Indirect Integration of Longitudinal and Transverse Wake Potentials for Unequal Beam Pipes and Arbitrary Beam Velocity
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 Ez and a Neumann Poisson problem driven by cBz, 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.
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BibTeX
@misc{zagorodnov2026indirect,
title = {{Indirect Integration of Longitudinal and Transverse Wake Potentials for Unequal Beam Pipes and Arbitrary Beam Velocity}},
author = {Igor Zagorodnov and Dmitry Bazyl},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05196},
eprint = {2609.05196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05196},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Indirect Integration of Longitudinal and Transverse Wake Potentials for Unequal Beam Pipes and Arbitrary Beam Velocity
AU - Igor Zagorodnov
AU - Dmitry Bazyl
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05196
UR - https://arxiv.org/abs/2609.05196
AB - 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.
ER -