# What is the exact mathematical definition of oscillation/vibration?

My question is basically is what criteria need to be fulfilled to decide wether a motion is osciliiation/vibration or not.

I found two definitions, def1: "moving around an equilibrum", def2: "periodical motion", but, the examples contradict to them.

Ex1: "bouncing ball". The equilibrum - as far as I know - the gound, and the motion is not around this point, because the ball won't go under the ground (does not fulfill def1), and its motion is not periodic (amplitude is changig, dumped motion), does not fulfill def2.

Ex2: "vibrating string". It fulfills def1, but not def2 (dumped motion - I think any of the dumped vibrations does not fulfills def2, because they are not periodic mathematically)

Ex3: I pick up a book, and I'm holding it in my hands for 1 hour, after that I put it down to the ground, and the book stays there for one hour. Then, I pick it up - exactly the inverse way as I put it down - and hold it in my hands for one hour, and I repeat this motion. This is absolutely periodic motion. Is it oscillation/vibration?

Ex4: I measure a signal for a $dt$ time. In the beginning its value is zero, during the $dt$ time, it goes up to $1$ like a sine wave, then I stop measuring. It is a vibration/oscillation?

• There's no rigid definition of oscillation, because English words just aren't that precise. What we mean when we talk about oscillations is conveyed in the equations, not the words. Apr 30, 2016 at 7:01

All the example you have quoted can be regarded as oscillatory in that in some way the motion repeats itself be it with changing amplitude and/or with changing period.

As an example would one use the term oscillatory for the motion of a pendulum? I think for most people the answer would be "Yes" even though both the period and the amplitude of the pendulum would vary with time.

So I think that it is reparation which is important as well as context. I do not think that I have ever heard the word oscillation used when somebody is doing press ups? This is your example 3.

Your bouncing ball does overshoot the equilibrium position when in contact with the ground but again the word oscillatory is not used in this context.

A string is vibrating and its motion is oscillatory to most people.