These are triangulation artifacts in the presence of distorted UV. We can show the same thing occurring on distorted UV without the use of any UV projection:
We can see that the (implicit) triangulation skews our wave texture to one side in the middle of each face. If we were to triangulate in the opposite way (perhaps with fixed alternate triangulate modifier) the skew would flip.
If we triangulate manually and look at a face like this more closely, we can see why:
I've triangulated the quad, then subdivided that line of triangulation. This is the center of the UV (as seen on the UV map as well.) Yet, because the face is a trapezoid, this point does not correspond to the center of the face. Our triangulation leaves us with two faces of uneven sizes in our world, but even sizes in our UV.
The same thing happens (in reverse, kinda) when we use a UV project with perspective. Now, we're mapping square faces to trapezoidal UV faces. Now the line of triangulation creates evenly sized world faces, but unevenly sized UV faces. Again, the actual center of the face does not correspond to the center of the UV.
So then, why does subdivision help this problem? Because we subdivide before triangulation, so each time we subdivide, the deviation of UV center from face center grows smaller:
So what are the solutions to this problem? The first solution is, don't use distorted UV. This is exactly why people try to minimize UV distortion. And the second solution is, just use a ton of subdivision levels to minimize the problem.
But there's a third solution, which is to bake the distortion into the image used by that UV:
Here, we can see that I made a texture using different coordinates (object polar coords) and when I bake it, that earlier distortion just goes away-- but it didn't go away entirely, it just got baked into the image texture instead. (This will also occur with texture painting; drawing a straight line on this mesh will give me a wiggly line of the 2D image.)
So one relatively straightforward solution here is to bake the texture from a heavily subdivided copy, to both minimize the number of verts and to minimize the size of triangulation artifacts.