\begin{aligned}s_{1} & = x_{1} + x_{2} \\ s_{2} & = x_{2} − x_{1} \end{aligned}

\begin{aligned}s_{1}(t) & = A_{1} cos(ω_{1}t + ϕ_{1}) \\ s_{2}(t) & = A_{2} cos(ω_{2}t + ϕ_{2}) \end{aligned}

\begin{aligned}s_{1}(t) & = L cos(ω_{1}t) \\ s_{2}(t) & = L cos(ω_{2}t) \end{aligned}

s_{1} − s_{2} = (x_{1} + x_{2}) − (x_{2} − x_{1}) = 2x_{1}

x_{1} = \frac{1}{2}(s_{1} − s_{2})

x_{2} = \frac{1}{2}(s_{1} + s_{2})

x_{1}(t) = \frac{1}{2}L [cos(ω_{1}t) − cos(ω_{2}t)]

= \frac{1}{2}L [cos (\sqrt{\frac{k}{m}}t) − cos (\sqrt{\frac{3k}{m}} t)]

x_{2}(t) = \frac{1}{2}L [cos(ω_{1}t) + cos(ω_{2}t)]

= \frac{1}{2}L [cos (\sqrt{\frac{k}{m}}t) + cos (\sqrt{\frac{3k}{m}} t)]