remains at the SWL for all values of t and other points (antinodes) where 

 the water particle excursion at the surface, is 2 H^ 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 2-61. Since water motion at the anti- 

 nodes 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 2ttx/L = (x = 0) since there is an antinode at that location. 



2.53 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 oscillat- 

 tions between two vertical walls can be described by locating the two 

 barriers so that they are both at antinodes; for example, barriers at 

 X = and tt or x = and 2tt, etc. represent possible modes of oscillation. 

 If the barriers are taken at x = and x = it, there is one-half of a wave 

 in the basin or, if Iq is the basin length, Iq - L/2. Since the wave- 

 length is given by Equation 2-4 



gT^ , /27rd\ 

 L = ^- tanh — , 

 2jr \ L / 



the period of this fundamental mode of oscillation is, 



(2-80) 



T = 



47^4 



gtanh (jrd/^) 



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

 the basin (barriers at x = and x = 2ir) and the next mode when there are 

 3/2 waves in the basin (barriers at x = and x = 37t/2, etc. In general, 



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



manmade basin 2.^, is fixed and the wavelength of the resonant wave con- 

 tained in the basin will be the variable; hence, 



in 



L = — j = 1,2,----, (2-81) 



may be thought of as defining the wavelengths capable of causing resonance 

 in a basin of length l^. The general form of Equation 2-80 is found by 



2-115 



