Dynamically increase noise on a Particle's Trail

Hello,

I’m firing bullet as Particles with a trail and I’m trying to have a dynamic trail behind them.
I want my particle to go straight forward and to have its trail which is affected by a noise effect over time in order to distort it, but I can’t figure out a way to access to it nor to put some noise on it.

I tried different parameters to slowly increase the noise of my particle but as I want it to stay straight, I can’t get a proper effect.

Do you have any ideas about how to figure this out ?
Here is an exaggerated example of the kind of effect I’m looking for :

Thanks in advance !

Or change the value type of strength from “constant” to “curve”. The curve can either be the 2-point preset from zero to one, or 3-point 0-1-0 which looks like a smooth hill. Be careful that a curve bug can lead you to edit the wrong curve (of different axes even if “separate axes” is unchecked) and you need to move away the irrelevant curves until the curve graph reports that the currently highlighted curve represents “strength”.

(I am using mobile so I can’t “reply” to comment.)