Further Work

7.2 Further Work

There is a great deal of work to be done before an X-independent numerical method for this transmission problem is developed and the ‘gap’ mentioned in Chapter 2 is bridged. In order to achieve this, we must obtain a better understanding of diffraction through penetrable obstacles. To reach such an understanding, further experimentation with the RTA and BEM would prove beneficial. In particular, it would be instructive

to examine how u d changes as the angle of the incident light is altered. It would also

be worthwhile considering in detail obstacles of different shapes. In this thesis, we studied the square crystal in most detail. This was done for

clarity of presentation since the expected symmetry was identifiable and, using the same shape throughout, it was easier to draw reliable conclusions about the convergence and diffraction patterns. However, in order to gather data about the diffraction of penetrable corners, studying the triangle would be more helpful since it is the shape most like a single wedge. In fact, to better isolate diffraction, considering the infinite penetrable wedge as in [16] (see Figure 7.1), would be useful.

Figure 7.1: Diffraction by an infinite penetrable wedge.

Although this is moving away from ice crystal shapes (since it is not a polygon), I

64 CHAPTER 7. CONCLUSIONS AND FUTURE WORK

believe this would be the best way to establish the form of diffraction, as conjectured in (6.9). We could again use the RTA and BEM to identify u d on Ω in Figure 7.1. Here Ω is an imaginary line which is an arc of the circle centred at O. In order to do this, the BEM and RTA may have to be adapted slightly and perhaps it may prove better to employ a FEM instead of a BEM. To avoid adaptation of the codes one could also consider a very large triangle as an approximation to the open wedge.

Another feature of this problem to investigate would be the effect of diffraction on the ‘far-field pattern’, i.e. the field some distance away from the scatterer as opposed to the boundary data. It may even be noticed that the influence of diffraction is negligible in the far-field, although I believe this to be unlikely. A study may show that the

behaviour of u d in the far-field is easier to model or it may provide some insight into the issues discussed in this thesis, such as the functional form (6.9). Also, considering the application to light scattering by cirrus cloud, it is the far-field that we are interested in ascertaining so why not study this from the outset? Of course, there are some very good reasons for studying the effects on the boundary, such as Green’s representation theorems. However, it may prove a faster, although perhaps less elegant, route to understanding the scattering properties of cirrus clouds, to develop a theory for the approximate far-field behaviour which may be implemented soon.

Evidently, there is a large scope for possible future work on the transmission prob- lem, not only in furthering the numerical approximation of the solution but also search- ing for the analytic solution to the problem of transmission through a penetrable wedge. Naturally, a discovery of the latter would aid the progress of the former.