and the transmission coefficient will depend upon the geometry and compo- 

 sition of a structure and the incident wave characteristics such as wave 

 steepness and relative depth d/L, at the structure site. 



2.52 REFLECTION FROM IMPERMEABLE, VERTICAL WALLS (LINEAR THEORY) 



Impermeable vertical walls will reflect almost all incident wave 

 energy unless they are fronted by rubble toe protection or are extremely 

 rough. The reflection coefficient x is therefore equal to approximately 

 1.0 and the height of a reflected wave will be equal to the height of the 

 incident wave. Although some experiments with smooth, vertical, impermeable 

 walls appear to show a significant decrease of x with increasing wave 

 steepness, Domzig (1955), Coda and Abe (1968) have shown that this paradox 

 probably results from the experimental technique, based on linear wave 

 theory, used to determine x- The use of a higher order theory to describe 

 the water motion in front of the wall gives a reflection coefficient of 

 1.0 and satisfies the conservation of energy principle. 



Wave motion in front of a perfectly reflecting vertical wall subjected 

 to monochromatic waves moving in a direction perpendicular to the barrier 

 can be determined by superposing two waves with identical wave niombers, 

 periods and amplitudes but traveling in opposite directions. The water 

 surface of the incident wave is given to a first order (linear) approxi- 

 mation by Equation 2-10, 



(2-10) 



and the reflected wave by. 



H. 



T? = cos 



' 2 



+ 



L T 



Consequently, the water surface is given by the sum of n^ and n^ °^- 

 since H^ = H^, 



H, 



T? = 7?,- + T?^ 



/27rx 2jrt\ /27rx 27rt^ 



cos — + cos + 



\L T/ I L T , 



which reduces to 



27rx 2-nt 



■n = H. cos cos -— - 



' L T 



(2-79) 



Equation 2-79 represents the water surface for a standing wave or olapotis 

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

 both cos(2ttx/L) and cos(2TTt/T) equal 1. The water surface profile as a 

 function of 2ttx/L for several values of 27rt/T are shown in Figure 2-61. 

 There are some points (nodes) on the profile where the water surface 



2-113 



