DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

First-order local impedance condition at an artificial plane

Definition

Use the outgoing time convention $e^{- i ω t}$ . On an artificial boundary with outward unit normal $n$ , the first-order local impedance condition is $\partial_{n} u - i k u = 0$ . For the plane $x = R$ with $n = + e_{x}$ , this is $\partial_{x} u - i k u = 0$ . It is the exact outgoing relation at normal incidence and an approximation for oblique propagating modes; no exactness or uniform small-reflection claim at grazing incidence is part of this definition.

Depends on

Sound-soft exterior Helmholtz problem for a spherical obstacle

Used by

Dependency tree · one level

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Sources

Galkowski, Lafontaine, and Spence, Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves (standard reference, not scraped)
NIST Digital Library of Mathematical Functions, Chapter 10: Bessel Functions (standard reference, not scraped)