I can't understand one step in the derivation of the Lorentz transformation. I'm speaking of the standard configuration and LL solution. I understand perfectly well the spherical wave front and the hyperbolic functions. But I have a problem with this little detail: Consider a boost in $x$ direction. Then the spherical wave front becomes $c^2t^2 - x^2 = 0$. From this it follows that $x = ct$. But now we fix the coordinate system with the other observer, and claim that $x = vt$. What am I missing out here?
$x=v\,t$ defines the motion of the relatively moving frame's origin when the origins of the two frames match up at $t=0$ (we can always arrange this latter condition through a translation). This is by definition what we mean when we say that the relative motion is $v$: of choose any point at $t=0$ and its motion during any time $t$ is $v\,t$. The hyperbolic functions are simply a "funky" way to transform the common spherical wavefront condition, and have been of course chosen at the outset so that things work out smoothly when you impose the condition describing the origin's motion.
I must say, I don't recall that derivation, but it's a particularly elegant one: typically concise and fast as most things in Landau and Lifshitz.