 Two-mass three-spring normal modes

Two positive masses linked in series between fixed walls by three identical springs of spring constant $k$ have equations of motion $m_1{\ddot{x}}_1=-2k{x}_1+k{x}_2$ and $m_2{\ddot{x}}_2=k{x}_1-2k{x}_2$. A normal mode $x_j=A_j\cos \omega t$ exists when $(2k-m_1{\omega }^2)(2k-m_2{\omega }^2)-k^2=0$, giving the displayed angular frequencies. Its amplitude ratio is $A_2/A_1=(2k-m_1{\omega }^2)/k$: positive for the lower angular frequency, negative for the higher one. Equal masses give amplitude ratios $+1,-1$ and squared angular frequencies $k/m,3k/m$.