= Solution
The right <spring> changes length by $u_{n+1}-u_n$ and pulls <Chemical atom>[atom] $n$ to the right with <force> $C(u_{n+1}-u_n)$. The left <spring> contributes $-C(u_n-u_{n-1})$. Adding these <Restoring force>[restoring forces] and applying <Newton's second law> gives
$$
m\ddot u_n=C(u_{n+1}-2u_n+u_{n-1}).
$$
Uniform <displacement> makes both <spring> extensions vanish, as <Translational symmetry>[translational invariance] requires. Search for a <normal mode> $u_n=u_0e^{i(kna-\omega t)}$. Substitution yields
$$
-m\omega^2=C(e^{ika}+e^{-ika}-2)=-4C\sin^2(ka/2),
\qquad\boxed{\omega(k)=2\sqrt{C/m}\,|\sin(ka/2)|.}
$$
One representative <first Brillouin zone> is $-\pi/a\leq k\leq\pi/a$. This is one positive-frequency <Acoustic phonon branch>[acoustic branch]: near zero <wavevector>, $\omega\simeq a\sqrt{C/m}|k|$, so the <Speed of sound>[sound speed] is $v_s=a\sqrt{C/m}$. Opposite signs of $k$ label opposite propagation directions, not a second polarization branch.
The model assumes small <Displacement>[displacements], <Harmonic oscillator>[harmonic] nearest-neighbor <Force>[forces], one scalar <displacement> per <Chemical atom>[atom] and uniform equilibrium spacing. It excludes <damping>, <Anharmonicity>[anharmonic] expansion and transverse polarizations. In a three-dimensional monatomic <crystal> there would normally be three acoustic polarizations; that counting cannot be imported into this explicitly one-dimensional scalar model.
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