\begin{aligned}\psi &= A \sin k_1 ( x + v_1 t) \cos k_2 ( x + v_2 t) \\ &= \frac{A}{4i} \left( e^{ i k_1 ( x + v_1 t)} - e^{ -i k_1 ( x + v_1 t)} \right) \left( e^{ i k_2 ( x + v_2 t)} + e^{ -i k_2 ( x + v_2 t)} \right) \\ &= \frac{A}{2} \left( \sin ((k_1 + k_2) x + (k_1 v_1 + k_2 v_2 ) t) + \sin ((k_1 - k_2) x + (k_1 v_1 - k_2 v_2 ) t) \right) \\ &= \frac{A}{2} \left( \sin \left( (k_1 + k_2) \left(x + \frac{k_1 v_1 + k_2 v_2 }{k_1 + k_2} t\right) \right)+\sin \left( (k_1 - k_2) \left(x + \frac{k_1 v_1 - k_2 v_2 }{k_1 - k_2} t\right) \right)\right)\end{aligned} 