What is the vibration?

Answer 1

There is no generic "vibration" equation, as many different things can vibrate with many different boundary conditions. There is, however, a generic wave equation which, as I just hinted at, can be used to formulate equations for specific vibrations.

Given a function u(x,y,z,t) where x, y, and z are spatial coordinates in Euclidean space and t is time, the wave equation is given as:

∂2u/∂t2 = vp2∇2u,

where vp is the phase velocity of the wave and ∇2 is the Laplacian.

For the specific example of a vibrating string with a small amplitude, the wave equation becomes:

∂2y/∂t2 = v2∂2y/∂x2,

where y(x,t) and v is the velocity of the wave.

The remarkable thing about the wave equation is how often Mother Nature uses it. The "u(x,y,z,t)" can describe the vibration of a drum head, the electromagnetic fields of light, the ripples on water, the sound of your voice and much more.