How does a electric quadrupole oscillate? I know that in static a electric quadrupole is made of two positive charges and two negative charges, distributed as in the following figure:

What I don't understand is when it oscillates and radiates, does it oscillate like two antiparallel electric dipoles, or it oscillates like that shown in the attached figure, with four dipoles? 
 
Image source
 A: You can make a quadrupole oscillate an infinite number of ways. I think what you are asking about is a motion that causes the quadrupole moment to oscillate, without there being also a changing dipole moment. To keep the dipole moment zero, just make sure that as the charges move or change size, at all times the charge either side of any line through the middle of the distribution is balanced, or, to be precise
$$
\sum_i q_i {\bf r}_i = 0.
$$
Here are two examples: (1) leave the charges at the same locations, but have their magnitude oscillate, with diagonally opposite pairs at all times equal. (2) Leave the sizes of the charges fixed, but have them move in the directions along the diagonals, with diagonally opposite pairs at all times at equal distances from the centre. As I say, these are just examples. You can also have quadrupoles of other shapes, such as linear, and also three-dimensional shapes such as a ring and two dots.
A: If you want a truly quadrupolar oscillation, then it needs to oscillate as in the four-dipoles model, i.e. there is a current linking all adjacent pairs of localized charges, which is crucial to the formation of the correct quadrupolar symmetry for the radiation pattern.
