# Wave second order derivative equation

Whenever the second order derivative of any physical quantity is related to its second order space derivative a wave of some sort must travel in a medium, why this is so?

## Answers and Replies

blue_leaf77
Science Advisor
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Evanescent wave which does not propagate also satisfies wave equation.

Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
2021 Award
It's not so.

$$\frac{\partial^2 u}{\partial t^2} = v^2 \nabla^2 u$$

is a wave, but

$$\frac{\partial^2 u}{\partial t^2} = -v^2 \nabla^2 u$$

is not. The solution of the top equation is in sines and cosines, and the second is sinh and cosh.

Thanks.

And this is just for a linear relation. They can be "related" in more complicated ways.
Generalizations are tricky.:)