Complex Modal Decomposition for Estimating Wave Properties in One-dimensional Media

A method of complex orthogonal decomposition is summarized for the time-domain, and then formulated and justified for application in the frequency-domain. The method is then applied to the extraction of modes from simulation data of sampled multi-modal traveling waves for estimating wave parameters in one-dimensional continua. The decomposition is first performed on a transient nondispersive pulse. Complex wave modes are then extracted from a two-harmonic simulation of a dispersive medium. The wave frequencies and wave numbers are obtained by looking at the whirl of the complex modal coordinate, and the complex modal function, respectively, in the complex plane. From the frequencies and wave numbers, the wave speeds are then estimated, as well as the group velocity associated with the two waves. The decomposition is finally applied to a simulation of the traveling waves produced by a Gaussian initial displacement profile in an Euler-Bernoulli beam. While such a disturbance produces a continuous spectrum of wave components, the sampling conditions limit the range of modal components (i.e. mode shapes and modal coordinates) to be extracted. Within this working range, the wave numbers and frequencies are obtained from the extraction, and compared to theory. Modal signal energies are also quantified. The results are robust to random noise.