= Long-wavelength limits
Put $S=m_A+m_B$ and use $\sin qa\simeq qa$. Expanding the square root gives
$$
\sqrt{S^2-4m_Am_B\sin^2qa}\simeq S-\frac{2m_Am_B}{S}a^2q^2.
$$
Thus, to leading nonzero orders,
$$
\boxed{\omega_+\simeq\sqrt{\frac{2\alpha(m_A+m_B)}{m_Am_B}}},\qquad
\boxed{\omega_-\simeq|q|a\sqrt{\frac{2\alpha}{m_A+m_B}}}.
$$
The absolute value ensures a nonnegative <angular frequency> for either direction. The <speed of sound> is the initial acoustic slope $v_s=a\sqrt{2\alpha/(m_A+m_B)}$.
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