Deriving 3D Wave Eq.: Assumptions & Considerations

[pardesi]

when we derive the wave equation for a an o0ne dimensional wave moving at constant speed we assume that the wave move losslessly that is a plot of $$\psi(x,t)$$ with $$x$$ at any time $$t$$ is same to the extent that one can be obtained from the other by translation.
similarly what are the assumptions when we get the general three dimensional wave equation
$$\Nabla ^{2} \psi=\frac{1}{v^{2}} \frac{\delta^{2} \psi}{\delta t^{2}}$$

 

[OOO]

when we derive the wave equation for a an o0ne dimensional wave moving at constant speed we assume that the wave move losslessly that is a plot of $$\psi(x,t)$$ with $$x$$ at any time $$t$$ is same to the extent that one can be obtained from the other by translation.
similarly what are the assumptions when we get the general three dimensional wave equation
$$\Nabla ^{2} \psi=\frac{1}{v^{2}} \frac{\delta^{2} \psi}{\delta t^{2}}$$

Looks like you are referring to a very special derivation of the wave equation. There are lots of derivation, probably the most clear ones are from physical principles (e.g. Maxwell's equations, continuum mechanics, special relativity...). I think there is no such derivation from translation stuff in more than 1 spatial dimension.

 

[Claude Bile]

A key assumption is linearity, that the wave behaviour is invariant with respect to wave amplitude.

Claude.