MATHEMATICS AND WAVES

Wave Phenomena-A Mathematical excursion…A Tribute to Heinrich Hertz
J.V.Ramana Raju (Hon. Academic Coordinator, MSI)
The Analysis of wave phenomena fits properly in the study of nonlinear analysis. Ever since physicists, Huygens, Hertz and others began understanding the light waves, there is an ever-increasing demand to understand waves better. This is partly due to practical considerations and partly for pure mathematical fascination. We shall briefly look at the two viewpoints: There is some deep Mathematics involved in making a portion of a scene invisible to a portion of the horizon. The understanding of this phenomenon involves some intricate Mathematics and Physics (which the author does not yet understand). Another direction is that of Resonances. One generally imagines musical instruments, Tacoma Bridge disaster and such other stories to relate to resonances. However the right kind of Mathematical framework leads us to eigenvalues. Roughly speaking, eigenvalues of some self adjoint operators describe among other things energies of bound states. These are states that remain forever, but it is difficult to see it in the real world. From the knowledge of certain atomic spectra one looks at the steady states of stars and their composition. In many other examples, damping leads to either decay or escape to infinity.
The simplest case to begin such a study is the case of an oscillating string. Let X=[0,π], and P=-∂2x be the Laplacian on X acting on functions satisfying the dirichlet boundary conditions, u(0)=0=u(π). The position of the string at any time ‘t’ can be understood by the wave equation. One can then generalize to the case of eigenvalues of operators on a compact manifold. Distribution of eigenvalues at high energies remains a fascinating Mathematical story.
Another practical consideration:- Tsunami’s are water waves that start in the deeper ocean, most probably due to an earthquake beneath. Initially they start with a small amplitude giving a feeling that they are like wind waves. However the wave propagation is such that one observes wavelengths as high as 500 kms. The dynamics is essentially governed by the so called shallow water wave equations. The speed of its propagation is approximated by v≈√gb, where g is the gravitational force and b is the depth of the ocean. What leads to a chaos is that as the tsunami approaches the shore, while the depth component decreases a phenomena called wave shoaling forces the amplitude to increase hugely since the amplitude here is inversely proportional to b-1/4.A simplified model to explain the horrifying features of the tsunami can be given by using the incompressible euler equation.
For more mathematical details refer to the blog by Terence Tao ( On shallow water wave equation). The resonance part is adapted from the AMS Notices article by M.Zworski.