The alternate stretching and squashing casued by a gravitational wave is an example of a quadrupole oscillation. This is another word that probably means very little to most readers, and, unless you like maths, Wikipedia isn’t going to help you, so I’ll explain.
Let’s start with a monopole. You get a monopole when you put ‘stuff’ somewhere. Here, ‘stuff’ can mean almost anything you like – mass, electric charge, nematodes…(cafe scientifique last night was about nematodes, or ’roundworms’ – such is their world domination that they deserve to be used to illustrate these physics ideas…) So talking about the number of nematodes in a centimetre cubed of soil would be a describing their distribution in a monopole form.
Dipoles are the next step here. The most obvious example of a dipole is a bar magnet. As well as having a certain strength, it has a North Pole and a South Pole. So it is directional. A bar magnet pointing horizontally is not the same as one pointing vertically, at least not in this description. Magnetic dipoles are easily described by vectors – with a length (the strength of the dipole) and a direction (from south to north). We could add to our monopole description of our nematodes a dipole description too. In this case, we could describe in what way the nematodes were travelling (on average) in a dipole manner. For example, they could on average be moving westwards at a speed of half a millimetre a second - so they have a strength (half a millimetre a second) and a direction (west).
Now for the quadrupole. This is less obvious, and I’m not sure I can do it with nematodes. The quadrupole moment can relate preferences in more than one direction. Imagine two roads, each with equal traffic flows in both directions, intersecting at a roundabout. More cars want to travel straight on than turn. What tends to happen, is that, say, the east-west flow will dominate for a while (the flow stops the north-south traffic entering the roundabout), then a north-south car muscles in and stops the east-west, and the north-south dominates. There is no dipole moment (as many cars flow north to south as south to north, and east to west as west to east, but there is a quadrupole – at a given moment in time one route (e.g. north-south) will tend to dominate over the other one.
In a gravitational wave we get a regular swapping between the dominating directions – e.g. first the up-down direction dominates the left-right (and we stretch and get thin); then the left-right dominates over the up-down (and we shrink and get fat).
We can build up more multipoles too – e.g. the next one is the octupole; and so it goes on. But physically it gets rather complicated to describe, and, for most things, the monopole, dipole, quadrupole series will cover what we need.