Mode interactions at Unstable fluid interfaces induced by a shock wave (Richtmyer-Meshkov Instability) are studied both analytically and numerically. The analytical approach is based on a potential flow model with source singularities in incompressible fluids of infinite density ratio. The potential flow model shows that a single bubble has a decaying growth rates at late time and an asymptotic constant radius. Bubble interactions, bubbles of different radii propagates with different velocities and the leading bubbles grow in size at the expense of their neighboring bubbles, are predicted by the potential flow model. This phenomenon is validated by full numerical simulations of the Richtmyer-Meshkov instability in compressible fluids for initial multi-frequency perturbations on the unstable interface.