**6. Conclusion**

In this paper we have described numerical schemes and their implementation for the solution of scattering of a plane wave by two different cylindrical structures: a single-cornered structure and a second structure with two corners, each with three different boundary conditions imposed on their surfaces - soft, hard and an impedance boundary condition. We have numerically demonstrated that the field scattered by the rounded structure converges, in both the *L*<sup>2</sup> and *L*<sup>∞</sup> norm, to that scattered by the corresponding sharp cornered object as the radius of curvature in the vicinity of the corner tends to zero.

It is important to use an appropriate quadrature scheme - a graded mesh - in order to obtain numerical results efficiently, for both the scatterer with sharp corners and for the scatterer with rounded corners possessing small radii of curvature. We anticipate that improvements to the graded mesh employed for the two-cornered object will match the rate of convergence demonstrated for the single-cornered lemniscate.

Our results show that for the soft boundary condition, the *L*<sup>∞</sup> norm difference between the near or far scattered field of the single-cornered scatterer and that of the rounded scatterer is less than 4% when the radius of curvature is restricted so that *kρ* ≤ 3*π*/50. This percentage reduces to 3% or 2% respectively, when the boundary condition is replaced by the Neumann boundary condition or the impedance boundary condition (with *λ* = 1 + *i*), respectively. More precise measures of the difference are given in Table 4. Similar results were obtained for the the two-cornered object, and are displayed in Table 8.

Our approach provides a relatively simple yet efficient and accurate method for computing near and far-fields scattered by sharp cornered objects of diameter *D* up to a few wavelengths in extent. Accuracy was of paramount importance in this study in assessing the effects of rounding a corner. Our calculations rigorously examined the regime 1 ≤ *ka* ≤ 10 corresponding to 0.318 ≤ *D*/*λ*<sup>0</sup> ≤ 3.18, where *λ*<sup>0</sup> is wavelength.

A more sophisticated approach to the scattering from soft cylindrical structures with sharp corners is given in [7]. It employs the so-called recursively compressed inverse preconditioning method, and as the authors note in their survey of the two dimensional scattering literature, it alone addresses the problem of accurate near-field evaluation in scatterers with corners.

In conclusion, this paper provides some precise quantification and assessment of the impact that the rounding of the corner of a sharp cornered object has on the scattering of acoustic waves. The method would seem to be extendible considerably beyond the wavelength range examined, constrained mainly by computer resources.
