  This paper studies the quasinormal mode spectrum of the scalar perturbation
on the background of the rotating accelerating black holes. The quasinormal
frequency $\omega$ and the separation constant $\lambda$ are calculated using
two methods: the continued fractions method and the directly integration
method. The spectrum is found to include three families of modes: the photon
sphere modes, the acceleration modes, and the near extreme modes. We
investigate the effects of back hole parameters such as spin and acceleration.
Empirical formulas are presented for the numerical results, specifically for
the acceleration modes in the small black hole limit or the near extreme modes
in the extreme black hole limit. An interesting phenomenon known as eigenvalue
repulsion is observed when the acceleration modes intersect with the near
extreme modes at certain parameter values. The strong cosmic censorship
conjecture for spinning C-metric is respected.
