with the uncle rTfjrater contours » These changes in wave characteristics are 

 known as wave refraction, (See Figure 11). 



23 c. The velocity of a wave C in any depth of water d is given by 



„ / gL tanh 2Trd 



G - V 2iF "IT (1) 



In deep water j, 2vd/L becoitES very large, the hyperbolic tangent of 2Tfd/L 

 approaches unity 5, and the wave velocity approaches: 



^ ^ 2ir (2) 



In shallow water , 2r<d/L becomes very small, tanh 2nd/L approaches 2vd/L and 

 the wave velocity approaches: 



G = \W~' (3) 



Those depths over which equation (l) applies unchanged are knoT-m, as 

 transitiona l water, 



2I1.0 The total energy per unit crest width in a wave is 



E^ =(l/8|p gH^L (h) 



where p is the water density and pg = w, the weight of water per cubic 

 foot. Of thiSj only a paz't is transmitted forward with the x\rave form,, This 

 transmitted part may be determj.ned as 



E = n E^ (^) 



ii.Tfd n 



where n =(l/2j 1 + L 



L HnFlt-ir d/rj (6) 



In deep water sinh ittrd/L becomes very large and the expression for n becoiiBs 

 n = 1/2. (The wave length in deep water L may be found from equation (2) 

 by substituting the general relationship L = CT_, and in feet is given by 

 L ^ 5«12T o Table 1 of Appendix D lists values of 2vd/L^ iiTrd,/L, tanh 2Tid/L 

 and sinh hvd/L as functions of the ratio d/tL ) . 



25 » Re_fra.ction o f Waves. - >Jhen wave refraction is considered, it is 



assumed that in a wave aivancing toward shore 5 no energy flews laterally 



along a wave crest. That is^, between two lines drawn perpendicular' to a 



wave crest as it passes over changing topography, the transmitted energy 



remains constanto The wave energy transmitted forward between any two such 



lines in deep water is E = 1/2 b E^ , wliere b is the spacing of the two 



c o Tq o 



lines in deep water, (The subscript o always refers to deep water con- 

 ditions )„ It may therefore be equated to the energy transmitted fortjard 

 between the same two lines in shallow water (E = n b E ), where b is the 



17 



