Wave second order derivative equation

[shiromani]

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?

 

[blue_leaf77]

Evanescent wave which does not propagate also satisfies wave equation.

 

[Vanadium 50]

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.

 

[shiromani]

Thanks.

 

[nasu]

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