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Bessel Function of Order Zero

Bessel Function of Order Zero

A common physical example of wave propagation with radial symmetry is the ripple formed when a stone is dropped into a still pond. The disturbance originates at a central point and travels outward as a circular wave. As the radius of the wavefront increases, the same initial energy is distributed along a progressively larger circumference. Consequently, the amplitude, or height, of the wave decreases with distance from the center. This decay behavior cannot be captured by simple sine or cosine...

Singularity Functions for Bending Moment

Singularity Functions for Bending Moment

Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented using a...

Trigonometric Substitution

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals that contain square root expressions involving quadratic forms. It is particularly effective when the integrand includes terms resembling those found in standard geometric equations, such as circles or ellipses.Molniya satellites follow highly elliptical orbits, repeatedly sweeping out the same regions of space as they revolve around Earth. To estimate the area enclosed by such an orbit, the path is modeled as an ellipse...

Integrals of Powers of Sine and Cosine

Integrals of Powers of Sine and Cosine

Trigonometric integrals involve the integration of expressions containing powers of sine, cosine, and related functions. They are common in calculus problems and have applications in physics and engineering. The method for integrating expressions of the form sinm(x)cosn(x) depends on whether the exponents are odd or even.If the power of sine is odd, one sine factor is separated from the integrand, leaving an even power of sine. The remaining sine terms are rewritten in terms of cosine using the...

Beams with Symmetric Loadings

Beams with Symmetric Loadings

The moment-area method is an analytical tool used in structural engineering to determine the slope and deflection of beams under various loads. Consider a cantilever with a concentrated load and moment at the free end. The first step is constructing a free-body diagram to calculate the reactions at the fixed end. Next, the bending moment diagram is plotted to visualize how the bending moment varies along the beam's length, focusing on points where the bending moment equals zero.
The M/EI...

Deflection of a Beam

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Accurately determining beam deflection and slope under various loading conditions in structural engineering is crucial for ensuring safety and structural integrity. Singularity functions offer a streamlined approach to analyzing beams, especially when multiple loading functions complicate the bending moment equation.
Singularity functions, described in an earlier lesson, are powerful mathematical tools that represent discontinuities within a function commonly encountered in structural loading...

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Related Experiment Video

Updated: Jun 16, 2026

Scattering And Absorption of Light in Planetary Regoliths

Scattering And Absorption of Light in Planetary Regoliths

Published on: July 1, 2019

Generating bessel functions in mie scattering calculations using continued fractions.

|February 19, 2010

Summary

A novel algorithm efficiently computes Bessel functions for Mie theory calculations. This method enhances accuracy without requiring extensive computational resources, simplifying complex scientific modeling.

Area of Science:

Computational physics
Electromagnetic theory
Numerical analysis

Background:

Mie calculations are essential for understanding light scattering.
Bessel functions are critical components of Mie theory.
Existing methods for Bessel function computation can be resource-intensive.

Purpose of the Study:

To present a new, accurate method for generating Bessel functions.
To improve the efficiency of Mie calculation inputs.
To offer a robust algorithm for Bessel function evaluation.

Main Methods:

Developed a novel algorithm for evaluating continued fractions.
The algorithm starts evaluation from the beginning of the fraction.
Incorporated a built-in error-checking mechanism.

More Related Videos

Controlled Synthesis and Fluorescence Tracking of Highly Uniform Poly(N-isopropylacrylamide) Microgels

Controlled Synthesis and Fluorescence Tracking of Highly Uniform Poly(N-isopropylacrylamide) Microgels

Published on: September 8, 2016

In situ Grazing Incidence Small Angle X-ray Scattering on Roll-To-Roll Coating of Organic Solar Cells with Laboratory X-ray Instrumentation

In situ Grazing Incidence Small Angle X-ray Scattering on Roll-To-Roll Coating of Organic Solar Cells with Laboratory X-ray Instrumentation

Published on: March 2, 2021

Related Experiment Videos

Last Updated: Jun 16, 2026

Scattering And Absorption of Light in Planetary Regoliths

Scattering And Absorption of Light in Planetary Regoliths

Published on: July 1, 2019

Controlled Synthesis and Fluorescence Tracking of Highly Uniform Poly(N-isopropylacrylamide) Microgels

Controlled Synthesis and Fluorescence Tracking of Highly Uniform Poly(N-isopropylacrylamide) Microgels

Published on: September 8, 2016

In situ Grazing Incidence Small Angle X-ray Scattering on Roll-To-Roll Coating of Organic Solar Cells with Laboratory X-ray Instrumentation

In situ Grazing Incidence Small Angle X-ray Scattering on Roll-To-Roll Coating of Organic Solar Cells with Laboratory X-ray Instrumentation

Published on: March 2, 2021

Main Results:

Achieved improved accuracy in Bessel function generation.
Eliminated the need for extended precision or large storage.
Successfully presented continued fraction representations for spherical Bessel functions and their ratios.

Conclusions:

The new method provides an accurate and efficient way to compute Bessel functions.
This advancement simplifies complex Mie scattering simulations.
The algorithm offers a valuable tool for researchers in computational physics and optics.