II. RAYLEIGH METHOD AND ITS ENHANCEMENT
A. Generalized radiation condition
In the canonical problem of diffraction gratings, a periodic surface, S, denoted by y=f(x)=f(x+p), separates space into two regions. It is assumed that the electric vector for s polarization is parallel to the z-axis and the z-component of the E-field is denoted by u, with subscripts i, s or t signifying incident, scattered or transmitted fields.
Fig. 1: A periodic surface.
With the time factor exp(it) being omitted, the incident field ui(x, y)= p1/2exp(i xiy) (1) in which =k0sini, =k0cosi, i the incident angle measured counter clockwise from the positive y-axis as shown in Fig. 1, k0=2/=/c, and c the wavelength and velocity of the radiation propagating in vacuum.
To formulate this problem, in the reference[3] M. Cadilhac considers a function u in the half plane y > max f(x) with the three assumptions:
a) u is bounded for large y.
b) u is a solution of Helmholtz equation 2u + k2u = 0 with real .
c) For fixed y, u can be Fourier-analyzed as a function of x, in some sense (u is for example square integrable in x; more generally, it is a tempered distribution).
To solve the boundary value problem is to determine the function u which satisfies the Helmholtz equation in the domain : y>f(x) instead of the half plane y>max f(x), and preassigned values on the boundary : y=f(x) (Dirichlet problem). Therefore we have to invoke RH which based on the postulate that one and the same representation describes the field in the 'outer' zone y>maxf(x) as well as in the 'inner' zone f(x)≤y≤maxf(x)of the periodic surface.
The so-called Rayleigh expansion is widely used in periodic surface problems abd it has the form as ur(x,y)... ... middle of paper ...
...ential functions in (8) into Fourier series exp(i f(x))=l Vlexp(i2lx/p) (9a) and exp(imf(x))=l Mlmexp(i2lx/p) (9b) with Vl= p10p exp(i f(x))exp(i2lx/p)dx (9c) and Mlm= p10p exp(imf(x))exp(i2lx/p)dx (9d) substituting (9a) and (9b) into (8) then multiplying by n*, taking integrals over period p on the both sides, owing to the orthogonality of m, it obtains that
VnmMmn, mRm=0 (10)
The matrix-vector equation (10) can be used to determine unknown vector Rm, and us(x, y) in turn, can be calculated.
Similarly, based on (7) we can formulate ERM
Vnm N mn, mJm=0 (11) in which
Nlmp10p(Zm/2)exp(im(f(x)+h))exp(i2lx/p)dx
=(Zm/2)exp(imh)Mlm (12)
(11) is the matrix-vector equation to determine unkown vector Jm, and uR(x, y) in turn, can be calculated.
2. The motion of these quanta are governed by a set of materialistic principles constituting the
The purpose of the boundary layer is to all the fluid to change its velocity from the upstream value of U to zero on the surface [6]. Therefore, V=0 at y=0 and V=UÎ at the edge of the boundary layer with the velocity profile of u=u(x,y), bridging the boundary layer thickness [6]. This boundary layer characteristic is true for a variety of different flow situations [6].
Now we can use techniques from linear regression to solve the problem. After the transformation, the least squares method will be used to predict those unknown betas. The core concept of the least squares method is to make the sum of (y-ye)2 the least (Jia, 2011). There is no need to calculate them by ourselves because the process is complex. We usually use computer to assist us to get the results of the least squares method.
In the story “The Adventure of the Speckled Band” the author Conan Doyle portrays many different narrative techniques. He uses these techniques to provide meaning for the audience as well as help them visualize the situation that was being described. The different techniques used are narrative discourse, suspense, conflict, close, and focalization in his story. Narrative discourses are events, to describe a series of events that are contingent one on another. Typically focuses on one or more performers of actions. Personal experience can influence narrative discourse due to the explaining of significant events in the life of the narrator. Narrative discourse is made up of setting, developing
Spherical aberration is the blurred image of an object due to parallel light rays passing through t...
If we introduce two new vectors S0 and S1 and let them equal n0 Q0 and n1Q1 respectively we will get
The independent variables will be denoted by x and y and the dependent variables denoted by z. The partial differentials coefficients are denoted as follows:
The solutions using in integro-differential equations have an important role in lots of engineering fields, also in financial problems, physics theories. The major area of integro-differential –equations are especially mechanical engineering, electric-electronic engineering, economics.[5] Boundary conditions are very important for Volterra equations in order to make them more visual. Furthermore the benefit working on boundary conditions is to see excellent satability properties and high accuracy for Volterra equations.[1] In addition, while evaluating integro differential equations, we should consider the situations about nonlinear integro-differential equations. Nonlinear integro differential equations are essential also in several fields. For instance, fluid dynamics, polymer science, population dynamics, thermoelasticity, chemical engineering can be researching area.[2]
West, Beverly H., et al., eds. The Prentice-Hall Encyclopedia of Mathematics. Englewood Cliffs, NJ: Prentice Hall, Inc , 1982. Print.
Substituting the value of 〖S ̅^x〗_(0 i,j) and 〖S ̅^y〗_(0 i,j) from the equations (2.11) and (2.12) in the equations (2.6), (2.7) and (2.8) , finally we will get the following three equations.
... around to fit your needs and find what variable is missing I the equation. There are many other ways that you can calculate the variables but the equation given above would be considered the easiest and fastest way for solving for an unknown variable.
[3] F.P.Beer,E.R.Johnston and D.F.Mazurek, “Vector Mechanics For Engineers” in McGrow-Hill, 10th ed in SI units., pp.35-37
whereβ the intercept 0 and β the slope 1 are unknown constants and ε is a random error component .
4. Question d: Explain the variables you should take into account when assessing page 4
The era of radio telescopes along with radio astronomy begun with Karl Guthe Jansky's discovery of an astronomical radio source in 1931. In the 20th century, many types of telescopes were developed for a wide range of wavelengths from radio to gamma-rays. (infrared telescopes--700 nm/ 0.7 µm - 1000 µm/1 mm, ultra-violet teles...