waves, so that the transfer numbers derived are valid only in the areas 

 from the entrance line up to the position of the breaks. In these situa- 

 tions, recourse to analysis and to experimental data relative to runup, 

 reflection, and reformation is necessary to make the proper adjustment 

 of the model indicated in the critical areas. 



b. Distorted Model (Long Waves, < d/X < 0.05) . In the case of 

 long waves the relationship of times for the corresponding events in a 

 distorted model and its prototype is also readily established by differ- 

 ential equations. A departure from the system of equations used above 

 for undistorted models is indicated for ease in development. In long 

 waves, where wavelengths are large compared with water depths, vertical 

 accelerations are negligible and pressures in the liquid are hydrostatic 



P = Pg(T? + z) . 



(2-55) 



The velocity components 

 tions of X, y, and t. 



u and V are independent of z and are func- 

 The dynamic surface conditions are 



9u _ dju 9v _ _ bri 



9t "^ 9x ' at ^ ay 



(2-56) 



These are equivalent to equation 2-34. Taking the equation of the free 

 surface in the form 



F(x, y, z, t) = z - 17 , z = Tj 



(2-57) 



the surface kinematic condition, stating that a particle on the surface 

 remains on the surface, is 



|2 +u|2 + v|2- w = 0, 



at ax ay 



Z = TJ 



(2-58) 



This corresponds to equation 2-35 if second-order terms are neglected. 

 Taking the equation of the rigid bottom surface in the form 



F(x, y, z) = z + d, z = -d 



(2-59) 



the bottom surface condition, stating that particles on the bottom surface 

 move along that surface, is 



ad , 9d , _ f. _ , 



u -r- + V ^r- + w = 0, z = -d , 



ax ay ' 



(2-60) 



which corresponds to equation 2-33. Taking the condition of continuity 



au . av , aw 



ax ay az 



(2-61) 



41 



