Derivation of modified Smyshlyaev's formulae using integral transform of Kontorovich-Lebedev type Valyaev V. Yu, Shanin A.V Moscow State University Days on Diffraction 2010

The aim Valyaev, Shanin Derivation of modified Smyshlyaevs formulae Our aim is to derive modified Smyshlyaevs formulae for the problem of diffraction of plane wave by quarter plane We are looking for the directivity of the scattered field

2. Problem on unit sphere with cut 1. Separation of radial variable 5. Reconstruction of far-field asymptotic of scattered field Valyaev, Shanin Derivation of modified Smyshlyaevs formulae Smyshlyaevs formula receiver source cut 3. g can be represented as a series over eigenfunctions 4. The series can be transformed into integral over

1. Integral over diverges 2. It can be regularized 3. There is a set of directions, Oasis, where can be deformed to make the integral convergent 4. Oasis is zone free of reflected waves Properties of Smyshlyaevs formula Valyaev, Shanin Derivation of modified Smyshlyaevs formulae Oasis

What is modified Smyshlyaevs formula? Valyaev, Shanin Derivation of modified Smyshlyaevs formulae From Greens function on sphere we switch to edge Greens functions We can express the directivity in terms of edge Greens functions: Benefits: For each edge Greens function depend on only one point on the unit sphere (not on 2, as for g in Smyshlyaevs formula) Oasis is wider source cut receiver

Where does MSF come from? Valyaev, Shanin Derivation of modified Smyshlyaevs formulae We introduce edge Greens functions in 3D space and their directivities: source receiver Then using a trick we get embedding formula: Modified Smyshlyaevs formula Kontorovich-Lebedev-Smyshlyaev transform

What is KLS transform? Valyaev, Shanin Derivation of modified Smyshlyaevs formulae For function of one variable: For function of two variables: We need the following properties of transformants: 1. They are even functions 2. Their only singularities are isolated poles on the real axis 3. They are regular at 4. They decay exponentially as

Properties of KLS transform Valyaev, Shanin Derivation of modified Smyshlyaevs formulae Plancherel theorem: Convolution theorem: How do we use them: We also have more complicated embedding formulae for which we need the convolution theorem.

The proof Valyaev, Shanin Derivation of modified Smyshlyaevs formulae 1. Use well-known formula 2. Split double integral over into two: 3. Transform the contours using evenness and compute inner integrals by residues:

Resume Valyaev, Shanin Derivation of modified Smyshlyaevs formulae 1. Introduced KLS transform 2. Plancherel and convolution theorems are proven 3. These theorems can be used for derivation of MSF