EM fields from an accelerated charge: something doesn't add up?

I have seen on several books that the expression for the E field generated by an accelerated charge, at enough distance and in the non-relativistic aproximation, is something like that (taken from Jackson):




where "β with the dot above" is the acceleration divided by c, n is a vector pointing from the charge to the calculation point, and R is the distance to the charge. Books use this expression (and another one for the B field) to calculate de power irradiated in different directions with the Poynting vector, assuming that we have a radiation field, but... wait a moment!

According to that expression, if the acceleration of the charge oscillates, so does the field, but if the acceleration is constant or varies monotonically, the E field does the same (consider for example an electron accelerating in a rectilinear motion). Does it have any sense?. Can a radiation field be described by a field that is constant (or at least not oscillating)?. There should be something I have misunderstood.

By the way, there is a tendency to interpret always the Poynting vector as an energy flux density in that direction, but this cannot be done in general. Consider for example a static charge near a magnet: the Poynting vector is not zero, but there is no energy flux.