Computation of the Motion of Long Water Waves 



DIRECTION OF MASS TRANSPORT 



STILL WATER 

 LEVEL 



BACK CURRENT 



?1.ZS 11. so 11.75 IE. 00 IE. 25 



12. SO 



13.S0 13. 7S in. 00 



Fig. 15. Motion of fluid particles 



The paths of the fluid particles are plotted in Fig. 15. We 

 selected three fluid particles which lie on the vertical plane x = 12,0 

 at t = 0.0, Their initial vertical positions are y = 0.0, 0.5 and 

 1.0, respectively. The instantaneous particle positions are plotted 

 at every 5 6t's (6t = 0,05). Each particle moves in an oscillatory 

 pattern which completely differs in nature from the translation motion 

 in a solitary wave. The surface particle travels in a quasi- elliptic 

 orbit but never returns to its original position. Thus, there is a net 

 mass transport in the direction of wave propagation near the free 

 surface. At half water depth the scale of the orbits is smaller and 

 the current (mass transport) is opposite to the wave direction. On 

 the channel bottom the particle merely goes back and forth hori- 

 zontally and the "backward current" is also larger there. Because 

 the wave channel in our simulation is a closed system, the fluid 

 carried along by the surface waves must return in the opposite 

 direction In the lower fluid layers. 



Finally, a comparison was miade with the numerical solutions 

 of Fangmeler [ 1967] . The qualitative agreement was good, as was 

 the agreement in the wave phase; however, the SUMMAC method 

 gave a much better treatment of the free surface that markedly re- 

 duced the height of the largest of the waves as compared to 

 Fangmeler's simulation. 



183 



