dorsal/arxiv
View SchemaDiffraction by a Right-Angle Penetrable Wedge: Closed-Form Solution for General Refractive Index
| Authors | Jonas Matuzas |
|---|---|
| Categories | |
| ArXiv ID | 2601.07070vv1 |
| URL | https://arxiv.org/abs/2601.07070 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
We consider the two-dimensional time-harmonic transmission problem for an impedance-matched ($\rho=1$) right-angle penetrable wedge at general refractive index ratio $\nu>1$. 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 $\Sigma_\nu: Y^2=\nu^2 t^4+2(\nu^2-2)t^2+\nu^2$. We uniformize $\Sigma_\nu$ by Jacobi/Weierstrass elliptic functions on a torus $\mathbb{C}/\Lambda$ 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 $\nu^2=2$ 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 $\nu=3/2$.
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"abstract": "We consider the two-dimensional time-harmonic transmission problem for an impedance-matched ($\\rho=1$) right-angle penetrable wedge at general refractive index ratio $\\nu\u003e1$. 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 $\\Sigma_\\nu: Y^2=\\nu^2 t^4+2(\\nu^2-2)t^2+\\nu^2$. We uniformize $\\Sigma_\\nu$ by Jacobi/Weierstrass elliptic functions on a torus $\\mathbb{C}/\\Lambda$ 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 $\\nu^2=2$ 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 $\\nu=3/2$.",
"arxiv_id": "2601.07070",
"authors": [
"Jonas Matuzas"
],
"categories": [
"math-ph",
"math.MP"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Diffraction by a Right-Angle Penetrable Wedge: Closed-Form Solution for General Refractive Index",
"url": "https://arxiv.org/abs/2601.07070",
"version": "v1"
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