The third-integer coupling resonance at ${ensuremath{u}}_{x}ensuremath{-}2{ensuremath{u}}_{z}=ensuremath{ell}$, known as the Walkinshaw resonance, is important in high-power accelerators. We find that, when the betatron tunes ramp through a Walkinshaw resonance the fractional emittance growth (FEG) is a universal function of the effective resonance strength: ${G}_{1,ensuremath{-}2,ensuremath{ell}}sqrt{{?}_{xi}}|ensuremath{Delta}({ensuremath{u}}_{x}ensuremath{-}2{ensuremath{u}}_{z})/ensuremath{Delta}n{|}^{ensuremath{-}1/2}$, where ${G}_{1,ensuremath{-}2,ensuremath{ell}}$ is the resonance strength; ${?}_{xi}$ and ${?}_{zi}$ are the initial horizontal and vertical emittances, respectively; and $|ensuremath{Delta}({ensuremath{u}}_{x}ensuremath{-}2{ensuremath{u}}_{z})/ensuremath{Delta}n|$ is the resonance crossing rate per revolution. At large effective resonance strengths, the FEG reaches an asymptotic maximum value $(mathrm{FEG}{)}_{mathrm{max}?}ensuremath{sim}2{?}_{xi}/{?}_{zi}$ for ${?}_{xi}ensuremath{gg}rac{1}{2}{?}_{zi}$ or ${?}_{zi}/(2{?}_{xi})$ for ${?}_{xi}ensuremath{ll}rac{1}{2}{?}_{zi}$. There is little emittance exchange at ${?}_{xi}=rac{1}{2}{?}_{zi}$, which can be used to minimize emittance growth in crossing a Walkinshaw resonance.
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