c. Scale Effects . 



(1) Short-Period Waves and Undistorted, Linear-Scale Models . 

 Although harbor models with the problem of short -period waves are de- 

 signed in accordance with Froude's law and are constructed geometrically 

 similar to the prototype, the conditions of similitude are not met com- 

 pletely in most instances because friction forces cannot be modeled 

 correctly. Waves are attenuated by surface tension, internal friction, 

 and friction in the bottom boundary layer. Friction effects also reduce 

 the amount of wave energy that is transmitted through pervious coastal 

 structures such as rubble-mound breakwaters and jetties. Bottom friction 

 and the energy loss as waves are transmitted through rubble-mound break- 

 waters are exaggerated in harbor models and these phenomena constitute 

 the major scale-effect problems in the design of short -period, wave 

 action models. If the linear-scale ratio (Lj.) is too small, surface 

 tension can affect the wave velocity, resulting in errors in wave refrac- 

 tion, and internal friction can considerably reduce the wave heights. 

 The effects of surface tension on wave velocity are shown in Figure 4-1. 

 According to Keulegan (1950a), the expression for the variation of wave 

 height with time, due to internal friction, is 



i ^ 87r2j;t/x2 



(4-25) 



and, if t' is the time required to reduce the wave height 50 percent, 



t' - 0.0088 — . (4-26) 



In terms of wave period (T) and the distance of wave travel in time 

 t', (x) , and with a temperature of 21° Celsius (70° Fahrenheit) 

 (v = 1.059 X 10"^), equation (4-26) reduces to 



Xp = 11 1,750 T^ (tanh^^ . (4-27) 



The effects of internal friction on the reduction of wave height for 

 relatively small wave periods are shown in Figure 4-2. The figure shows 

 that the effects of surface tension and internal friction can be made 

 negligible in harbor wave action models by the proper selection of linear 

 scale. The law of variation of the wave height with distance due to fric- 

 tion in the viscous boundary layer for a train of progressive oscillatory 

 waves in a rectangular channel of uniform cross section, is (Keulegan, 

 1950b) 



H 



= p -ax 



= e "-^D . (4-28) 



217 



