# How does vibration transform after rotation of the vibrating object?

I can't find the math of that.

Imagine a vibrating object, like a circular membrane, with vibration modes $$\phi_{ij}(x,y)$$. Let's express the initial vibration state like

$$u(x,y,t)=\sum_{ij}a_{ij}e^{i\omega_{ij} t}\phi_{ij}(x,y)$$

Now imagine that this membrane is physically rotated by an angle $$\theta$$ while vibrating. What becomes the vibration state? In other words, what become the different $$a_{ij}$$?

If the membrane is initially vibrating with some privileged direction, this direction must be conserved with respect to an inertial frame of reference, by analogy with the Foucault effect. So the $$a_{ij}$$ should change but how?

Thanks a lot!