How can the invariance of the wave equation be shown without using tensors?

[loonychune]

The problem is, rather briefly:
Show that the wave equation is INVARIANT
The equation is given as:

[the Laplacian of phi] - 1/(c^2)*[dee^2(phi)/dee(t^2)]

dee being the partial derivative.. phi is a scalar of (x, y, z, t)


Now, i want, and think i should be able, to solve this problem without resorting to tensors.

What I've tried to do is this:
Given a 4-vector, X, X.X = X'.X' would imply that whatever makes up that 4-vector is invariant. So, i have to write the above equation as a 4-vector, apply the Lorentz transformation to get X' and then check to see if X.X = X'.X' ?
i can't actually write that equation as a 4-vector!
Given that the 4-velocity involves manipulating r = (x, y, z, ict) to get u = gamma(u, ic) - i am thinking, what do i do to x, y and z and to r to get the equation ? - this line of thinking seems to have me stumped.
..perhaps using the chain rule to get the laplacian in terms of dee x / dee t(proper time) as it relates to velocity ...
Perhaps i need to look at tensors, and relativity of electrodynamics ? But i am assuming that I'm over complicating things given the detail of the book i got this problem from (it's actually Marion & Thornton: Classical Dynamics chapter 14 problem 1)

I hope I've given enough to warrant a reply, as, even though i probably wouldn't be asked this on the examination (it's more likely to be applications in relativistic kinematics), I'm pretty aggrivated as to why i can't seem to even get close to an answer given the study of the relevant chapter...

 

[e(ho0n3]

What does it mean for an equation to be invariant?

 

[loonychune]

Well, i think it means that under a transformation (Lorentz invariance concerns rotations of x1-x4, so in this case, under a rotation) the equation, relative to its new axis, remains unchanged. So measuring the Xn components yields the same answer in all inertial frames...

That's where i was going with that equation at least, I'm just not able to work out 'its components'...

 

[loonychune]

not sure that makes sense in saying 'measure the Xn components'... the equation will be the same with x replaced with x' and y, y' z, z' t, t'

 

[loonychune]

...help :(

 

[cristo]

Well, the first thing I would do is to reduce it to the 1D wave equation, to save time, since we can take the lorentz transformation to be only in the x direction. Now, consider the lorentz transformation $$\bar{x}=\gamma(x-vt) \hspace{2cm} \bar{t}=\gamma\left(t-\frac{vx}{c^2}\right)$$.

Now, the equation is $$\frac{\partial^2\phi}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}=0$$

You want to show that this is invariant under the above transformation, so you need to calculate $$\frac{\partial \bar{\phi}}{\partial t}$$ (and the other derivatives) in terms of the barred coordinates, using the chain rule, and substitute into the equation to show that is invariant.

I should also point out that there is not really any physics needed to do this-- its simply an exercise on evaluating partial derivatives with respect to some transformed coordinates.

 

[loonychune]

Thanks, i did look roughly along those lines once i found a thread showing why the equation ISN'T invariant under galilean transformations, but found the algebra a bit messy so i spose now i know where I'm going - bob's my uncle.. thanks