High-frequency limit of the longitudinal impedance of an array of cavities

Abstract

The longitudinal impedance of an array of cylindrically symmetric cavities connected by side pipes is estimated in the high-frequency limit. The expression for the impedance is obtained for an arbitrary number of cavities. The transition from the case of a single cavity to a periodic structure is studied. The impedance per cell decreases with frequency \ensuremath{\omega} as ${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}1/2}$ for a small number of cells. For a large number of cells the impedance decreases as ${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}1/2}$ or as ${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}3/2}$ depending on a certain relation between the frequency and the number of cells. The parameter which governs the transition from one regime to the other is found. In particular, for the infinite periodic structure there is only the second regime and the impedance decreases as ${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}3/2}$ for all frequencies.