Wave second order derivative equation

AI Thread Summary
The discussion centers on the relationship between second order derivatives of physical quantities and wave propagation in a medium. It asserts that when the second order time derivative is directly proportional to the second order spatial derivative, a wave must travel through the medium. The equations presented clarify that while the first equation represents a wave, the second does not, as it yields different mathematical solutions. The conversation also touches on the complexity of generalizing these relationships beyond linear cases. Overall, the nuances of wave equations and their implications for physical phenomena are emphasized.
shiromani
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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?
 
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Evanescent wave which does not propagate also satisfies wave equation.
 
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.:)
 

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