Tuesday, July 5, 2011

Nonlinear wave propagation

Just a comment about harmonics and nonlinear waves. Whatever their physical origin in the machinery, vibroseis harmonics can be considered as generated by asymetric (nonlinear) up and down motion of the baseplate interacting with the earth. I have written about this elsewhere. We can say this is nonlinear interaction between the machine and the earth, although linear wave propagation occurs in the air and earth near surface. This kind of nonlinearity is interesting and common, but there is an other kind of much greater interest.

When wave equations are nonlinear, then we say the medium is nonlinear. New kinds of phenomena emerge not present in linear wave propagation. For example, when two linear waves meet the amplitude is the sum of the two waves. Not so with nonlinear waves. The primary feature of a nonlinear wave is that it's velocity depends on the peak amplitude. Linear wave velocity does not depend on peak amplitude. There are many examples, a classic being the Korteweg & de Vries equation describing solitons (non-dissapating waves) in a narrow water channel.

Why is all of this nonlinear stuff is of interest to geophysicists? Because the theory of poroelasticity (as developed originally by M. A. Biot in the 1950s) leads to wave equations that are nonlinear. when such a medium is probed by a harmonic signal (single frequency), asymmetries will develop in the propagated wave. The response will repeat the same number of times per second as the input, but because it is no longer a perfect sine or cosine a Fourier transform will reveal the original frequency and a series of harmonics. Encoded in the spacing and amplitude of the harmonics is key information about the medium, such a fluid viscosity and permeability. But it is an open question how to do in situ experiments to reveal this phenomena and, if we could, it is not known how to unravel the important information contained therein.

P.S. Steven Wolfram has written a fascinating interactive nonlinear wave equation explorer. A couple of plots are given below.

Linear waves interact by superposition just as we think they should.

Interaction of nonlinear waves is much more complicated and the details depend on the nature of the wave equation.

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