# Invariance of wave equation

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....

What does it mean for an equation to be invariant?

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'......

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'

...help :(

cristo
Staff Emeritus
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.

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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