Amplitude function mismatch

We develop the general equation of motion of an amplitude function mismatch in an accelerator lattice and look at its solution for some interesting cases. For a free P wave oscillation the amplitude of the mismatch is written in terms of the determinant of a single matrix made up of the difference between the new Courant-Snyder parameters and their ideal values. Using this result, once one calculates the mismatch of the amplitude function and its slope at one point in the lattice (at the end of a nearly matched insertion, for example, then the maximum mismatch downstream can be easily computed. The formalism is also used to describe emittance growth in a hadron synchrotron caused by amplitude function mismatches at injection. While most of the content of this paper is not new to the accelerator physics community, we thought it would be useful to place this important, basic information all in one place. Besides the classic work of Courant and Snyder (1958), our sources include other papers, internal reports, and numerous discussions with colleagues.< >