Wave second order derivative equation

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SUMMARY

The discussion centers on the mathematical representation of wave equations, specifically highlighting the conditions under which a wave propagates in a medium. It establishes that the equation \(\frac{\partial^2 u}{\partial t^2} = v^2 \nabla^2 u\) represents a wave, while \(\frac{\partial^2 u}{\partial t^2} = -v^2 \nabla^2 u\) does not. The solutions to the former are sinusoidal functions, while the latter yields hyperbolic functions. The conversation emphasizes the complexity of generalizing these relationships beyond linear cases.

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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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