The answer to you question depends somewhat on which equations you are considering for your problem.
First, let us assume that you are taking the linear case, that is:
As you can verify, a solution to this equation is a plane wave that propagates with velocity $c$. While a single plane wave is not really useful (you mentioned yourself, these don't really exist) we know that this is a linear differential equation, which means that the sum of two solutions is also a solution.
Using the linearity we can create much more general functions through the addition of plane waves. In particular, any periodic function can be expressed through a Fourier sum, whereas arbitrary functions can be thought of "sums" of plane waves through the use of the Fourier transform.
If you are using a more general form of equation (for example the compressible Navier-Stokes equation with a boundary condition on the pressure) then it is not linear, but we can still take plane waves (more generally, Fourier modes) as initial conditions to evaluate the stability of the solution (you are going to have to solve this numerically) in what is known as Von Neumann stability analysis.
Finally, for more general physical problems the plane waves are relevant because one frequently uses "periodic boundary conditions" where the solutions are always Fourier sums. For instance, I am currently working in a project on triphasic fluid flow and we use pseudo-spectral methods (Fourier) with periodic boundary conditions (more Fourier... Man, he is everywhere). I have also seen this being used in light propagation.
I hope this helped.