I've just started learning Blender for my games. I had been trying for the past 5 hours, to make curvy waves on plane.
I wanted the first picture to have the wave like as seen on the second.
I just created the plane below in a simple manner using loop cuts and a displacement modifier.
Firstly, you need to create the geometry for what you want. To do this, I made a plane, scaled it up only along the x axis, and then pressed Ctrl+R to create loop cuts. I then pressed 20 on the keyboard for 2 cuts, and got the following result:
Then all we need to do is move each of these loops up or down. This could be done manuall, but I instead used a displacement modifier. Go into the modifiers tab on the right, and add a displacement modifier. Create a new texture for it, go into the textures tab, and change the type to wood. This will create a wavy pattern, but you will notice it is not smooth like the one you want.
To make it smooth, I went back to the modifiers tab, and added a subdivison surface modifier. This makes it smoother, but you can still see all the individual faces. To fix this, I used the "Shade smooth option", and got the result I showed at the beginning.
I hope this helps!
Here ya go: also, correct me if i'm wrong, but I think there's a randomization value you can mess with to make it less of a repeating pattern - https://docs.blender.org/manual/en/latest/modeling/modifiers/deform/wave.html