= Solution
Let adjacent equal masses $m$ be separated by $a$, with alternating strong and weak spring constants $K$ and $\kappa$, $K\gg\kappa>0$. The pattern repeats after $2a$, even though the masses are identical. Thus the first <Brillouin zone> is
$$
\boxed{-\frac{\pi}{2a}\leq q\leq\frac{\pi}{2a}.}
$$
Choose the strong bond inside each two-atom <primitive unit cell>. With displacement amplitudes $(u,v)e^{i(2naq-\omega t)}$, the <normal mode> equations are
$$
m\omega^2\begin{pmatrix}u\\v\end{pmatrix}
=\begin{pmatrix}K+\kappa&-K-\kappa e^{-2iqa}\\-K-\kappa e^{2iqa}&K+\kappa\end{pmatrix}
\begin{pmatrix}u\\v\end{pmatrix}.
$$
The <alternating-spring chain dispersion relation> is consequently
$$
\boxed{\omega_\pm^2(q)=\frac{K+\kappa\pm\sqrt{(K-\kappa)^2+4K\kappa\cos^2(qa)}}m.}
$$
The lower <acoustic phonon branch> has in-phase motion within the strongly bound pair and $\omega_-(0)=0$. Its small-$q$ slope is the <speed of sound>,
$$
c_s=a\sqrt{\frac{2K\kappa}{m(K+\kappa)}}\simeq a\sqrt{\frac{2\kappa}{m}}.
$$
The upper <optical phonon branch> has opposite motion within each pair, with $\omega_+(0)=\sqrt{2(K+\kappa)/m}$. Here “optical” names the gapped branch; equal neutral masses need not couple to light through an electric dipole.
In the <strong-bond limit of the alternating-spring chain>, the pairs act almost as rigid dimers in the low-frequency branch, while their internal stretch gives a high, nearly flat branch:
$$
\omega_-^2\simeq\frac{2\kappa}{m}\sin^2(qa),\qquad
\omega_+^2\simeq\frac{2K}{m}+\frac{2\kappa}{m}\cos^2(qa).
$$
At the zone edge the two frequencies are $\sqrt{2\kappa/m}$ and $\sqrt{2K/m}$, separated by a large forbidden frequency interval. Neighbouring cells are out of phase there. Both branches have horizontal tangents at the edge.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2021-q3-alternating-springs.png]
{title=Phonon branches and displacement patterns of a strongly dimerized equal-mass chain}
{description=Original calculation for $\kappa/K=0.06$. The horizontal coordinate uses the two-atom-cell Brillouin-zone boundary. Strong solid bonds and weak dashed bonds identify each dimer. Arrows show relative displacements at the zone centre and edge; their overall sign and amplitude are arbitrary.}
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