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Heat Conduction
Published in Greg F. Naterer, Advanced Heat Transfer, 2018
The method of conformal mapping can also be applied to the transformation of a standard coordinate system to other orthogonal curvilinear coordinates, for example, elliptic cylindrical coordinates or bipolar coordinates (see Figure 2.16). An orthogonal coordinate system is a system of curvilinear coordinates wherein each family of surfaces intersects the others at right angles. Elliptic cylindrical coordinates are an orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular z-direction. As a result, the coordinate surfaces become prisms of confocal ellipses and hyperbolae. Orthogonal curvilinear coordinates are useful in solving the heat conduction equation when boundaries are more closely aligned with these coordinates than Cartesian, cylindrical, or spherical coordinates.
Directional wavemakers
Published in Robert A. Dalrymple, Physical Modelling in Coastal Engineering, 2018
Robert A. Dalrymple, Michael Greenberg
Along the centerline of the wavemaker (i.e., the x axis) the wave motion can be shown to be the same as for the Case I situation (using Fresnel approximation) ; however, at a large value of y (with respect to x) the two types of wavemakers differ, as the Case II wavemaker does not make waves along the y axis. (A more elaborate mathematical treatment of the problem, including the waves generated behind the paddle, would involve the use of an elliptic coordinate system).
Anti-plane dynamic analysis of a quarter-space piezoelectric material with an elliptic notch in the corner
Published in Mechanics of Advanced Materials and Structures, 2021
Fuqing Chu, Hui Qi, Guangqian Liu, Jing Guo, Zhiyu Fan, Guohui Wu
In elliptic coordinates, the governing equation can be expressed as where and are radial and angular coordinates of the elliptical coordinate system, respectively. The equations in Eq. (2) are Helmholtz equation and Laplace equation in elliptic coordinate system, respectively.
Dynamic Stress Analysis of a Shallow Unlined Elliptical Tunnel under the Action of SH Waves
Published in Journal of Earthquake Engineering, 2022
Fuqing Chu, Hui Qi, Jing Guo, Guohui Wu
In contrast to the study of Tsaur (Tsaur and Chang 2018), due to the presence of a double series in this paper, further validation of the accuracy of the results is necessary. Therefore, our results are compared with the results of the half-space circular tunnel obtained by Liu and Lin (2002) (Liu and Lin 2002). The characteristic of the elliptical coordinate system that the ellipse approaches a circle when approaches infinity is used. When the minor-to-major axial ratio () of the elliptical tunnel tends to 1, the model in Fig. 1 can be approximated as a circular tunnel in half space. Figure 4 illustrates the distribution of the DSCFs under the combinations of , , or , when the elliptical tunnel is almost circular (). Compare them with the results of the half-space circular tunnel obtained by Liu and Lin (2002) (Tsaur and Chang 2018), respectively. It is observed that they are in excellent agreement with each other. It is worth noting that the conversion between polar coordinates and elliptical coordinates is involved in the comparison of the results, and the relationship between them are and . Since the angular coordinate in the elliptic coordinate system can be regarded as the angular coordinate in the polar coordinate system when , no distinction is made between and here.
Airy transform of Ince–Gaussian beams
Published in Waves in Random and Complex Media, 2022
Haiqi Huang, You Wu, Zejia Lin, Danlin Xu, Junjie Jiang, Zhenwu Mo, Haobin Yang, Dongmei Deng
The optical field of an IGB passing through the ATOS is given by the following formula [26]: where and are horizontal coordinates in the input plane and the output plane, and are transverse numerical scales of the ATOS in two transversal directions, respectively. The subscripts p and m denote orders and degrees, and the superscripts e and o denote even modes and odd modes, respectively. For even and odd modes, p and m have the same parity and , but for even modes while for odd modes. is the position vector, where and . is the ellipticity parameter, where and respectively represent the semifocal separation and the beam width in the waist plane z = 0. is the Airy function and can be expressed by [29]. The elliptic coordinate system is established and the z-axis is taken as the propagation direction of the beam. The general expressions of even modes and odd modes of IGBs in the source plane are [2] where A and B are normalized constants. denotes Ince even (odd) mode polynomial.