How to solve this wave equation using Fourier Transform? I have the following wave equation:
$$\frac{\partial}{\partial x}\left(\frac{1}{l(x)}\frac{\partial}{\partial x}V(x,t)\right)=c(x)\frac{\partial ^2}{\partial t^2}V(x,t)$$
where $l(x)$, $c(x)$ and $V(x,t)$ can be taken to be inductance, capacitance and voltage respectively. For physical intuition, it is an equation for the photonic crystal waveguide. The main change being that $l(x)$ and $c(x)$ are not constant to solve like the well known wave equation. They both are space dependent. I am aware of the standard method of solving the standard wave equation using the Fourier Transform (FT) method where you get to get a homogeneous equation which is easy to solve.
What is the approach for this equation?
 A: While you can't use a spatial Fourier transform because of $l(x)$ and $c(x)$ not being constant, you can still use a Fourier transform in time to turn the PDE into an ODE. So using the Fourier transform $$\tilde{V}(x,\omega) \equiv \int_{-\infty}^\infty dt~V(x,t) e^{-i\omega t},$$
your equation transforms to
$$\frac{\partial}{\partial x}\Big(\frac{1}{l(x)}\frac{\partial}{\partial x} \tilde{V}(x,\omega)\Big) +\omega^2c(x)\tilde{V}(x,\omega)=0.$$
This is a linear second-order ODE for $\tilde{V}(x,\omega)$, which is readily written in Sturm-Liouville form. The solution really depends on the functional form of $l(x)$ and $c(x)$, i.e. there is no closed form solution for arbitrary $l(x)$ and $c(x)$.
For example, if $l(x)=l_0/x$ and $c(x) = c_0(x-1/x)$, the ODE turns into the Bessel equation, with the general solution
$$\tilde{V}(x,\omega) = A~J_{\omega\sqrt{c_0l_0}} \big(\omega\sqrt{c_0l_0} x\big) + B~Y_{\omega\sqrt{c_0l_0}} \big(\omega\sqrt{c_0l_0} x\big),$$
where $J_n(x)$ and $Y_n(x)$ are Bessel functions of the first and second kinds, respectively, and $A$ and $B$ are determined by initial conditions.
On the other hand, if, e.g., $l(x) = l_0/(1-x^2)$ and $c(x) = c_0$, the solution can be written in terms of Legendre functions
$$\tilde{V}(x,\omega) = A~P_{k(\omega)}(x) + B~Q_{k(\omega)}(x),$$
where $k(\omega)\equiv\frac{\sqrt{4c_0 l_0 \omega^2+1}-1}{2} $, and $P_n(x)$, $Q_n(x)$ are Legendre functions of the first and second kinds, respectively.
So in conclusion, writing the solution in terms of arbitrary $l(x)$ and $c(x)$ functions is a difficult, if not impossible task. But depending on the problem, you may find analytic solutions for specific functions.
