Given the wave equation for a material wave:

$$\frac{\partial^2 \phi}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} $$ We can apply the Galilei transformation $x'=x-Vt$ and $t'= t$ which result in:

$$(1-\frac{V^2}{c^2})\frac{\partial^2 \phi}{\partial x'^2} = \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t'^2} - \frac{2V}{c^2} \frac{\partial^2 \phi}{\partial x' \partial t'} $$

Which, we can conclude is clearly not invariant under a Galilei transformation.

I wonder how this is possible when looking at the derivation of the wave equation. In the derivation of the wave equation we use Newton's second law $F = ma$. It is often claimed that Newton's 2nd law is Galilean invariant by proofing that the accelerating $a$ is invariant under a Galilean transformation.

Now how is it possible that the wave equation is not invariant under a Galilean transformation but the basic ingredient in deriving the same wave equation, Newton's 2nd law, is Galilean invariant?

Given the wave equation for a material wave: $$\frac{\partial^2 \phi}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2},$$ we can apply the Galilean transformation $x'=x-Vt$ and $t'= t$ which results in $$\left(1-\frac{V^2}{c^2}\right)\frac{\partial^2 \phi}{\partial x'^2} = \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t'^2} - \frac{2V}{c^2} \frac{\partial^2 \phi}{\partial x' \partial t'}$$ from which, we can conclude is clearly not invariant under a Galilean transformation.

I wonder how this is possible when looking at the derivation of the wave equation. In the derivation of the wave equation, we use Newton's second law $F = ma$. It is often claimed that Newton's 2nd law is Galilean invariant by proofing that the acceleration $a$ is invariant under a Galilean transformation.

Now, how is it possible that the wave equation is not invariant under a Galilean transformation but the basic ingredient in deriving the same wave equation, Newton's 2nd law, is Galilean invariant?