There is a very celebrated mathematical theorem that when a mathematical condition known as “linearity” is combined with a kind of symmetry of the universe known as “translational symmetry”, oscillatory or periodic phenomena result. Since many laws of physics exhibit linearity (exactly or approximately), this implies that almost everything will vibrate when excited!
This famous theorem, known variously as the “eigendecomposition”, “modal decomposition” or “spectral decomposition” theorem, is the reason for the convergence of such diverse phenomena as waves on water, the carrying of sound through air, vibrations in a building or machine, electricity surging through wires and the beautiful sounds of musical instruments.
Naturally, modal decomposition and associated methods, are some of the most important tools in an applied mathematicians arsenal, and can be brought to bear upon problems involving (among others):
- Mechanical vibrations
- Wave propagation
In fact it would be difficult to imagine a field where the concepts of vibrations, waves, oscillatory or periodic phenomena are not important!
Although, as suggested above, almost all projects ever performed by applied mathematicians/physicists will require some sort of modal analysis (see above), following are some projects where modal analysis played a very important role:
ADAPTIVE SIGNAL PROCESSING. Many adaptive algorithms get thrown of by very highly correlated input signals. For example, acoustic echo cancellation algorithms usually get thrown off by music or similar highly regular sounds. In this project, we built a model explaining why this happens, and also figured out a work around for certain cases.
INTERMODULATION. Non-linear systems usually exhibit “intermodulation distortion” where more than one input oscillations combine in inharmonic ways to produce new frequencies in the spectrum. Thus, if a system has some non-linearity, it's output will show many spurious peaks in the frequency spectrum. In this project, we created a method of detecting, from a spectrum with a large number of frequency peaks, which frequency peaks are fundamental, and which are intermodulations.
ELECTROMAGNETIC FEM. Built our own FEM EM solver to cross-check Mie scattering calculations, and to be able to perform scattering calculations for irregular shaped objects.
BIREFRINGENCE. Created models of uniaxial and biaxial birefringence. The biaxial birefringence model requires a modal decomposition of a non-linear equation, which was solved using a novel non-linear Rayleigh quotient method devised by us. These models were used to find birefringence parameters of actual samples. Read more…