Anyway, "you talk about visibly propagating waves", to me this seems like the size of your transition/interpolation area is too big, i.e, it should be perhaps 10-20% at most. Put it higher than that and you sacrifice quality (for the given performance hit) and the terrain always looks like it's changing.
Also, note that heightmaps are not well-suited to high-frequency detail. I used real world data in my implementation of geoclipmaps which was somewhat high-frequency, I tried everything from low-quality to high-quality meshes and flew over the terrain at high speeds, and even then I could not really say that I felt bothered by the transitions other than in the worst high-frequency areas. So I would say it's a problem in your implementation (or heightmap resolution).
Also! I recommend that you don't compute the normals when sampling, sample them from a separate texture instead, hides the transitions a bit better and with the added bonus that you can use higher resolution normal textures (produces really good visuals even with low quality meshes).
Another thing, I don't remember the specifics at the moment and am too dumb to make sense of it right now, but in your transition area remember that it applies to triangles, you can't just sample in the middle if you want exact results, you need to do multiple samples (if this doesn't make any sense, just forget it). Otherwise you will find that certain the triangle pops in certain higher-frequency areas (i.e an /\ -shape will pop significantly and not interpolate), but it might not be warranted depending on your terrain.