Visualisasi Fungsi Legendre dan Perilaku dari Permukaan Bola Harmonik dengan Menggunakan Bahasa Pemrograman Matlab

Telah dilakukan penelitian yang berhubungan dengan medan potensial gravitasi di permukaan bumi yang merupakan persamaan harmonik bola. Harmonik bola tersebut merupakan penyelesaian dari persamaan Laplace, dalam sistem koordinat bola. Persamaan ini dikenal dengan Associated Legendre Functions. Pada penelitian ini dikembangkan visualisasi Associated Legendre Functions atau disebut juga dengan Spherical Harmonic Functions dalam tampilan 1D, 2D, 3D dengan menggunakan pemrograman MATLAB. Telah berhasil dikembang visualisasi dalam koordinat bola untuk fungsi r + Pnm(x) dengan r = 5 dan Pnm(x) yaitu Associated Legendre Functions untuk derajat n = dari 1 sampai 7 dan dengan orde m ≤ n.

Construction of regional geoid using a virtual spherical harmonics model

The spherical cap harmonics (SCH) method can be used in regional geoid modeling. The core of this approach is the computation of its associated Legendre functions (ALF) with non-integer degree. However, it is unlikely to obtain a large number of zero-root values for the non-integer ALF. To overcome this problem, a new approach called virtual spherical harmonics (VSH) is proposed in this paper to transform the cap range into the whole sphere so that unlimited numbers of zero-root values can be obtained. The new approach was tested using four cap ranges with the radii of {30^{\circ }}, {15^{\circ }}, {10^{\circ }} and {5^{\circ }}, and geoid undulations for each of the regions are calculated from EGM2008. For each of the regions, the geoid undulations were used to construct three models with three different degrees of 20, 30 and 40. Numerical results showed that with the increase in the degree of the VSH model, the value of the maximum error decreases; and the maximum error of the model was less than 1 mm while the maximum degree is 40.

Fast computation of sine/cosine series coefficients of associated Legendre function of arbitrary high degree and order

In order to accelerate the spherical/spheroidal harmonic synthesis of any function, we developed a new recursive method to compute the sine/cosine series coefficient of the 4π fully- and Schmidt quasi-normalized associated Legendre functions. The key of the method is a set of increasing-degree/order mixed-wavenumber two to four-term recurrence formulas to compute the diagonal terms. They are used in preparing the seed values of the decreasing-order fixed-degree, and fixed-wavenumber two- and three-term recurrence formulas, which are obtained by modifying the classic relations. The new method is accurate and capable to deal with an arbitrary high degree/ order/wavenumber. Also, it runs significantly faster than the previous method of ours utilizing the Wigner d function, say around 20 times more when the maximum degree exceeds 1,000.

Brinkman Flow Generated by the Rectilinear Oscillations of an Oblate Spheroid Along its Axis of Symmetry

In this paper, we consider an impervious Oblate spheroid placed in a fully saturated porous medium, where in the flow is governed by Brinkmann flow equation. We assume that the spheroid is performing rectilinear harmonic oscillations along the axis of symmetry with a speed u. The flow is studied under the Stokesian approximation. The expressions for the velocity and pressure fields are obtained in terms of Legendre functions, associated Legendre functions and Radial and Angular spheroidal wave functions. We obtain an expression for the drag experienced by the spheroid, and numerically study its variation with respect to the flow parameters and display its variation through graphs.