The ubiquitous Planewave Ansatz

Question

In physics, the planewave ansatz (meaning: an educated solution guess) is very ubiquitously used, when solving differential equations, in different domains of physics. E.g. to solve the dispersion relation of magnons or phonons in solids, also very commonly in QM e.g. in semiclassical approximations, and so on. The examples above all correspond to systems of 2nd order linear differential equation.

My question:

• Is the use of the planewave ansatz purely motivated by mathematical reasons? (i.e. form/type of differential equation).

• Knowing that planewaves don't really exist in reality, as they would require infinite propagation, how is it that it still remains a theoretically suited ansatz most often (for describing physical phenomena)? (e.g. phonons to propagate to infinity surely) I guess this last question could be formulated differently: In which cases one can be sure that a planewave ansatz (type solution) would absolutely fail?

Answer 1

In many cases our systems are described by linear differential equations, and these have the property that any linear combination of solutions to the differential equation is also a solution to the differential equation.

This is useful because usually any arbitrary solution can be Fourier transformed to express it as a sum of plane waves. So if we can find plane wave solutions to our LDE we can combine them to create more complicated solutions.

Answer 2

John has answered this partially, however the fundamental mathematical idea is missing:

If we think of a function being member of some vectorspace, then basisvectors exist. This concept will surely be familiar to you from Quantum mechanics.

But also from there we remember, that problems were alot easier to handle, if we know the Eigenbasis of the Operators that we throw on functions. Then we just change the functions basis into the operators Eigenbasis and we're done.

Just how do we do this? How do we construct the Eigenspace for differential operators that make up differential equations? Let's think about it... First let's state the Eigenvalue Problem in abstract notation $$ \hat{O} f=cf $$ and then put in that we use $\hat{O} = d/dx$: $$\frac{df}{dx} = c f$$ And we immediately see, that this is a known equation! For which the solution is $f \sim exp(cx)$

Thus we know that if we deal with linear differential operators, an exponential basis will always faciliate our calculations. All we need to do is to use proper arguments $c$ to fit the order of the equation (i.e. add some $i$s or $\pm$).

The freedom in choosing $c$ for this method still to work has then given not only rise to the Fourier transform and its by John described properties, but also to the Laplace-transform.

By chosing $c$ appropriately we are ofc making also an 'educated guess' about the form of our exponential solution. For example a DEQ $$\partial_t^2 f + \partial_x^2 f=0$$ will give oscillatory solutions for $c=i(kx \pm \omega t)$, but something that has a 'starting' point and then decays or grows for real c's. That's the idea of a Laplace-transform. You can also look up the examples on the wikipage (scroll down).

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Concerning your other question:

The plane-wave-ansatz surely fails as soon as nonlinearities enter the differential equation, like this maximally simplified 1-D Navier-Stokes-Equation: $$\partial_t v + v \partial_x v = 0$$ Already any attempt to use a Fourier-transform of $v$ will not result in an algebraicly solvable equation.
Numerically modellers adress this problem e.e. by pseudospectral methods, where all the equation is fourier-transformed except for nonlinearities. Those are calculated in real-space and then later numerically transformed.

The other problem you mentioned of infinity, can be also seen as problem of scale: With a box-size $L$ on which you want to solve your problem you will always get discretization effects of the waves that wobble around in this box. However if the wavenumbers $k << L$ then your density of waves is high enough to still be able to obtain a physically meaningful result. The correctness then increases the better the above condition is fulfilled.

Answer 3

There are some situations in which the plane wave ansatz is useful. It is a solution to many wave equations. Plane waves are also familiar and we have mathematical techniques to handle them, such as Fourier series.

However, in some situations you can quite badly wrong using the planewave ansatz. Notably when the wave interacts with a structure that has features of the order of the size of the wavelength you may have a problem. I will discuss the case of diffractions gratings where this issue has been discussed in detail. Some of this may be applicable to whatever problem you're interested in.

Consider the case of an electromagnetic field incident on a diffraction grating with period similar to that of the incident field and a depth similar to its wavelength. The field near that structure often has a planewave expansion that doesn't behave nicely. Suppose the plane wave expansion has amplitudes $\alpha_j$ for $j = 1\dots\infty$. The sequence of $\alpha_j$ is convergent if $\lim_{j\to\infty} = \alpha$ for some finite $\alpha$. If you take a finite set of those terms and add them up then you have a series $A_N = \sum_{j=1}^N\alpha_j$. That series is convergent if $\lim_{N\to\infty} = A$ for some finite $A$. In general the sequence can converge while the series diverges or vice versa. Or they can converge to different values. See

J. P. Hugonin, R. Petit, and M. Cadilhac, "Plane-wave expansions used to describe the field diffracted by a grating", JOSA, Vol. 71, Issue 5, pp. 593-598 (1981)

for more discussion.

In cases where plane wave methods for calculating quantities of interest won't work, other methods may be useful. For example, the Chandezon method for calculating the field diffracted by a grating with a smooth profile regardless of the depth of the grating. The Chandezon method involves rewriting the Maxwell equations in a coordinate system in which the grating is flat and then expanding the wave in that coordinate system. For a discussion of how to implement the Chandezon method see

Lifeng Li, Jean Chandezon, Gérard Granet, and Jean-Pierre Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers", Applied Optics, Vol. 38, Issue 2, pp. 304-313 (1999).

For a comparison between the Chandezon method and the plane wave expansion method see

http://www.ip.univ-bpclermont.fr/Personnel/Kofi.Edee/rayleih_method_C.pdf.