Fourier sine series

This Fourier sine series will serve to represent odd functions on the interval (—7t, tc) or odd periodic functions of period 2n for all x. 

Example. The complete Fourier series for x is a sine series since x is odd. 

---

Consider the general class of laminated rectangular plates that are simply supported along edges x = 0, x = a, y = 0, and y = b and subjected to a distributed transverse load, p(x,y). In Figure 5-8. The transverse load can be expanded in a double Fourier sine series ... 

If the transverse loading is represented by the Fourier sine series in Equation (5.25), the solution to this fourth-order partial differential equation and subject to its associated boundary conditions is remarkably simple. As with isotropic plates, the solution can easily be verified to be... 

Hence, h s are the coefficients of the Fourier sine series expansion of Cq —Ci. The values of h s can be obtained by multiplying both sides by sm. mnx/L) and integrating from 0 to L using the following relationship ... 

We want to learn how to quantize the radiation field. As a first step, consider a continuous elastic system. Any classical continuous elastic system in one dimension can be treated by a normal-mode analysis. Consider an elastic string of length a [m], tied at both ends to some fixed objects, with density per unit length p [kg m ], and tension, or Hooke s law force constant kH [N m-1]. The transverse displacements of the string along the x axis can be described by a transverse stretch y(x, t) at any point x along the string and at a time t. One can describe the y(x, t) as a Fourier sine series in x ... 

This is a Fourier sine series, and the values of the C may be determined by expanding the constant temperature difference T2 - T, in a Fourier series over the interval 0 < x < W. This series is... 

To compute the interacting RPA density-response function of equation (32), we follow the method described in Ref. [66]. We first assume that n(z) vanishes at a distance Zq from either jellium edge [67], and expand the wave functions (<) in a Fourier sine series. We then introduce a double-cosine Fourier representation for the density-response function, and find explicit expressions for the stopping power of equation (36) in terms of the Fourier coefficients of the density-response function [57]. We take the wave functions <)),(7) to be the eigenfunctions of a one-dimensional local-density approximation (LDA) Hamiltonian with use of the Perdew-Zunger parametrization [68] of the Quantum Monte Carlo xc energy of a uniform FEG [69]. 

Although we are interested in the response of the cylinder to a perturbation of arbitrary shape, such a perturbation can be constructed as a Fourier sine series, and it is sufficient to look at the stability of a single mode for all possible values of k. If the shape-perturbation mode grows for any k, the system is unstable. Now, if / has the form (12-25), we see from the normal-stress condition (12-24) that p must be equal to... 

If the function f x) is an even function, all of the bn coefficients will vanish, and only the cosine terms will appear in the series. Such a series is called a Fourier cosine series. If f x) is an odd function, only the sine terms will appear, and the series is called a Fourier sine series. If we want to represent a function only in the interval 0 < a < L we can regard it as the right half of an odd function or the right half of an even function, and can therefore represent it either with a sine series or a cosine series. These two series would have the same value in the interval 0 < X < L but would be the negatives of each other in the interval — L < jc < 0. 

This is recognized to be in the form of a Fourier sine series over the interval 0 < Y < 7T. Drawing upon the Dirichlet integral technique to evaluate the coefficient ([Pg.327]

Thus, all the cosine contributions equal zero since cos n9 is symmetrical about 9 = TV. The square wave is evidendy a Fourier sine series, with only nonvanishing bn coefficients. We find... 

The series representing fix) is called a Fourier-Sine series. Similarly, we can also express functions in terms of the Fourier-Cosine series... 

To model the above mentioned situation, an arbitrary hydrofoil will be expanded in a Fourier-sine series. For one Fourier mode the problem can be sketched as in figure 2. Suppose a wavy-wall, n (x), with a thin bubbly layer on it, whereby t) (x) is defined in the complex notation as ... 

For the cases where the transverse load can be expanded in the double Fourier sine series, the load q (x, y) is given by ... 

Equation 6.21 can be recognized as the Fourier sine series representation off(x). That is... 

As a note, the quantity 1 - x can be represented in its Fourier sine series form to be... 

It is a necessary condition for the convergence of Fourier series that the coefficients become smaller and smaller and approach zero as n becomes larger and larger. If convergence is fairly rapid, it might be possible to approximate a Fourier series by a partial sum. Figure 11.2 shows three different partial sums of the Fourier sine series that represents the square-wave function... 

The linear combination shown in Eq. (14.3-24) is called a Fourier sine series. Fourier cosine series also exist, which are linear combinations of cosine functions. The Fourier coefficients a, a2,... must depend on t to satisfy the wave equation. With the initial condition that the string is passing through its equilibrium position at... 

Recently Gislason and Kosmas have suggested expanding 1(0) sin 0 in a Fourier sine series. Their procedure is more convenient to use than the Legendre expansion method. The uncertainty (noise) in the Fourier coefficients is known exactly, so it is relatively easy to determine where to truncate the series. In addition, they have... 

The reduced differential cross section can be expanded in a Fourier sine series as... 

The expansion of the differential cross section in a Fourier sine series allows one to obtain very accurate differential cross sections from quasiclassical trajectories. In turn, this permits precise comparison of the reaction dynamics on different potential energy surfaces and at different energies. We plan to extend this work to different energies and different systems, and we also hope to use similar techniques to obtain accurate double-differential cross sections. 

E. A. Gislason and A. Kosmas, Expansion of the differential cross section determined from a classical trajectory study in a Fourier sine series, to be published. 

---