Ions in an ion bunch trapped inside an Electrostatic Ion Beam Trap (EIBT) exhibit collective oscillations within the bunch under the influence of an external driving force. These internal oscillations have been measured explicitly using a new method with a particle detector outside the EIBT. In this approach, the evolving ion bunch is monitored along the entire trap length, in contrast to the localized single point measurements that are often carried out in other techniques. In the present study, quadrupole oscillations have been measured for the first time in an EIBT along with the dipole oscillations that were measured previously. The frequency of the quadrupole oscillation is found to be about twice the dipole oscillation frequency. This is in agreement with the prediction of a theoretical model.