Topic
Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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TL;DR: In this paper, the directional dependency of the vibrational wavefield in each component of an engineering structure is modelled by using a Fourier series and the resulting energy balance equations may be cast in the form of conventional SEA with the addition of non-direct coupling loss factors.
Abstract: In the statistical energy analysis (SEA) approach to high frequency dynamics it is assumed that the vibrational wavefield in each component of an engineering structure is diffuse. In some instances the directional filtering effects of structural joints can lead to highly non-diffuse wavefields, and in such cases SEA will yield a very poor estimate of the vibrational response. An alternative approach is presented here in which the directional dependency of the vibrational wavefield in each component is modelled by using a Fourier series. It is shown that, if required, the resulting energy balance equations may be cast in the form of conventional SEA with the addition of `non-direct' coupling loss factors. The method is applied to the bending and in-plane vibrations of various plate structures and a comparison is made with exact results yielded by the dynamic stiffness method. A significant improvement over conventional SEA is demonstrated.
71 citations
TL;DR: In this article, the authors extended the exact analysis for plane wave scattering from a sinusoidal surface to include the case of an arbitrary periodic surface and showed that the field amplitudes of the upgoing scattered (or evanescent) waves (valid above the highest surface excursion) are expressed as summation over these expansion coefficients multiplied by integrals over the surface function.
Abstract: We extend our exact analysis for plane wave scattering from a sinusoidal surface to include the case of an arbitrary periodic surface. Both Dirichlet (transverse electric polarization) and Neumann (transverse magnetic polarization) boundary conditions are considered. The method uses Green's theorem and our previous idea of expanding the surface fields in Fourier series multiplied by the Kirchhoff or physical optics approximation. The expansion coefficients solve a set of linear equations. The field amplitudes of the upgoing scattered (or evanescent) waves (valid above the highest surface excursion) are expressed as a summation over these expansion coefficients multiplied by integrals over the surface function. The Rayleigh hypothesis is not invoked. Some examples are presented. For an analytic surface, steepest descent methods yield the asymptotic values of the amplitudes. Using this and other asymptotic results, the convergence of the scattered wave expansion is studied as it is analytically continued into the surface wells, and a simple and explicit confirmation of the conditions under which the Rayleigh hypothesis is valid is presented as well as new results for other examples. The periodic surface examples include a sinusoid, an echelette, a quadratic surface, a log cosine surface, a vortexlike surface, a cycloid, a trapezoid, a full-wave rectified surface, and a surface of semicircular cylinders (bosses). The method is general and applies to a very broad class of physical problems.
71 citations
71 citations
TL;DR: In this article, principal component analysis (PCA) and Fourier decomposition (FD) were used for variable star light curve analysis and compared their relative performance in studying the changes in the light curve structures of pulsating Cepheids and in the classification of variable stars.
Abstract: Context. Ongoing and future surveys of variable stars will require new techniques to analyse their light curves as well as to tag objects according to their variability class in an automated way. Aims. We show the use of principal component analysis (PCA) and Fourier decomposition (FD) method as tools for variable star light curve analysis and compare their relative performance in studying the changes in the light curve structures of pulsating Cepheids and in the classification of variable stars. Methods. We have calculated the Fourier parameters of 17 606 light curves of a variety of variables, e.g., RR Lyraes, Cepheids, Mira Variables and extrinsic variables for our analysis. We have also performed PCA on the same database of light curves. The inputs to the PCA are the 100 values of the magnitudes for each of these 17 606 light curves in the database interpolated between phase 0 to 1. Unlike some previous studies, Fourier coefficients are not used as input to the PCA. Results. We show that in general, the first few principal components (PCs) are enough to reconstruct the original light curves compared to the FD method where 2 to 3 times more parameters are required to satisfactorily reconstruct the light curves. The computation of the required number of Fourier parameters on average needs 20 times more CPU time than the computation of the required number of PCs. Therefore, PCA does have some advantages over the FD method in analysing the variable stars in a larger database. However, in some cases, particularly in finding the resonances in fundamental mode (FU) Cepheids, the PCA results show no distinct advantages over the FD method. We also demonstrate that the PCA technique can be used to classify variables into different variability classes in an automated, unsupervised way, a feature that has immense potential for larger databases in the future.
71 citations
TL;DR: This work pads the set of spectral coefficients with zeros and takes an FFT of length 3N to interpolate the Chebyshev series to a very fine grid, and applies either the Mth order Euler sum acceleration or (2M + 1)-point Lagrangian interpolation to approximate the sum of the series on the irregular grid.
71 citations