Galilean transformation of the wave equation
Solution 1
You must first rewrite the old partial derivatives in terms of the new ones. A priori, they're some linear combinations with coefficients that could depend on the spacetime coordinates in general but here they don't depend because the transformation is linear. The rules $$ t'=t, \quad x'=x-Vt,\quad y'=y $$ get translated to $$ \frac{\partial}{\partial t} = \frac{\partial}{\partial t'} - V \frac{\partial}{\partial x'}$$ $$ \frac{\partial}{\partial x} = \frac{\partial}{\partial x'}$$ $$ \frac{\partial}{\partial y} = \frac{\partial}{\partial y'}$$ If you write the coefficients in front of the right-hand-side primed derivatives as a matrix, it's the same matrix as the original matrix of derivatives $\partial x'_i/\partial x_j$. If you don't want to work with matrices, just verify that all the expressions of the type $\partial x/\partial t$ are what they should be if you rewrite these derivatives using the three displayed equations and if you use the obvious partial derivatives $\partial y'/\partial t'$ etc.
If you simply rewrite the (second) derivatives with respect to the unprimed coordinates in terms of the (second) derivatives with respect to the primed coordinates, you will get your second, Galilean-transformed form of the equation. I've verified it works – up to the possible error in the sign of $V$ which only affects the sign of the term with the mixed $xt$ second derivative.
I guess that if this explanation won't be enough, you should re-ask this question on the math forum.
Solution 2
transformation rule for partial derivatives:
$$ \frac{\partial}{\partial x_{\mu}} = \sum_{\nu} \frac{\partial x'_{\nu}}{\partial x_\mu} \frac{\partial}{\partial x'_{\nu}}$$
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ComputerSaysNo
Updated on June 22, 2020Comments
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ComputerSaysNo over 3 years
I have this general wave equation:
\begin{equation} \dfrac{\partial^2 \psi}{\partial x^2}+\dfrac{\partial^2 \psi}{\partial y^2}-\dfrac{1}{c^2}\dfrac{\partial^2 \psi}{\partial t^2}=0 \end{equation}
And the following transformation : $t'=t$ ; $x'=x-Vt$ and $y'=y$
The solution to this has to be : $$\dfrac{\partial^2 \psi}{\partial x'^2}\left( 1-\frac{V^2}{c^2}\right)+\dfrac{\partial^2 \psi}{\partial y'^2}+\dfrac{2V}{c^2}\dfrac{\partial^2 \psi}{\partial x' \partial t'^2}-\dfrac{1}{c^2}\dfrac{\partial^2 \psi}{\partial t^{'2}}=0$$
This proves that the velocity of the wave depends on the direction you are looking at. I don't know how to get to this? If you just substitute it in the equation you get $x'+Vt$ in the partial derivative. Is there another way to do this, or which rule do I have to use to solve it? I was thinking about the chain rule or something, but how do I apply it on partial derivatives?
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Vijay Murthy over 11 yearsSee this. It does not provide an answer directly, but provides the needed input. Other related issues are also discussed.
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stupidity over 11 yearsIs the sign in the middle term, $-\dfrac{2V}{c^2}\dfrac{\partial^2 \psi}{\partial x'\partial t'}$ correct? Or should it be positive?
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Melian about 8 yearsIt should be positive.
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ComputerSaysNo over 11 yearsThaks alot! I've checked, and it works. I had some troubles with the transformation of differential operators. I apologize for posting this mathematical question in the physics category, although the meaning of the solution is appropriate.
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Santosh Linkha almost 10 yearsHi ... shouldn't $\frac{\partial }{\partial x'} = \frac{\partial }{\partial x} - \frac{1}{V}\frac{\partial }{\partial t}$?? could you elaborate why just $\frac{\partial}{\partial x} = \frac{\partial}{\partial x'}$ ??
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Alexander Cska over 7 years@SantoshLinkha because $\partial_x(\psi(x'))=\partial_x(\psi(x-vt))=\partial_{x'}\psi * \partial_x(x-Vt)=\partial_{x'}\psi $
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Noldorin over 2 yearsIn case anyone else accidentally falls into the same trap @SantoshLinkha (easily) did, a slightly more obvious way to see the mistake is that using the chain (transformation) rule for partial derivatives we we get a term that is $\frac{\partial t'}{\partial x}$, which is actually $0$, since $x$ does not depend directly on $t'$, rather only on $t$ (it would be $1/V$ if we were considering the total derivative, of course).