Simulating a 1-dimensional wave on (a segment of) an infinite line

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?

• Seems more of a computational problem than a physics problem. However, have you tried setting your boundary oscillators to have the same value as the proceeding one (i.e., $u_N=u_{N-1}$ and $u_1=u_2$)? Sep 6, 2013 at 10:59
• This question is better suited for scicomp.stackexchange.com/?as=1 Sep 6, 2013 at 12:41
• Although ja72's answer shows that there is no boundary condition to do exactly what you want, there are approximations. I suggest you look at the Beam Propagation Method for ideas: much work is done on the problem of waves bouncing back from the boundaries. A slowly increasing loss at the boundary can absorb a wave without reflexion (a graded loss profile rather than an abrupt beginning to a lossy region prevents Fresnel scattering) and also wide angle numerical "antireflexion multilayer coatings" can be used too. So try @ja72 's damping loss idea, but do it in many layers, beginning with ... Sep 7, 2013 at 12:57
• ...a very small loss for a thin section, then another section with a bit higher loss and so on. Also randomize the thickness of the sections to prevent Bragg resonances from the boundaries. Sep 7, 2013 at 13:00

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):

1. Fixed Ends: $u(0)=0$, $u(L)=0$
2. Mirror Ends: $u'=0$, $u'(L)=0$
3. Free Ends: $u''=0$, $u''(L)=0$
4. Periodic Displacements: $u(0)=u(L)$, $u'(0)=u'(L)$
5. 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.

• Thanks for your reply. Althought it may be impossible to obtain this kind of behaviour by means of boundary condition on $u$ and its derivatives maybe it's still possible to think about different (and possibly cheap) strategies. Creating very long chains increases the number of iterations a lot, is it really necessary? Adding dampening seems not to work: the point where the dampening is high acts as a boundary and reflects the wave. Sep 6, 2013 at 13:52
• Maybe can make the invisible oscillators behave symmetrically wrt the visible ones in order to make the wave cancel when it meets its specular counterpart... Sep 6, 2013 at 14:24
• @Marco I used to work with vibration on transmission lines, where waves would need to dissipated at the ends or they will bring the structures down, and the typical solution involved mass tuned dampers. A single DOF damper with matching mechanical impedance would purely absorb all the energy of one frequency wave only. For your case you have multiple frequencies to deal with. My experience tells me that coulomb dampers work best in this case. Sep 6, 2013 at 14:25
• The best dampening I can do is to set v=0 at a node, but this will make the node a reflector of the wave. Sep 6, 2013 at 14:55
• @Marco read this paper on deciding how to apply a damper near the end points in order to absorb any incoming waves. Sep 7, 2013 at 14:50