{
  "abstract": "Indirect integration replaces the long uniform beam-pipe parts of a wakefield calculation by field problems in the pipe cross sections. Earlier ultrarelativistic methods were developed mainly for the longitudinal wake, whereas the transverse wake was usually obtained from the Panofsky--Wenzel theorem. We derive indirect formulas that complete a transverse Lorentz-force integral already accumulated in a time-domain calculation. At $β=1$, each semi-infinite tail is found from a Dirichlet Poisson problem driven by $E_z$ and a Neumann Poisson problem driven by $cB_z$, describing the TM and TE contributions, respectively. We obtain both a fixed-time moving-window representation and a fixed-plane time-history representation for equal or unequal input and output pipes. The method is then extended to a rigid bunch moving with constant velocity $0<βc<c$. The longitudinal correction satisfies an anisotropic elliptic equation in $(x,y,s)$ and provides an additional source for the transverse TM problem. In a two-port finite-reference convention, the complete fields, including space charge, are integrated directly between two fixed planes, while the semi-infinite tails are calculated after subtraction of the stationary field in each pipe. The resulting Panofsky--Wenzel relation contains the difference of the stationary transverse electric fields at the two ports. Numerical tests for an unequal rectangular step-out at $β=1$ and $β=0.8$ confirm the transverse indirect integration and the unequal-pipe boundary term.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05196",
  "author_total": 2,
  "authors": [
    {
      "name": "Igor Zagorodnov"
    },
    {
      "name": "Dmitry Bazyl"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bhyd62949y6yj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05196",
  "fulltext_url": null,
  "id": "bhyd62949y6yj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05196",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "physics.acc-ph"
    ],
    [
      "openalex",
      "2202|Particle accelerators and beam dynamics"
    ]
  ],
  "openalex": "W7211913825",
  "pdf_url": "https://arxiv.org/pdf/2609.05196",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bhyd62949y6yj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:14Z",
      "native_id": "2609.05196",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05196"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.acc-ph",
      "subfield": "applied-instrumentation"
    }
  ],
  "title": "Indirect Integration of Longitudinal and Transverse Wake Potentials for Unequal Beam Pipes and Arbitrary Beam Velocity",
  "title_text": "Indirect Integration of Longitudinal and Transverse Wake Potentials for Unequal Beam Pipes and Arbitrary Beam Velocity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:14Z",
  "url": "https://research.draconitic.com/paper/bhyd62949y6yj",
  "venue": "arXiv",
  "versions": []
}