Hi,
The question seems trivial, but I want to check if I miss something.
I'm trying to analyze this article : Classical Analog of Electromagnetically Induced Transparency (http://arxiv.org/pdf/quant-ph/0107061.pdf) which gives a classical analog to EIT by two coupled oscillators, both damped and one is driven too.
The system looks like : ///------[m1]----[m2]----///
where m1=m2=m are the masses and the -- represent the springs. The strings which attach the masses to the walls have both spring constant k, and the spring which connects both masses has constant K. Mass m1 is driven by an harmonic force : [itex]Fe^{-i\omega_{s}t}[/itex] .
The equations given in the article are :
(1) [itex]\ddot{x_{1}}+\gamma_{1}\dot{x_{1}}+\omega^{2}x_{1}-{\Omega_{r}}^{2}x_{2}=\frac{F}{m}e^{-i\omega_{s}t}[/itex]
(2) [itex]\ddot{x_{2}}+\gamma_{2}\dot{x_{2}}+\omega^{2}x_{2}-{\Omega_{r}}^{2}x_{1}=0[/itex]
Where : [itex]x_{1}[/itex] and [itex]x_{2}[/itex] are the displacements of the masses from their equilibrium. [itex]\gamma_{1}[/itex] and [itex]\gamma_{2}[/itex] are the damping constants. And [itex]\omega^{2}=\frac{k}{m}[/itex] and [itex]{\Omega_{r}}^{2}=\frac{K}{m}[/itex].
My question is : shouldn't there be [itex]{+\Omega_{r}}^{2}x_{1} [/itex] term in eqn. (1) and [itex]{+\Omega_{r}}^{2}x_{2} [/itex] term in eqn. 2? For example if the two masses moved the same distance apparently the spring connecting them should be in it equilibrium length and not influence them.
I though that's an error in the article, but I see the same in other articles (for example : http://cms.bsu.edu/sitecore/shell/-/...s/Joe/06_1.pdf and http://arxiv.org/pdf/1006.5167v3.pdf), so what am I missing?
Thanks in advance,
Naftali
The question seems trivial, but I want to check if I miss something.
I'm trying to analyze this article : Classical Analog of Electromagnetically Induced Transparency (http://arxiv.org/pdf/quant-ph/0107061.pdf) which gives a classical analog to EIT by two coupled oscillators, both damped and one is driven too.
The system looks like : ///------[m1]----[m2]----///
where m1=m2=m are the masses and the -- represent the springs. The strings which attach the masses to the walls have both spring constant k, and the spring which connects both masses has constant K. Mass m1 is driven by an harmonic force : [itex]Fe^{-i\omega_{s}t}[/itex] .
The equations given in the article are :
(1) [itex]\ddot{x_{1}}+\gamma_{1}\dot{x_{1}}+\omega^{2}x_{1}-{\Omega_{r}}^{2}x_{2}=\frac{F}{m}e^{-i\omega_{s}t}[/itex]
(2) [itex]\ddot{x_{2}}+\gamma_{2}\dot{x_{2}}+\omega^{2}x_{2}-{\Omega_{r}}^{2}x_{1}=0[/itex]
Where : [itex]x_{1}[/itex] and [itex]x_{2}[/itex] are the displacements of the masses from their equilibrium. [itex]\gamma_{1}[/itex] and [itex]\gamma_{2}[/itex] are the damping constants. And [itex]\omega^{2}=\frac{k}{m}[/itex] and [itex]{\Omega_{r}}^{2}=\frac{K}{m}[/itex].
My question is : shouldn't there be [itex]{+\Omega_{r}}^{2}x_{1} [/itex] term in eqn. (1) and [itex]{+\Omega_{r}}^{2}x_{2} [/itex] term in eqn. 2? For example if the two masses moved the same distance apparently the spring connecting them should be in it equilibrium length and not influence them.
I though that's an error in the article, but I see the same in other articles (for example : http://cms.bsu.edu/sitecore/shell/-/...s/Joe/06_1.pdf and http://arxiv.org/pdf/1006.5167v3.pdf), so what am I missing?
Thanks in advance,
Naftali