Time-dependent transmission line for wave creation and propagation I am interested in the behaviour of the wave $A(t,x)$ propagation at $x>l$ where the wave was created by a time-dependent source $j(t,x)$ which is homogeneous and localized in space at $0<x<l$, i.e.,
$$
j(t,x)
=
\begin{cases}
j(t) & 0<x<l\\
0 & \text{otherwise}
\end{cases}
\tag{1}
.
$$
The system can be described using the Lagrangian
$$
L
=
\int_0^\infty dx
\left\{
  \frac{1}{2}
  \left[
    (\partial_tA)^2
    -
    (\partial_x A)^2
  \right]
  +jA
\right\}
\tag{2}
$$
where I am using natural units where the speed of the wave $c=1$.
The Euler-Lagrange equation yields
$$
0
=
\partial_t^2A-\partial_x^2A-j
\tag{3}
$$
which can be solved by inserting the Fourier representation of $A$ and $j$, e.g.,
$$
A(t,x)
=
\int_0^\infty dk\int_{-\infty}^{+\infty}d\omega\ A(\omega,k)e^{i\omega t-ikx}
\tag{4}
$$
which leads to
$$
(k^2-\omega^2)A(\omega,k)
=
j(\omega,k)
\tag{5}
$$
and then
$$
A(t,x)
=
\int_0^\infty dk
\int_{-\infty}^{+\infty}d\omega\ 
\frac{j(\omega,k)}{k^2-\omega^2}
e^{i\omega t-ikx}
\tag{6}
.
$$
From here, I am not sure how to proceed as I don't see an obvious strategy to further simplify eq. (6). We cannot use contour integration because we don't know the specific form of $j(\omega,k)$.
As a result I would expect something which implements causality and well the spectrum of the wave should be related to $j(\omega,k)$.
I am also unsure if we need to employ boundary conditions $j(t,0)=0=j(t,x)$ or the like.
 A: In more everyday terms the problem is about solving an inhomogeneous wave equation,
$$
\frac{\partial^2}{\partial t^2}A(x,t) - \frac{1}{v^2}\frac{\partial^2}{\partial x^2}A(x,t) = j(x,t).
$$
This needs to be supplemented by the boundary conditions, which are probably something like
$$
A(0,t)=A(l,t)=0,
$$
although they are not a priori clear from the OP.
There are multiple methods for dealing with this equations, which are covered in books on partial differential equations. One possibility to express a general solution for an unknown source is to resort to a Green's function, which is the solution of equation
$$
\frac{\partial^2}{\partial t^2}G(x,t;x',t') - \frac{1}{v^2}\frac{\partial^2}{\partial x^2}G(x,t;x',t') = \delta(x-x')\delta(t-t'),
$$
with the same boundary conditions. The particular solution of our equation is then
$$
A(x,t) = \int dx'\int dt'G(x,t;x',t')j(x',t').
$$
One can modify this solution by adding an arbitrary solution of the homogeneous wave equation. Note also that one needs to impose appropriate conditions on the Green's function at $t=t'$ - in particular, one may speak of retarded, advanced or time-ordered Green's functions. In many problems it is natural to consider the retarded one (the solution is zero for $t<t'$), since it reflects the causality (response to a perturbation follows the perturbation and not the other way around).  Incidentally, correctly treating time behavior is necessary for correctly integrating around poles, if using the Fourier expansion (Laplace transform is a useful option as well).
