# Towards microlocal analysis of waves

2/9/11

## Some Background from time series analyisis

1. The periodogram vs the power spectrum
2. functions cannot be band-limited in both time and frequency. But we CAN try to localize the function's support in both domains (approximately) [HW problem 1: prove the first statement.]
3. stationary versus non stationary signals.
4. time series versus space series: stochastic processes
5. The short time Fourier Transform
6. wavelets and multiresolution analysis

Let f(t) be a temporal function defined on the real line. Then we know that $f(t) = 1/\sqrt{2 \pi} \int _ {-\infty} ^ {\infty} F(\omega) \exp(i \omega t) d\omega$

Or let f(x) be a spatial function defined on the real line. Then we know that $f(x) = 1/\sqrt{2 \pi} \int _ {-\infty} ^ {\infty} F(k) \exp(i k x) dk$

Now when we make a measurement, ANY measurement, it is impossible to directly measure f(t) or f(x). What we actually say we measure is f at discretely sampled point: $f: t_1, t_2, t_3 ..... \,$

Now the power spectrum of the underlying function $f(t)\,$ is $|F(\omega)|^2\,$, which we don't (indeed cannot) know. On the other hand we can certainly calculate the discrete Fourier transform of our sampled data: $\hat{F} \equiv (F(\omega _1), F(\omega_2) ....) \,$. This is just the FFT or DFT of the data. But what is $|\hat{F}|^2\,$?

The absolute value squared of the FFT or DFT is call the periodogram. At best it is an estimate of the power spectrum. Only in the event that the number of sampling points goes to infinity does the periodogram approach the actual power spectrum of the function. Often the periodogram is a terrible representation of the power spectrum since we get it by assuming that the real time series is our finite chunk of data cloned over and over again (periodic extension). This introduces discontinuities in the periodogram associated with the abrupt change in the data at the end of each chunk.

An example of a nonstationary time series. The top figure is the time series, the bottom figure is the periodogram. Which moment is obviously varying with time? HW 2: explain the difference between the periodogram and the power spectrum.

An example of a spatially random process. No different fundamentally than a time series. In general we refer to such phenomena as stochastic processes, whether they are time series or space series

Here is some technical background on stochastic processes.

2/15/11

HW 3: compute the spectrogram of the tiltmeter data File:Judahlevine.txt. Investigate the influence of different tapers, window lengths, zero padding and overlap of the tapers.

## Numerical aspects of time-frequency analysis

Here are the top three free replacements for matlab. The first two are basically matlab clones. Python, on the other hand, is a general purpose programming language that is very easy to learn. NB, numerically intensive calculations in Python actually link to high-quality fortran and C libraries. So there should be no efficiency penalty for the luxury of easy programming.

this is our basic tool, the spectrogram Note in particular window argument is nontrivial.

Here are a python file showing how to compute spectrograms from audio files.

File:Spectrogram.py And here is the data file that goes with it. It's 4 MB. gong audio file . It's a wmv audio file, so you can listen to it and see how your intuition compares to the plots. Download the audio file and after you've installed the python libraries you should be able to do the following:

./spectrogram.py gong.wav


The ./ thing is only needed if . (the current working directory is not in your path).

[1] Here's a link to some wave files representing whistlers, which are dispersive atmospheric EM waves.

a more systematic approach

Or, more appropriately for classical time series analysis, the Wigner-Ville distribution

HW 4 Using the given Python implementation, wigner.py, compute and analyze the Wigner transformation of the tiltmeter [revised!] data.

## Review of Geometrical Optics and the eikonal equation

HW 5: reexamine the derivation of the scalar equation for electromagnetism. If you assume from the outset that the permittivity (take permeability to be 1) is a function of space and time, then what scalar 'wave equation' do you end up with?

## Local plane wave analysis

Here you see that even somewhat complicated wavefront may be locally a plane wave

via [2]

Does this always work? Not without careful generalization in cases for which multiple waves cross at the same point. Watch this video to see incoherent and coherent waves.

The answer is coherence prevails, but only in the far field, due to something called the van Cittert-Zernike theorem (vcz) which we will talk about later.

## Geometrical optics fails for multiple phases

via Engquist and Runborg 1996

via Benamou et al 2004

# Waves in Dispersive, Heterogeneous or Random media

Have a look at this movie. It involves waves propagating in a bounded random medium excited by a point source. These are data, no simulations. Your eye can pick up wavefronts buried in random looking junk that would be (manifestly) difficult to do computationally.

The big picture: waves in homogeneous media or piecewise homogeneous media are not too hard provided you have only a single wavefront. Even if you allow for dispersion (which is equivalent to allowing for attenuation), in which case there are some subtle Kramers-Kronig issues, the path to success is clear.

Once you allow for multiple wavefronts you run into the problem that Geometrical Optics (GO) fails at those places where wavefronts cross, since it ignores interference and the phase is not uniquely defined.

Worse yet, once you allow for highly heterogenous media (by which I mean that either the medium is continuously variable, or there are so many parts, that it's impractical to consider representing it as piecewise, then you're forced to deal with some rather elaborate mathematics, unless you are content with very coarse averages or bulk measurements.

The upshot is that in a random medium it is likely impossible to define things like phase and group velocity since there may not exist a local wavenumber.

## Microlocal analysis

definitions

from Wikipedia: In mathematical analysis, microlocal analysis is a term used to describe techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes generalized functions, pseudo-differential operators, wave front sets, Fourier integral operators, oscillatory integral operators, and paradifferential operators.

from Richard Melrose's lecture notes: Microlocal analysis is a geometric theory of distributions, or a theory of geometric distributions. Rather than study general distributions – which are like general continuous functions but worse – we consider more speciﬁc types of distributions which actually arise in the study of diﬀerential and integral equations.

An important concept is the singular support of a function (wikipedia): Singular support

In Fourier analysis in particular, it is interesting to study the singular support of a distribution. This has the intuitive interpretation as the set of points at which a distribution fails to be a function.

For example, the Fourier transform of the Heaviside step function can, up to constant factors, be considered to be 1/x (a function) except at x = 0. While this is clearly a special point, it is more precise to say that the transform qua distribution has singular support {0}: it cannot accurately be expressed as a function in relation to test functions with support including 0. It can be expressed as an application of a Cauchy principal value improper integral.

For distributions in several variables, singular supports allow one to define wave front sets and understand Huygens' principle in terms of mathematical analysis. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails - essentially because the singular supports of the distributions to be multiplied should be disjoint).

A generalization of this to distributions is the wave front set.

some examples. Here is a numerical determination of wave propagation directions (on the right) versus a full solution to the Helmholtz equation on the left.

via Benamou et al 2004

2/23/11

## even simple boundaries can give rise to chaotic scattering if the rays mix sufficiently

Even 4 scattereres can give rise to chaotic effects

The boundary created by the different colors is called a Wada Boundary

if you wait long enough

## scalar diffraction

In the following picture you see the circular standing waves on a back side of a dime with a point source on the opposite side. In other words these waves are in the geometrical shadow of the time and exists only because of diffraction effects (i.e., a breakdown of classical ray theory).

Here is a standing EM wave created on a glass slide cover.

Ayuffa

## Coherence

The van Cittert-Zernike theorem (vcz)