Gravitational waves are generated in a manner analogous to electromagnetic waves.
Classically, changing electric or magnetic fields can generate electromagnetic waves, a radio antenna being a good example, and also radiation is emitted by accelerating or decelerating particles. Maxwell's equations are "simple" enough as one is dealing with vector fields. This is reflected in the quantum mechanical carrier of the electromagnetic field, the photon, which has spin one.
In General Relativity the mathematics is more complex, still, gravitational waves are expected for "changing gravitational fields" , in quotes, because the gravitational field emerges from the space curvature posited by GR. Since one is dealing with tensor fields , the quantum mechanical carrier (in the effective quantizations of gravity used up to now) is the graviton of spin two.
gravitational waves transport energy as gravitational radiation. The existence of gravitational waves is a possible consequence of the Lorentz invariance of general relativity since it brings the concept of a limiting speed of propagation of the physical interactions with it. By contrast, gravitational waves cannot exist in the Newtonian theory of gravitation, which postulates that physical interactions propagate at infinite speed.
This illustrates the space distortions as the wave passes:
The effect of a plus-polarized gravitational wave on a ring of particles.
So, as with electromagnetic waves,
In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically symmetric (like an expanding or contracting sphere) or cylindrically symmetric (like a spinning disk or sphere). A simple example of this principle is a spinning dumbbell. If the dumbbell spins like a wheel on an axle, it will not radiate gravitational waves; if it tumbles end over end, as in the case of two planets orbiting each other, it will radiate gravitational waves. The heavier the dumbbell, and the faster it tumbles, the greater is the gravitational radiation it will give off. In an extreme case, such as when the two weights of the dumbbell are massive stars like neutron stars or black holes, orbiting each other quickly, then significant amounts of gravitational radiation would be given off.
So under certain conditions an accelerating mass can radiate gravitational waves. The effect on planetary orbits , though present due to the emission of gravitational waves, is very small , because of the great weakness of gravity.
Gravitational radiation is another mechanism of orbital decay. It is negligible for orbits of planets and planetary satellites, but is noticeable for systems of compact objects, as seen in observations of neutron star orbits.
In the matter of gravity vs. gravitational waves, I've always found it easier to think of (static) electric fields vs. EM waves.
Think of a static field "going out" from a charge. It isn't really going out. The field lines don't have "ends" that travel out at the speed of light. That's because they don't have ends at all. They all end at charges, and they stretch as far as needed. Charge is never created, it's always + - dipoles that are created (like electron-positron pair creation) so there's no problems with field lines that go out from an electron and go to the limits of the universe and end on some + charges out there. They've been stretched for that long, since the Big Bang. We don't think of how fast they move. They've been there since the beginning when the universe was small, and now they span it completely. A billion light years is nothing.
Similarly with static gravity. Mass-energy cannot just appear or disappear, so the field lines never have ends that have to move outward. They're always connected to mass and energy far away, and have had since the Big Bang to do so. There's no point in asking how fast static gravity moves. It's just "there" from here to the edge of the universe.
If you start to suddenly move, with respect to an already established static gravity or electric charge field-line, the field DIRECTION moves immediately with you, and so does the direction to the source. That's just Lorentzian relativity. The speed of light is not being violated. A source a billion light years away would suddenly start to look like it is moving, but that's because its field is already out to where you are, and the field where-you-are, tells you. It changes direction when you move. It responds immediately to relative uniform motion, via the mechanism of the field that is already extended to each.
But if the static or gravitational CHARGE moves (accelerates) then there is a "kink" or update that moves out from it at c. You don't see this at all from far away, until time d/c has passed. That's the EM wave or gravity wave. It's not a relative thing between source and viewer, because acceleration is not "relative" in relativity. You can't pretend that the observer accelerates and the source does not.
The Lienard-Weichert potentials for EM have two terms for this reason. One is the static one that depends only on relative velocity and points at the source (so long as relative velocity has been constant for long enough). The other one shows aberration (does not point at source), retardation, and is a disturbance in the field due to source acceleration (not observer acceleration).
The slow dance of Sun and Moon are a mix of both effects, in the near-field. The static effects point right at the sources, and are due to fields that already extend to infinity. They have no "speed." However, the second order effects due to small amounts of source acceleration (from orbital acceleration) are tiny, but they are genuine gravitational waves, and they move outward at speed c. They are retarded and would show aberration.
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These kind of effects apply to any kind of mass or energy. A massive particle moving at below the speed of light will carry a "subluminal gravitational wave". Dipolar radiation will carry a "dipolar gravitational wave" etc.
In principle, you can assign the label "gravitational wave" to any kind of gravitational field propagating along with some matter-energy object. But this is not usually done. A more purist viewpoint would be to call the free gravitational waves as the only gravitational waves. I.e., if it is a vacuum excitation of the space-time itself, it is a gravitational wave, if it is actually the gravity associated with some object, it is not. Such gravitational waves radiated from isolated sources can indeed be only quadrupole.
However, this line is harder to draw in the case of null (moving at speed of light) matter. In that case, you often do not know which part is the "free" wave and the one "carried along" because they would both move at the speed of light along the matter/radiation. And this is also a fundamental issue; there is no truly fundamental difference between the "space-time at rest" and the "vibrating space-time" through which the wave is passing. We are able to speak of the difference between a "wave" and a "non-wave" only thanks to a predefined meaning of a "still background". But when you are for example in the thin shell of gamma radiation travelling away from a source, the "correct background" is basically impossible to define.
Hence, people often talk about waves dragged with null matter simply as gravitational waves and do not make a distinction between the "free" and "dragged along" part (Griffiths & Podolský have a few chapters with examples). In the paper you link, the distinction could be made because one could show that there is no extra freedom in the polarization or strength of the wave. Simply put, the gravitational wave has no free properties and it is fully determined by the shell of gamma radiation (up to non-physical gauge transformations). So we could either call the mentioned excitation in the metric a "monopole wave" or a "dragged-along gravitational field", it is really just a question of a finer naming convention.