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Related Concept Videos

Propagation of Waves01:07

Propagation of Waves

When a wave propagates from one medium to another, part of it may get reflected in the first medium, and part of it may get transmitted to the second medium. In such a case, the interface of the two mediums can be considered as a boundary that is neither fixed nor free.
Consider a scenario where a wave propagates from a string of low linear mass density to a string of high linear mass density. In such a case, the reflected wave is out of phase with respect to the incident wave, however the...
Beams with Symmetric Loadings01:15

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...
Method of Superposition01:20

Method of Superposition

The method of superposition is a crucial technique in structural engineering, used to analyze the effect of multiple loads on beams. This approach involves calculating the deflection and slope for each load on a beam separately, and then summing these effects to determine the overall impact. It is applicable only when the beam material remains within its elastic limit, ensuring that deformations are linearly elastic.
When applying the method of superposition, each type of load—whether...
Castigliano's Theorem: Problem Solving01:14

Castigliano's Theorem: Problem Solving

The deflection of a simply supported beam that carries a central point load can be analyzed using structural mechanics principles, particularly by applying Castigliano's theorem. This theorem relates the displacement at the load application point to the partial derivatives of the strain energy in the structure. The simply supported beam with a point load at its center has symmetric reaction forces at the supports, each bearing half of the load. The bending moment at any point along the beam is...
Stokes’ Theorem and Its Applications01:24

Stokes’ Theorem and Its Applications

Stokes’ Theorem provides a fundamental connection between the circulation of a vector field along a closed boundary and the cumulative rotational behavior across the surface it encloses. For a smooth three-dimensional surface with an oriented boundary curve, this theorem offers a unified way to relate motion along the edge to local rotational effects distributed over the surface.Mathematical FormulationThe theorem states that the circulation of a vector field along a closed curve is equal to...
Shearing Stresses in a Beam: Problem Solving01:14

Shearing Stresses in a Beam: Problem Solving

A cantilever beam with a rectangular cross-section under distributed and point loads experiences shearing stresses. The analysis begins by identifying the loads acting on the beam. Then, the reactions at the beam's fixed end are calculated using equilibrium equations. The vertical reaction is a combination of the distributed and point loads, while the moment reaction is the sum of their moments. The shear force distribution along the beam, resulting from these loads, is established by creating...

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Related Experiment Video

Updated: Jun 20, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
12:14

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

Unconditionally stable procedure to propagate beams through optical waveguides using the collocation method.

A Sharma, A Taneja

    Optics Letters
    |September 25, 2009
    PubMed
    Summary
    This summary is machine-generated.

    A new stable method improves beam propagation in optical waveguides, overcoming limitations of earlier techniques for accurate simulations.

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    High-speed Continuous-wave Stimulated Brillouin Scattering Spectrometer for Material Analysis
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    High-speed Continuous-wave Stimulated Brillouin Scattering Spectrometer for Material Analysis

    Published on: September 22, 2017

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    Last Updated: Jun 20, 2026

    The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
    12:14

    The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

    Published on: August 12, 2013

    High-speed Continuous-wave Stimulated Brillouin Scattering Spectrometer for Material Analysis
    07:55

    High-speed Continuous-wave Stimulated Brillouin Scattering Spectrometer for Material Analysis

    Published on: September 22, 2017

    Area of Science:

    • Optics and photonics
    • Computational physics
    • Waveguide theory

    Background:

    • The standard collocation method for optical waveguide beam propagation becomes unstable with larger extrapolation intervals.
    • Existing methods struggle with accuracy and stability for extended simulation distances.

    Purpose of the Study:

    • To develop a novel, unconditionally stable numerical procedure for solving the matrix propagation equation in optical waveguides.
    • To enhance the accuracy and stability of beam propagation simulations, particularly for large extrapolation intervals.

    Main Methods:

    • A new procedure is presented to solve the matrix propagation equation.
    • The method utilizes the Fresnel approximation for enhanced stability.
    • Numerical performance is benchmarked against the original collocation method and the standard propagating-beam method.

    Main Results:

    • The new procedure demonstrates unconditional stability for arbitrarily large extrapolation intervals.
    • The formulation maintains accuracy comparable to existing methods while significantly improving stability.
    • Comparative analysis confirms the robustness of the new method for extended simulations.

    Conclusions:

    • The developed method offers a significant advancement in the numerical simulation of beam propagation in optical waveguides.
    • This unconditionally stable approach enables more reliable and extended simulations, crucial for optical device design and analysis.
    • The Fresnel approximation-based technique provides a robust alternative for addressing stability issues in waveguide propagation modeling.