- #1

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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....