 Solution

Since both initial displacements vanish, only sine terms are needed. Resolve the velocity $(0,u_0{)}^T$ into the two normal modes:Each modal initial velocity must be divided by its angular frequency to obtain its displacement amplitude. ThereforeExplicitly,At zero, both displacements vanish and differentiation gives $({\dot{x}}_1,{\dot{x}}_2)=(0,u_0)$. The motion combines the in-phase and out-of-phase modes, with opposite signed modal velocity coefficients. Their different frequencies produce changing constructive and destructive interference in the physical coordinates, so motion is transferred through the coupling spring. This does not require assuming equal masses or exact periodic transfer of all the energy.