An intermediate attempt at modelling interference graphically

This modelling work, which I have spent most of my weekend developing, stems from a very simple question which came up in my AS physics textbook, on the topic of wave interference. It went something along the lines of "Draw 6 diagrams, 1 second apart, to show how these two approaching waveforms interfere". I was surprised at how difficult it was to faithfully do this, even with two simple waveforms in wide discrete time jumps and it got me thinking - how could I use a computer to do this better? The result is in an 8-page investigation, where I have endeavoured to develop some theory about how to model passing and reflecting waves and how they interfere - it begins easy, with some simple linear interference by summation of instantaneous displacements, but soon I found that the mathematics became more complicated (particularly in stage 3, where I began to model how waves slowly escape from a trap, like a solid-state laser but without the quantum stimulated emissions).

I have tried to explain myself fully throughout, although not enough proof-reading has been done to keep it from being a little haphazard, and the last page of the document requires some following-up when I have a little more time. However, I am pretty proud of some of the derivations.

In the paper, I have referenced some equations with numbers in square brackets. These match-up with equations on the corresponding Desmos file (here: http://tinyurl.com/zlmzg9d), to illustrate exactly what each equation looks like in practice.

The paper can be found here: http://tinyurl.com/zwegna4

Refraction in the third spatial dimension

In my AS Physics classes, we have reached the wave mechanics point of the course - one of the most fundamental observable properties of waves travelling through media of differing densities is refraction; however, we only ever seem to consider refraction in a coplanar sense (where there is only an incident angle against the plane of the boundary in two dimensions). My investigation of the morning is to develop a little theory, which will hope to extend my understanding of refraction to apply to the third dimension, which up until this point in my education has been ignored.

Firstly, let's review Snell's law. This states that the product of the refractive index of one medium and the sine of the angle in that medium against the normal is constant throughout the refraction process. In algebraic terms:

This provides the basic relationship between the two angles on the diagram below, which demonstrates the coplanar refraction I am talking about:

Similarly, I could represent the effect of refraction on the third-dimension in the same way, by reducing the problem into a coplanar one and ignoring the second dimension in the above example. However, considering all three dimensions at once will involve some basic vector geometry. The direction of the light ray, where the perspex-air boundary is an arbitrary straight line with the equation x = C, can be represented by a 3-D vector U:

This vector can then be divided into two components, dealing with different planes - the x-y plane and the x-z plane. The angle in the y-z plane is irrelevant because it is parallel to the boundary plane.  I have worked through this problem, and here is my result:

From here, I used Desmos graphing calculator to demonstrate the result of this. I spent much time working with the z-axis, trying to produce the most convincing one possible by adjusting the positions of the x and y axis, all parametrically under the control of the rotation parameters A, B and C. I had some success with this, using the sinusoidal nature of rotational perspective to produce a set of coordinate axes which can be rotated about the origin (i.e. the real z-axis) in the x-y plane, and the image of the x-axis, allowing enough movement to achieve most angles on any 3-D subject. A current demonstration of this somewhat rudimentary and flawed model can be found here: http://tinyurl.com/hk4d3l7. In the future, I hope to be able to understand how not only full rotational freedom can be achieved through the parameters A, B and C, but also how I can faithfully project real points (x,y,z) onto these simulated axes while they can stand up to rotation.

However, since I have yet to truly crack the matter, I have for now settled with a simple pseudo-z-axis. However, it still demonstrates my mathematics fairly well. Furthermore I have paid attention to detail diagrammatically by programming Desmos to draw proportional arrows for the magnitudes of each component ray, as well as the resultant incident and refracted rays. The online version can be found here: http://tinyurl.com/jthre74.

Overall, I feel I have opened a metaphorical can of worms for myself when it comes to the true complexities of this physical phenomenon. Further research has yielded information regarding Fresnell's equations for transparent materials, which determine the relative amplitudes (and hence intensities) of reflected and refracted light at such a boundary between two media, which could be programmed into the graphing calculator. Also, I could isolate the mathematics from the assumption that the boundary is parallel to the y-z plane, such that refraction and reflection on a 3-D surface of differential-determined gradient could be modelled. Perhaps a follow-up to this will be required in the future.