be the variable; hence, 



2-113 



