As it was only slightly touched on this thread, I can add the bit of info I read somewhere, about making rivers in map editor, that certain river shapes and/or cross-section shapes tend to create the waving phenomena. One of the root causes is "wrong" water physics, which ends up having positive feedback loops, creating oscillation. It can be even worse when some player-made structures affect the flow, especially if that structure's effect can change over time (e.g. dam when it is still stabilizing after being built). Or the interplay of a dam and a water source.
I run in to this issue with the very first map I tried to make with the editor, having a relatively straight river oscillating already in the editor

It took quite a few non-natural looking changes to make it stable. And it is not only the rivers that oscillate, but also e.g. the interplay of ocean (edge water source/sinks) and large changes in water flow (huge dams on a river with large flow).
In addition to the two maps mentioned, the (only) official map I have been playing so far (Islands) has some waving in its rivers, but as the rivers in it are so deep, they rarely (if ever) show full dry spots. In it, the waving mostly causes issues with flooding or dam electricity generation.
I'd make an educated guess and say the issue for the two mentioned maps is the "wrong" water physics, combined with the map having the rivers (and water sources) done in a way that are susceptible to waving effect.
As a possible way to reduce those waves (without CO fixing the map and/or the water physics), if you have room on the map, you can add one (or both) of two kinds of changes:
1) A somewhat large side-path in the river that is about as long as the distance between dry spots / wave peaks, but starts at half that distance before a dry spot, ends at dry spot. A wave will split at the start point, travel forward along both paths.. at the time when a dry spot reaches the end of the side-path, the previous wave's peak comes out from the side-path, thus averaging out the dry and wave peak (with some sideways effects remaining). In order to keep the total flow somewhat unchanged, the original river could/should be narrowed roughly as much as the side path has cross-section, but this may be just tried out. (If narrowing the main river, total buildable area will not reduce as much.)
2) Suitably large side lake from the river. It should be roughly shaped as a sector of a circle or a cone, with the tighter end attached to river, and have radius or length outwards from the river of about 1/4th the distance between wave peaks (or much larger 3/4th). In this case a wave spills into the side lake, travels to the other end, reflects back, and half a wave period later (at river's dry point) flows back to river. The "backwalls" of the lake may need some adjustments to make the reflection suitable (sharp walls make the reflection sharp, smooth beach smooth the reflection). Or the backwall can be arranged in different distances, sectors reflecting the wave back at slightly different times, though this may also lead to changing the reflection angle, needing tweaks in the way the side-lake is attached to the river (e.g. two different points). Note that the side cone/sector does not need to be purely perpendicular to the river, slightly angled orientation should work, too.
Both solutions will likely need quite a bit of trial and error to make them work right, but I've used a smaller version of the side-path solution once to "steal" some of wave peaks and spread them over longer period, thus attenuating dual-dam waving (different cause, but just as annoying) much faster than what they would stabilize without side-paths. (After the river got stabilized, the side-paths could be removed, but for the case in this issue, which is caused by the game physics/river shapes, the solution needs to be left there for as long as the causes exist.)
Naturally, these solutions do not (usually) affect the problem upriver.
With a bit of creative touch, both solutions can be made somewhat natural looking, but not fully so, and they both need some extra space.