We have studied the elementary excitations in a two-component Bose-Einstein
condensate. We concentrate on the breathing modes and find the elementary
excitations lo possess avoided crossings and regions of coalescing oscillat
ions where both components of the condensates oscillate with the same frequ
ency. For large repulsive interactions between the condensates, their oscil
lational modes tend to decouple due to decreased overlap. A thorough invest
igation of the eigenmodes near the avoided crossings is presented.