We derive an analytical solution for the Josephson plasma resonance of superconducting multilayers. This analytical solution is derived mainly for low-${T}_{c}$ systems with magnetic coupling between the superconducting layers, but many features of our results are more general, and thus an application to the recently derived plasma resonance phenomena for high-${T}_{c}$ superconductors of the ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{x}$ type is discussed. Our approach allows us to give full details of the different plasma resonance excitations, and we also predict the existence of new nonlinear effects, so far only identified in single junctions.