# Circumferencial waves on a cylinder/sphere

I was wondering how we can introduce $$e^{ik.x}$$ terms associated with circumferencially propagating waves? In this case $$\hat{e}_\theta$$ is the direction of wave propagation. However, I was not able to express the exponent of exponential using a dot product because of polar coordinates. How can we introduce this dot product? Can spherical harmonics or Fourier transform help here?

• Polar coordinates don't matter. You can express your $k$ and $x$ vectors in whatever basis you want and evaluate the dot products between the basis vectors. For starters, try expressing $\hat{e}_\theta$ in the cartesian unit vector basis.
– smh
May 19 '20 at 10:58
• Have you checked this answer? May 19 '20 at 22:28
• I don't think you can write this term in the form you want. Could you provide more detail on why you're trying to do this? May 23 '20 at 13:23