gravity - Do Gravitational Waves cause a 'Rate of change of Acceleration'?

I was reading the answers to a question regarding the difference between Gravitational Waves and Gravitational Field on Quora:

Quora Question: What is the difference between Gravitational Field and Gravitational Waves?

If the Gravitational Waves represent a change or disturbance in a Gravitational Field, can we say that - like Gravitational Field causes acceleration in objects - Gravitational Waves would cause a 'rate of change of acceleration'?

Answer

From a "Newtonian" point of view, yes. However, one should keep in mind that the full theory from which they're derived - general relativity - doesn't feature "gravitational 'acceleration'" as a true "acceleration": that's the whole point, so from that point of view, the answer is no, because there is no such thing as "gravitational acceleration" to begin with.

You may be asking, also, then, what I mean by a "Newtonian" point of view, given that strictly speaking, "Newtonian" gravity doesn't feature "gravitational waves". You're right, so I perhaps should better say the "Newton-Maxwell view" of gravity, which is an approximation that works in the limit of relatively weak gravitational fields, similar to Newtonian mechanics, but permits delayed-action effects and is also consistent at least with special relativity, yet retaining the notion of "gravitational acceleration". This viewpoint is more commonly called "gravitoelectromagnetism", but I find that a somewhat misleading and kind of clunky, awkward term.

If we're talking about rather modest gravitational waves, i.e. within the Newtonian regime of strength (and not, say, very near the sources like black holes), this theory works decently well, hence it can be used to put an answer to your question about "changing acceleration".

In this theory, in addition to the standard Newtonian gravitational field $\mathbf{g}$, there is an additional "gravimagnetic field" $\mathbf{d}$ (I use this letter by analogy with $\mathbf{B}$ for electromagnetic magnetic fields, because $\mathbf{B}$ is 3 letters before $\mathbf{E}$, and likewise $\mathbf{d}$ is three letters before $\mathbf{g}$), and the governing equations for the two fields together have the same form as Maxwell's equations (hence "Newton-Maxwell") for electromagnetism, and likewise, so do the waves. In particular, in one dimension, a Newton-Maxwell one-frequency gravitational wave thus has the same mathematical form as an electromagnetic wave, so its $\mathbf{g}$-field component looks like (with suitable axes)

$$[\mathbf{g}(t)](x) := A \cos\left((kx - \omega t) + \phi\right)\ \mathbf{\hat{z}}$$

where as usual the constraint $\frac{\omega}{k} = c$ should hold. Hence you can differentiate to get the "rate of change of acceleration", or and I bet you're thinking of it, "gravitational jerk", given by

$$[\mathbf{j}_\mathrm{grav}(t)](x) = -A \omega \sin\left((kx - \omega t) + \phi\right)\ \mathbf{\hat{z}}$$

In other words, an object in a gravitational wave - to the limits of Newton-Maxwell gravity - experiences cyclically varying jerk with period equal to the wave, as one might expect, and this jerk, naturally, gets worse with increasing frequency (as the object is "yanked" - another name for the derivative of force, sure enough - back and forth by the alternating gravitational field).

(Note that I'm ignoring the gravimagnetic contribution to the acceleration while the test object is moving, but for suitably mild waves at least it should be much weaker than the gravitational acceleration.)

Also, by the way, the reason for taking this detour into this extra theory is because we can't just insert a waving gravitational field into Newtonian mechanics and assume it will be accurate at all as a description without justification, but must instead work from the full theory (general relativity) to as simple an approximation as we can get to analyze this.