TITLE: Is a function changed into another function by a change of variables?
QUESTION [9 upvotes]: If I have a function $ u(x,t) = p(x+ct) + q(x-ct) $ (which is the d'Alembert solution to the $1D$-wave equation), I can make the substitutions 
$$ \xi(x,t) = x + ct\\
\eta(x,t) = x - ct $$
So I am now left with 
$$ u(x,t) = p(\xi) + q(\eta) $$
I could then take that a step further and write
$$ u(x,t) = p(\xi) + q(\eta) = u(\xi, \eta) $$
but I suspect the latter equality simply is wrong... Is that the case? Should I instead be defining a new function, say $v(\xi, \eta)$, so that
$$ u(x,t) = v(\xi, \eta) $$
Even in that last equality, I suspect you can not really 'equate' two functions of different variables. What is the correct approach/terminology here, to make it clear that two functions are equal, save for an appropriate change of variables?

REPLY [8 votes]: $p(\xi) + q(\eta) = u(\xi, \eta)$ is incorrect, because you had already defined $u$ as "$\text{$p$([first_variable] + $c$*[second_variable]) + $q$([first_variable] - $c$*[second_variable])}$", while the function here written is "$\text{$p$([first_variable]) + $q$([second_variable])}"$
$u(x,t) = p(\xi) + q(\eta)$ is almost correct (or totally correct if we agree to this writing convention) becuase it is shorthand for $u(x,t) = p(\xi(x,t)) + q(\eta(x,t))$
$u(x,t)=v(ξ,η)$ is, again, almost correct because (supposing $v$ is defined adequately) it is short for $u(x,t) = v(\xi(x,t),\eta(x,t))$