# Spacial Fourier transform in wave equation with time-dependent speed of propagation

I am solving (numerically) a wave equation for a general relativity problem given by:

$$\frac{\partial^2 u}{\partial t^2} - c(t)^2 \frac{\partial^2 u}{\partial x^2} = 0.$$

I have already written a solution using RK4 for the Fourier transformed solution, $u(x,t) = \sum_k u_k(t) e^{-ikx}$, which gives (for each $k$):

$$\frac{\partial^2 u}{\partial t^2} + c(t)^2 k^2 u = 0.$$

Since the Fourier transform ignores $t$, I don't see any problem with this. However all of the sources that I look at switch to conformal time $t \rightarrow \eta$, which makes the equation more or less time independent:

$$c(\eta)^2 \left( \frac{\partial^2 \tilde{u}}{\partial t^2} - \frac{\partial^2 \tilde{u}}{\partial x^2} \right) = 0,$$

before performing a Fourier transform on the equation.

Is this just convention? I haven't studied enough Fourier analysis to see why this might be a problem, and a cursory glance at the first equation listed tells me that the equation is separable to begin with. However, this might just be naive of me.

• What sort of behavior---say, at short/long times---does $c(t)$ exhibit? – Semiclassical Apr 17 '17 at 21:34
• $c(t)$ is asymptotically constant, with a rapid change near t=0. So, $c(t) = tanh(at)$ will do, but I am more curious about general solutions... (Sorry for the late reply) – user109527 Jul 19 '17 at 20:18