I'm trying to numerically simulate a 1-dimensional with a chain of linked harmonic oscillators as described here (the result can be seen here). The simulation behaves like a wave on a finite line segment with fixed or free boundaries (depending on the boundary conditions you set): when the wave reaches the border it bounces back (as it should by by conservation of energy). What I would like to do is to make my queue of oscillator behave like a segment of an infinite queue of oscillators, that means there must be no bounces and the energy must flow away throught the boundaries. I tried some strategies which turned out not to work such as adding invisible extensions of the queue and making it more dampened (actually it turned out to be equivalent to not adding the extension at all!). Any Idea?
So what should the boundary conditions be for a segment of an infinite line. Let us explore the options ($u$ is displacement, $u'$ is slope and $u''$ is curvature):
- Fixed Ends: $u(0)=0$, $u(L)=0$
- Mirror Ends: $u'=0$, $u'(L)=0$
- Free Ends: $u''=0$, $u''(L)=0$
- Periodic Displacements: $u(0)=u(L)$, $u'(0)=u'(L)$
- Periodic Tensions: $u'(0)=u'(L)$, $u''(0)=u''(L)$
None of them seem to work to simulate an infinite line. In fact the only way to simulate an infinite line is to assume there is some kind of periodicity in the shape and apply boundary conditions to enforce it. So the answer is you cannot do it.
Remember the old rule of physics, that every symmetry is a manifestation of some kind of conservation law. The energy is not conserved in the segment so there cannot be a symmetry to apply.
The best solution is to create some kind a really long segment, with free end conditions, add dampers and light springs on the end, and only display a short part of that away from the ends.