The videos below were generated by this Mathematica notebook and then compressed by
a video editting SW from 150MB to 2MB.
The formula for the scalar wave reads:
$$
\psi_{lm} = {\rm const.} \frac{ P^m_l(\cos\theta)~\tilde r^l}{\Big[\Big(1+\tilde r^2-\tilde t^2\Big)^2+4\tilde t^2\Big]^{\frac{l+1}{2}}}\cos\left[ m\varphi - \lambda(t,r) \right],
$$
$$
\lambda(t,r) = (l+1) \arctan\frac{2\tilde t}{1+{\tilde r}^2-{\tilde t}^2}.
$$
Both spatial and time scales are givem by the parameter a so
$$
\tilde t = \frac{r}{a} ~~~~~{\rm and}~~~~~ \tilde t = \frac{t}{a} .
$$
The animations on this page show the wave amplitude \(\psi_{lm}\) in the equatoreal plane with
$$
a = 1, ~~~~ l = m = 10,~~~~ \theta=\pi/2.
$$