Rotation alone is not sufficient.
A good way to illustrate that, I think, is a type of telescope mirror that is referred to as 'liquid mirror'.
The simplest implementation is a dish filled with mercury, with the dish spinning at a constant angular velocity. When the Mercury is co-rotating with the dish the Mercury comes to rest completely. The shape is very steady, steady enough for the surface of the Mercury to serve as a mirror that is good enough for astronomical observations.
The state of a fluid in uniform rotating motion is called 'solid body rotation'. The fluid is not actually solid, of course, the 'solid' in that expression refers to the property of all the parts of the fluid being motionless with respect to each other.
As the assembly is rotating: the dish doesn't need to have a profile that matches the surface profile of the fluid. Also: it isn't necessary for the dish to have itself a smooth surface; bumps in the surface make no difference; any bumps of the surface of the dish do not transfer to the surface profile of the rotating liquid. The liquid, being a liquid, accomodates any lump and/or depresssion of the surface of the dish.
In the case of a planet: over geologic time scale the body of the Earth's is deformable; the stone is in effect a highly viscous fluid. Over geologic time scale the solid body of a planet will deform, the final shape is the same shape that it would have if the celestial body would consist of a low viscocity fluid.
For instance, there are reconstructions that indicate that when the Earth first formed out of a proto-planetary disk it had a rotation period of about 6 hours. (And the event that formed the Moon happened early.) Ever since interaction with the Moon has been slowing down the Earth's rotation.
We have that the Earth's solid mass is in hydrostatic equilibrium. But even if the solid Earth would not be in hydrostatic equilibrium: the distribution of fluid over the Earth would not be affected by that. The surface of the fluid will still be the equilibrium shape, independent of the shape of the solid Earth
Going back to the example of the Mercury mirror. When the dish is rotating with a uniform velocity: then all of the fluid has the same angular velocity. The fluid at the outer rim has a larger speed, but it is in precise proportion, such that all of the fluid has the same angular velocity.
So: in and of itself shape of a rotating celestial body cannot produce any form of wave pattern, or more generally: in and of itself shape of a rotating celestial body cannot induce any form of motion.
In order for any form of churning to arise some form of energy gradient must be present. In the case of a planet orbiting a Sun there is a larger influx of heat at the equator than at the poles, and that energy gradient leads to convective flows.