the terms in equation 2-27a are divided by g, it is then in the dimen- 

 sionless form 



^^ + sin = . (2-27b) 



gdt2 



Let T = t/T, where T is the period o£ the pendulum and t is dimen- 

 sionless, from which dt = T dx, dt^ = T^ dx^, and, by substitution, 



-^ ^ + sin = . (2-27C) 



gT^ dr^ 



In equation 2-2 7c each term is dimensionless. This equation shows that, 

 for any two pendulums governed by this equation, the solution for G 

 will be the same, i.e.. 



at corresponding times if 



^m m 



(2-28) 



or for two pendulums in the same location (from eq. 2-16), 



T \2 



The following examples relate to fixed-bed harbor wave action models. 

 Other examples of the use of differential equations to obtain the require- 

 ments for similarity between scale models and their full-scale counterparts 

 are given by Langhaar (1951), Duncan (1953), Keulegan (1966), and Young 

 (1971). 



a. Undistorted Model (Intermediate Depth Waves, 0.05 < d/X < 0.5) . 

 The system of differential equations underlying the wave motion appli- 

 cable to waves of small amplitude and moderate periods in intermediate 

 depths, is considered to determine the conditions for similarity between 

 model and prototype. In this examination, the frictional effects, which 

 are nearly always small in nature and can usually be made negligible in 

 the model, or can be accounted for by analytical or experimental means, 

 are ignored and the flow everywhere is assumed irrotational. Thus, it 

 will be assumed that motions are created from rest and, accordingly, a 

 velocity potential ((i exists in a three-dimensional domain of x, y, z 

 points. Various texts show that ii> = <l> (x,y,z,t), which gives the veloc- 

 ity components 



(2-29) 





30 





90 





d0 



u - ■ 



3x' 



V - - 



ay' 



w - ■ 



dz 



35 



