 Long-wavelength limits

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2017/paper-2/section-c/c10/long-wavelength-limits

Put

S = m_{A} + m_{B}

and use

sin q a ≃ q a

. Expanding the square root gives

S^{2} - 4 m_{A} m_{B} sin^{2} q a ≃ S - \frac{2 m _{A} m _{B}}{S} a^{2} q^{2} .

Thus, to leading nonzero orders,

ω_{+} ≃ \frac{2 α ( m _{A} + m _{B} )}{m _{A} m _{B}}, ω_{-} ≃ ∣ q ∣ a \frac{2 α}{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 2 α / (m_{A} + m_{B})

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