The link between the Wiener-Hopf and the generalised Sommerfeld Malyuzhinets methods: Lecture 3

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• Guido Lombardi (Politecnico di Torino; Politecnico di Torino); J.M.L. Bernard (ENS de Cachan)
• Thursday 08 August 2019, 14:15-15:30
• Seminar Room 1, Newton Institute.

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WHT - Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications

The
Sommerfeld Malyuzhinets (SM) method and the Wiener Hopf (WH) technique are
different but closely related methods. In particular in the paper “Progress and
Prospects in The Theory of Linear Waves Propagation” SIAM SIREV vol.21, No.2,
April 1979, pp. 229-245, J.B. Keller posed the following question “What
features of the methods account for this difference?”. Furthermore
J.B. Keller notes “it might be
helpful to understand this in order to predict the success of other methods”.

We
agree with this opinion expressed by the giant of Diffraction. Furthermore we think that SM and
WH applied to the same problems (for instance the polygon diffraction) can determine a helpful synergy. In the past
the SM and WH methods were considered disconnected in particular because the SM
method was traditionally defined with the angular complex representation while
the WH method was traditionally defined in the Laplace domain.

In
this course we show that the two methods have significant points of similarity
when the representation of problems in both methods are expressed in terms of
difference equations. The two methods show their diversity in the solution
procedures that are completely different and effective.
Both similarity and diversity properties are of advantage in “Progress and Prospects in The Theory of
Linear Waves Propagation”. Moreover
both methods have demonstrated their efficacy in studying particularly complex
problems, beyond the traditional problem of scattering by a wedge: in
particular the scattering by a three part polygon that we will present.

Recent
progress in both methods:

One
of the most relevant recent progress in SM is the derivation of functional
difference equations without the use of Maliuzhinets inversion theorem.
One
of the most relevant recent progress in WH is transformation of WH equations
into integral equations for their effective solution.

This talk is part of the Isaac Newton Institute Seminar Series series.

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