Streetj Chan and Fromm 



Experiments were also performed on the problem of generating 

 periodic waves by pressure pulse (Sec. 4.2). Instability at the free 

 surface became explosive after 600 time steps when the particle 

 method was used. Using the forward implicit nnethod, we were able 

 to calculate up to more than 3000 steps ajid there were still no signs 

 of instability. 



The numerical tests described above indicated that it is ad- 

 vantageous to use the second- order upstream difference method to 

 compute the convective contributions to Uj^i; and v":^! . For the 

 free surface calculations, the forward implicit scheme Is best. 

 However, the particle method of computing the free surface need not 

 be dismissed altogether. The Eulerian method is restricted to waves 

 in a channel whose two ends are vertical walls. If the water surface 

 has an advancing front, such as a solitary wave climbing on a slope 

 [Chan and Street, 1970b] , the particle method is the only choice. 

 When 6t is small enough and the particle velocities are evaluated 

 at the n+1*^ time step, the particle method does provide a stable 

 solution. However, the particle method should not be used in the 

 simulation of periodic waves over long periods of time. 



4, 2. Results and Discussion 



As an example, periodic pressure pulses were used to 

 generate a train of oscillatory waves in a channel of constant depth 

 (Fig. 9). The fluid is entirely at rest at t = 0. Then, the pressure 

 distribution 



Fig. 9. Setup of pressure pulse problem 



178 



