After giving an analytical expression for the single bunch short range wake potential of a storage ring, we have discussed separately the roles of linear and nonlinear terms of the Taylor expansion of the wake potential on the bunch longitudinal motion. The equations describing bunch lengthening and increase in energy spread are established. Applications to different operatingmachines are made.

Starting from a single resonant rf cavity, disk-loaded travelling(forward or backward) wave accelerating structures'properties, such as the coupling coefficient K in the dispersion relation, group velocity vg, shunt impedance R, wake potential W (longitudinal and transverse), the coupling coefficient β of the coupler cavity and the coupler cavity axis shift δr which is introduced to compensate the asymmetry caused by the coupling apeerture, can be determined by rather simple analytical formulae

By considering delta function sextupole, octupole, and decapole perturbations and using di!erence action-angle variable equations, we "nd some useful analytical formulae for the estimation of the dynamic apertures of circular accelerators due to single sextupole, single octupole, single decapole (single 2m pole in general); and their combined e!ects are derived based on the Chirikov criterion of the onset of stochastic motions. Comparisons with numerical simulations are made, and the agreement is quite satisfactory.

nalytical formulae for the coupling coefficient β of a waveguide-traveling wave structure coupling system have been established and compared with experimental results. The dimension of the coupling aperture on the coupler cavity of a traveling wave structure can be determined easily. The relation between the coupling coefficient β and the other parameters. such as aperture dimension, wavelength, group velocity, wall thickness and coupler dimension, is explicitly revealed. These formulae include that for the caes of a waveguide-single cavity coupling system which has been established in ref. [1]