A complex orthogonal decomposition for wave motion analysis

Abstract

A method is presented for decomposing wave motion into its principle components. The basic idea is a complex generalization of proper orthogonal decomposition. The method involves the representation of real oscillatory signals as complex analytic signals. The relationship between complex modes and wave motion is explored. From an ensemble of complex signals, a complex correlation matrix is formed, and its complex eigensolution is the basis of the decomposition (like a complex singular value decomposition).The complex eigenvectors contain standing and traveling characteristics. A traveling index is proposed to quantify the relative degree of traveling and standing in a waveform. A method of dissecting a wave mode into its traveling and standing parts is also proposed. From the complex modes and modal coordinates, frequencies, wavelengths, and characteristic wave speeds can be obtained. The method is applied to traveling and standing-wave examples.

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models

“…From the decomposition given in (21) can be seen that the imaginary part is zero when the time is in phase with the extremum of the cosine or sine, that is, the sum of the two components is zero; at this time instant both are symmetrical matrices (Feeny, 2008). The imaginary part of (21) measures the degree of asymmetry when the sum of both matrices is different from zero; this is used to define the existence of arbitrary variations into the space, *(x)≠0; this feature is used to define the existence of travelling wave components in the space-time varying fields and to determine leading seismic wave propagation components.…”

Complex Modal Extraction for Estimating Wave Parameters in One-Dimensional Media

“…A "traveling index" has been defined as the reciprocal of the condition number of the matrix whose two columns are the real and imaginary components of the complex mode [1]. Pure traveling waves will have orthogonal components of the same magnitude, leading to a condition number of 1, and hence a traveling index of one.…”

“…The method of COD [1] was developed as a complex generalization of proper orthogonal decomposition [2][3][4]. Proper orthogonal decomposition (POD), or similary Karhunen-Loeve decomposition or principal components analysis, is now a standard tool that has been applied to turbulence, structures, and many other types of systems.…”

A nonsymmetric state-variable decomposition for modal analysis

“…The basic idea is a generalization of proper orthogonal decomposition (POD) [1][2][3], with similarity also to smooth orthogonal decomposition (SOD) [4] and complex orthogonal decomposition (COD) [5]. Below, the application of these methods to modal analysis is reviewed, and the niche of the new method is staged.…”

“…(Cast this way, the method might aptly be called "frequency decomposition.") Another decomposition technique is the complex orthogonal decomposition (COD) [5]. Here, the ensemble X of oscillatory signals is expanded into the complex domain to form an ensemble Z of complex analytic signals.…”