Embedding formulae for scattering by three-dimensional structures

On the diffraction of acoustic waves by a quarter-plane

“…These figures also show that in some cases the third formula can cover the whole sphere ( Figures 17 and 19 ), but this is not always the case as shown in Figure 20 and 22. In order to demonstrate that these results agree with the results published in [18], Figure 18 gives the value of the Dirichlet diffraction coefficient along the black dashed line of Figure 17. Figure 18 also emphasises the perfect agreement between the first and the third Modified Smyshlyaev formulae in this case.…”

“…Figure 17 shows the result only for the Dirichlet case, as in this case the incident wave direction is in the same plane as the quarter-plane so that the Neumann diffraction coefficient is zero everywhere. In [18], Shanin et al published the value of the diffraction coefficient given by the first formula along just the dashed black line in Figure 17 (they did not give any results linked to the third formula) and didn't implement the coordinate equation method numerically (they used an integral equation instead). In all the figures the superiority of the third modified Smyshlyaev formula, and the limitations of the first two formulae, are clear.…”

“…However, the detailed proof for both cases requires a multitude of details, and will be presented in [30]. In a recent publication [18], Shanin et al derive some similar formulae for an arbitrary corner angle in the Dirichlet case, but for definiteness we will here consider only the quarter-plane. In order to compute the diffraction coefficient, it is essential to be able to compute the spherical edge Green's functions in an efficient way.…”

“…The main advantage of this method compared to the one mentioned previously is that in this case the formulae giving the diffraction coefficient are "naturally convergent" in the sense that they don't require a special treatment to regularise or accelerate their convergence. This method is mainly based on the theory of embedding formulae, introduced by Williams [15] and developed by Craster, Shanin and others [16,17,18].…”

Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane

“…The first motivation is the inconsistency of the representation of the diffraction coefficient of the corner. Indeed, for example, two very important papers on the subject by Shanin (2005b) and Skelton et al (2010) give a different representation of this coefficient. The second motivation is the conjecture formulated at the end of Assier & Peake (2012), where the possibility of an ultimate modified Smyshlyaev formula was expressed.…”