Exploring a physical-mathematical problem

Arjen Markus (16 november 2012) I was testing an idea about how to mimick a flow field and the transport of particles in that flow field (I do not want to solve the actual hydrodynamic equations, just get some sort of rough approximation, especially with regards to the statistical properties) and the almost simplest realisation of that idea is embodied by the following two equations:

    dx/dt = sin( ky + wt + phi )
    dy/dt = -cos( kx + wt + phi )

That is: a particle is moving along with two propagating waves, one in the y-direction (with the water moving back and forth in the x-direction) and one perpendicular to that in the x-direction (the water moving back and forth in the y-direction).

I expected a particle to move around a bit in some closed cycle. But to my surprise it is moving on the long run along the line x=y. Depending on the phase (phi) it will move (in loops of various shapes) in the positive or the negative x-direction. The program below illustrates this. Note the inverted y-axis that is inherent to the canvas - I did not bother correcting it.

# twocompflow.tcl --
#     Using two perpendicular components, what is the resulting
#     particle trajectory?
pack [canvas .c -width 400 -height 400]

set x 0.0
set y 0.0

set dtime 0.1
for {set step 0} {$step < 10000 } {incr step } {
   set time [expr {$dtime * $step + 20.0}]
   set u    [expr {sin(0.03*$y+0.03*$time)}]
   set v    [expr {-cos(0.03*$x+0.03*$time)}]

   set xold $x
   set yold $y

   set x    [expr {$x + $u * $dtime}]
   set y    [expr {$y + $v * $dtime}]

   .c create line $xold $yold $x $y

.c move all 200 200