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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
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Assembly of 3D Parts Using Javascript Web Graphics Library 3D Compared to Javascript 3D Graphics 3D Graphics Tutorial Beginnings Proof of Pythagoras a^2+b^2=c^2
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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
Chapter 4
Chapter 5
Chapter 6
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Chapter 8
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Probability of Birthday Match Random Creation of Probability Function Profiles Two Dies Probability Distribution Three Dies Probability Distribution
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Scattering from a Crystalline Target 2D Scattering from a Crystalline Target 3D
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Energy Conduction in a 2D Numerically Modeled Solid Lattice Unbalanced Gas Dynamics:Energy Flow in a Gas
Chapter 4
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Time Dependent Wave Equation 1D

Introduction

This is the time-dependent analog of resonator1D page which found the stationary states of an electromagnetic wave (optical) resonator. Basically we release an optical pulse and observe its motion inside a 1D domain.

I recently discovered a very thorough paper on use of finite difference methods to solve the wave equation. In great detail it goes through boundary conditions, etc. Here I will use some of its methods to compute the propagation of a wave function.

Review of the finite difference method for the wave equation

For the wave equation, we have the second derivatives in both time and space:

`(d^2E)/(dt^2)=c^2(d^2E)/(dx^2)`

where c is the speed of propagation of light in vacuum. Translated to finite difference (FD) expressions this bocomes

`1/(deltat)^2(E_i^(n+1)-2E_i^n+E_i^(n-1))=(c^2)/(deltax^2)(E_(i+1)i^(n)-2E_i^n+E_(i-1)^(n))`

where index n denotes the time increment and index i denotes the space increment. If we multiply both sides of this equation by `deltat^2` we have:

`(E_i^(n+1)-2E_i^n+E_i^(n-1))=c^2(deltat^2)/(deltax^2)(E_(i+1)i^(n)-2E_i^n+E_(i-1)^(n))`

We can now solve for `E` at the next time increment, `E_i^(n+1)`

`E_i^(n+1)=2E_i^n-E_i^(n-1)+c^2(deltat^2)/(deltax^2)(E_(i+1)i^(n)-2E_i^n+E_(i-1)^(n))`

We can get the difference `deltaE_i^(n+1)=deltaE_i^(n+1)-E_i^n` from this equation:

`E_i^(n+1)-E_i^n=E_i^n-E_i^(n-1)+c^2(deltat^2)/(deltax^2)(E_(i+1)i^(n)-2E_i^n+E_(i-1)^(n))`

We see that

`deltaE_i^(n+1)=deltaE_i^n+c^2(deltat^2)/(deltax^2)(E_(i+1)i^(n)-2E_i^n+E_(i-1)^(n))`

Therefore, if the second derivative in space is not zero, the increment in `(dE)/dt`, which is `deltaE_i^(n+1)-deltaE_i^n`, which is really just `(d^2E)/(dt^2)` and will also not be zero.

For the case of electromagnetic wave propagation the equation becomes

`deltaE_i^(n+1)=deltaE_i^n+c^2(epsilon_0)/epsilon(deltat^2)/(deltax^2)(E_(i+1)i^(n)-2E_i^n+E_(i-1)^(n))`

where `epsilon` is the dielectric constant of the slab and `epsilon_0` is the dielectric constant of vacuum. Then epsilon/epsilon_0` is the square of the index of refraction of the slabs.

Explanation of Graphics

We start with a simple sine pulse with gaussian rolloff in the center of the domain (canvas). The index of refraction of the domain is 1.0 except for two fixed slabs to the left and right side of the sine pulse. Since the pulse is just a standing sine wave, it will split into two sub-pulses and propagate to the left and right until it reaches the edge of the domain. As you can see, the sub-pulses slow down, their amplitude is diminished, and their spatial frequency becomes much larger in the slabs. The edges of the slabs have swaged index rolloff which reduces reflections of the sub-pulses.

Description of Longer Term Propagation

On the first cycle of the pulse propagation where the sub-pulses return to the center of the domain, the sub-pulses usually merge (reconstruct) into almost exactly the same pulse that was started. You can confirm this by use of the Single Step checkbox when the pulses are almost merged. However, on susbsequent cycles, depending on the slab index and its swaged rolloff width this reconstruction will get less and less perfect due to reflections from the slab boundaries.