A brief review of time‐dependent theories of resonant mode coupling of Alfven waves in space plasmas is presented. A well behaved solution of the hydromagnetic wave equation for a general class of initial conditions is then discussed to understand the ringing, phase‐mixing, and variable eigenperiods of the field line oscillations in the magnetosphere. This solution is further exploited to discuss the initial transient amplification of Alfven waves in sheared magnetospheric plasma. The features of free oscillations of field lines studied here are shown to compare well with previous numerical studies and observations made for particular types of impulse excitations of the magnetosphere.