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Diffraction by a Right-Angle Penetrable Wedge: Closed-Form Solution for General Refractive Index

Jonas Matuzas

TL;DR

The paper tackles 2D time-harmonic diffraction by a penetrable right-angle wedge with impedance-matched transmission and general refractive-index ratio $\nu>1$. It develops an explicit genus-one Riemann–Hilbert framework via elliptic uniformization of the Snell surface $\Sigma_\nu$, solving a $2\times2$ RH problem for general forcing data, with the Meixner edge condition enforced by a residue-sum constraint. Extending the lemniscatic case ($\nu^2=2$) to general $\nu>1$, it provides a closed-form solution, a practical evaluation recipe, and a degeneration limit to jet-polynomial regimes, plus a complete Sommerfeld representation linking spectral data to the physical field. The approach yields explicit expressions for the exterior and interior fields, including the far-field diffraction coefficient, and delivers a reproducible symbolic example, making it a versatile building block for more complex wedge and polygon diffraction problems using elliptic-function techniques.

Abstract

We consider the two-dimensional time-harmonic transmission problem for an impedance-matched ($ρ=1$) right-angle penetrable wedge at general refractive index ratio $ν>1$. Starting from Sommerfeld spectral representations, the transmission conditions on the two wedge faces yield a closed spectral functional system whose unknowns live on the Snell surface $Σ_ν: Y^2=ν^2 t^4+2(ν^2-2)t^2+ν^2$. We uniformize $Σ_ν$ by Jacobi/Weierstrass elliptic functions on a torus $\mathbb{C}/Λ$ and solve the resulting $2\times 2$ genus-one Riemann--Hilbert problem in closed form for general finite forcing data. The Sommerfeld radiation condition and the Meixner edge condition are enforced by a simple residue-sum constraint. The construction extends the special lemniscatic case $ν^2=2$ treated in arXiv:2601.04183 and yields a practical evaluation recipe expressed in theta/sigma products and explicit triangular factor matrices. We include the complete Sommerfeld representation connecting the spectral solution to the physical field, explicit forcing data for plane wave incidence, and a worked symbolic example at $ν=3/2$.

Diffraction by a Right-Angle Penetrable Wedge: Closed-Form Solution for General Refractive Index

TL;DR

The paper tackles 2D time-harmonic diffraction by a penetrable right-angle wedge with impedance-matched transmission and general refractive-index ratio . It develops an explicit genus-one Riemann–Hilbert framework via elliptic uniformization of the Snell surface , solving a RH problem for general forcing data, with the Meixner edge condition enforced by a residue-sum constraint. Extending the lemniscatic case () to general , it provides a closed-form solution, a practical evaluation recipe, and a degeneration limit to jet-polynomial regimes, plus a complete Sommerfeld representation linking spectral data to the physical field. The approach yields explicit expressions for the exterior and interior fields, including the far-field diffraction coefficient, and delivers a reproducible symbolic example, making it a versatile building block for more complex wedge and polygon diffraction problems using elliptic-function techniques.

Abstract

We consider the two-dimensional time-harmonic transmission problem for an impedance-matched () right-angle penetrable wedge at general refractive index ratio . Starting from Sommerfeld spectral representations, the transmission conditions on the two wedge faces yield a closed spectral functional system whose unknowns live on the Snell surface . We uniformize by Jacobi/Weierstrass elliptic functions on a torus and solve the resulting genus-one Riemann--Hilbert problem in closed form for general finite forcing data. The Sommerfeld radiation condition and the Meixner edge condition are enforced by a simple residue-sum constraint. The construction extends the special lemniscatic case treated in arXiv:2601.04183 and yields a practical evaluation recipe expressed in theta/sigma products and explicit triangular factor matrices. We include the complete Sommerfeld representation connecting the spectral solution to the physical field, explicit forcing data for plane wave incidence, and a worked symbolic example at .
Paper Structure (43 sections, 3 theorems, 147 equations)

This paper contains 43 sections, 3 theorems, 147 equations.

Table of Contents

  1. Introduction and setup
  2. Physical formulation (for orientation)
  3. Spectral reduction and contribution of this paper
  4. Correct inversion quadratic and substitution check
  5. Half-period shifts for the spectral and Snell maps
  6. Shifts for Y and an involution dictionary on (t,Y,s)
  7. Partner map and domain swap
  8. Kernel expansion at the edge point: sign and absence of a constant term
  9. Residue constraint: necessity and sufficiency under the Meixner edge condition
  10. Main theorem (general refractive index closed-form RH solution)
  11. Parameters, uniformization, spectral map, and corrected inversion
  12. Branch cuts and continuation for the inversion zeta to u
  13. Reduced kernel scalars, factor matrices, and jump matrices
  14. RH problem with general forcing and partner points
  15. Formal Riemann--Hilbert problem statement (analyticity, growth, poles, normalization)
  16. ...and 28 more sections

Key Result

Theorem 2.1

Let $\nu>1$, and define $k,K,K',\omega_1,\omega_2,\tau,\Lambda$ as in §subsec:params. Define $t(u),Y(u),s(u),\zeta(u)$ by eq:tYuniform--eq:zetalift, and let $u(\zeta)$ be given by the corrected inversion eq:Xquad--eq:uofzeta on the physical branch eq:Xphys. Define $\delta_\pm$, $\beta$, $\gamma$, $G with prescribed simple poles at $u=u_m\in D_+$ and $u=u_m^\sharp\in D_-$, satisfying and with the

Theorems & Definitions (5)

  • Theorem 2.1: General-$\nu$ closed-form genus-one RH solution
  • Lemma 6.1: Sommerfeld nullity / uniqueness
  • proof : Proof (reference)
  • Lemma 6.2: Sommerfeld densities from the RH solution
  • proof : Proof sketch