I have recently been studying a structure for high sound absorption. There are a lot of literature on similar design, where all of them are using the plane wave as an input to the structure. I have the question (so does my supervisor) why people want to use plane wave for the study anyways? I know it's easy to study but it seems pure plane wave does not exist in real life and not even close.

So would anyone help me on clarifying this?

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    $\begingroup$ Speculation: any solution to the wave equation can be viewed as the superposition of plane waves; that's the whole point of Fourier analysis. So if the response of the system is linear (which may be a big "if"), then you can extrapolate from the system's response to a plane wave to its response to any type of wave. $\endgroup$ – Michael Seifert Mar 27 '19 at 13:31
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    $\begingroup$ Light coming from the sun can be, for nearly any purpose, treated as a plane wave. Similarly, sound coming from a distance much larger than the wavelength is also pretty much a plane wave. Plus, the wave equation is much simpler, and as Michael has pointed out, if the medium is linear you can always do the messy bits on the results of the wave equation, not in the equation itself. $\endgroup$ – Jon Custer Mar 27 '19 at 14:21
  • $\begingroup$ @MichaelSeifert Thank you for the comment. Now that Salvador has formulated that into an answer I will accept his answer. $\endgroup$ – Zhang Ze Mar 29 '19 at 10:41

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:

$$\nabla^{2}p-\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^2}=0$$

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.

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    $\begingroup$ I would change "the solution to this equation" into "a solution to this equation". $\endgroup$ – JTS Apr 11 '19 at 20:31

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