Sommerfeld Malyuzhinets (SM) method and the Wiener Hopf (WH) technique are

different but c losely related methods. In particular in the paper &ldquo\;Progress and

Prospects in The Theory o f Linear Waves Propagation&rdquo\; SIAM SIREV vol. 21\, No.2\,

April 1979\, pp. 229-245\, J.B. Kel ler posed the following question &ldquo\;What

f eatures of the methods account for this difference ?&rdquo\;. Furthermore

J.B. Keller notes &ldquo \;it might be

helpful to understand this in ord er to predict the success of other methods&rdquo\; . We

agree with this opinion expressed by the g iant of Diffraction. Furthermore we think that SM and

WH applied to the same problems (for instan ce the polygon diffraction) can determine a helpfu l synergy. In the past

the SM and WH methods we re considered disconnected in particular because t he SM

method was traditionally defined with the angular complex representation while

the WH me thod was traditionally defined in the Laplace doma in.

In

this course we show that the two met hods have significant points of similarity

when the representation of problems in both methods ar e expressed in terms of

difference equations. T he two methods show their diversity in the solutio n

procedures that are completely different and effective.

Both similarity and diversity proper ties are of advantage in &ldquo\;Progress and Pros pects in The Theory of

Linear Waves Propagation &rdquo\;. Moreover

both methods have demonstrat ed their efficacy in studying particularly complex

problems\, beyond the traditional problem of s cattering by a wedge: in

particular the scatter ing 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 equ ations without the use of Maliuzhinets inversion t heorem.

One

of the most relevant recent progress in WH is transformation of WH equations