  In this paper we present a quantitative measurement of the change in
frequency (tune) with intensity of four transverse resonances in a high
intensity Gaussian beam. Due to the non-linear space charge forces present in
high intensity beams, particle motion cannot be analytically described. Instead
we use the Simulator of Particle Orbit Dynamics (S-POD) and the Intense Beam
Experiment (IBEX), two linear Paul traps, to experimentally replicate the
system. In high intensity beams a coherent resonant response to both space
charge and external field driven perturbations is possible, these coherent
resonances are excited at a tune that differs by a factor $C_{m}$ from that of
the incoherent resonance. By increasing the number of ions stored in the linear
Paul trap and studying the location of four different resonances we extract
provisional values describing the change in tune of the resonance with
intensity. These values are then compared to the $C_{m}$ factors for coherent
resonances. We find that the $C_{m}$ factors do not accurately predict the
location of resonances in high intensity Gaussian beams. Further insight into
the experiment is gained through simulation using Warp, a particle-in-cell
code.
