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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
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Wave on a Linear Array

First I should say that the wave we are considering here is one dimensional and therefore a longitudinal wave. It involves both compression and extension relative to the equilibrium distance force between particles. Instead of displaying physical springs as in a mass-spring array we ›will use forces due to potentials between particles. The differential equation for the acceleration of the `ith` mass is:

`(d^2x_i)/(dt^2)=-1/m(del V(x))/(del x_i)\ \ \ (1)`

where V(x) is the potential defined by:

`V(x_i)=sum _(j=1) ^(j=n) v(x_i-x_j)`

where `x_i` and `x_j` are particle positions. For example a compressive potential expression for `v(u)` might be:

`v(u)=v_1/u \ \ \ (2)`

where `v_1` is an energy scaling factor. Then the acceleration of `x_i` becomes:

`(d^2x_i)/(dt^2)=v_1/m sum _(j=1) ^(j=n) (x_i-x_j)/|x_i-x_j|^3\ \ \ (3)`

so that, when `x_i>x_j`, `x_i` gets accelerated to the right since `x_j` is pushing `x_i`. The potential in equation 2 is a bit oversimplified because potentials between particles usually result in equilibrium distances between the particles. The Lennard-Jones potential (LJ) does result in equilibrium distances since it is repulsive at close distance and attractive at longer distances:

`v_(LJ)(u)=4 epsilon((sigma/u)^12-(sigma/u)^6) \ \ \ (2a)`

where `epsilon` takes the role of `v_1` in equation 2 and `sigma` is a scaling distance. The equilibrium distance between particles for the LJ potential is

`u_(LJ)= 2^(1/6) sigma \ \ \ (4)`

Since we desire smooth accelerations of the particles on our linear array, our nominal particle distance will be that given by equation 4.

I've chosen to make the initial displacements relative to the equilibrium spacings given by equation 4 to be in the form of small sinusoidal variations.

`x_i = i delta x_(0i) + A_0 sin(i K x_(0i))`

where i is an integer over the number of masses, `delta x_(0i)` is from equation 4, the amplitude `A_0` is much less than `delta x_(0i)` and `K=(2pin_c)/L` is the initial wave vector where `n_c` is the number of cycles, and L is the length of the array. I've also chosen to keep the mass elements at each end of the array fixed in position. One might think that that initial sinusoidal waveform would be repeated as time goes on. That does not work out because of a simple fact: When we chose a potential that gave equilibrium distances, we also obtained a whole new set of particle vibration frequencies. The simplest way to understand this is the frequency, `omega_2`, between just two particles with the potential given by equation 2a. For very small displacements from equilibrium, the two particles exhibit much higher frequencies than the frequencies of the entire linear array. So what happens is that the initial large amplitude, low speed, group displacements at low frequency, `omega_g` gets converted to much smaller amplitude displacements at higher speeds and frequencies. All this happens while the sum of potential energy and kinetic energy remains constant.

If you observe the behavior of the group displacement wave for long enough, you will notice that the wave almost repeats its initial sine wave displacement. This occurs at times, `tau`, when `omega_2 tau` is approximately an integer multiple of `omega_g tau` while `omega_g tau` is approximately `2lpi` where `l` is an integer. An alternative way that I could have induced a wave is to sinusoidally drive the mass at the left end of the array. However this adds energy with each cycle so the wave amplitude grows and it is hard to interpret the results. The time and space dependence of the wave displacement amplitude is qualitatively:

`A(x,t)=A_0 cos(omega_2t)sin(Kx-omega_g "t")+a sin(kx-omega_2t)`

so that, when `cos(omega_2t)=2mpi` and `omega_g "t"=2lpi` where `m` and `l` are integers, the displacement amplitude returns to equation 5 except for the small ripple amplitude `a` at a much higher spatial frequency `k`.

Comments and Critique

The initial conditions of the array allow for a sustained wave albeit not what one might at first expect. I could have set to attractive part of the LJ potential to zero and the wave would not be very different. However, if I set the repulsive part or the LJ potential even a bit lower while keeping the attractive part the same, the wave amplitude quickly becomes very large. This is because the particles can come together with the only restriction that the end particles have to stay fixed. So the particles all come to the middle of the array. This seeming dichotomy is due to fact that the attractive force is much weaker than the repulsive force.

The learner may find that certain combinations of the selectable parameters do not result in a stable array. If that is the case, a restart can be accomplished by pressing the "Refresh Page" button. Then the learner should choose less aggressive parameters and press "Start" again.