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Equations of Wave Motion

Equations of Wave Motion

Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a period T. The wave on the string is sinusoidal and is translated in the positive x-direction as time progresses. Sine is a function of the angle θ, oscillating between +A and −A and repeating every 2π radians. To construct a wave model, the ratio of the angle θ and the position x is considered.

Unsymmetric Loading of Thin-Walled Members: Problem Solving

Unsymmetric Loading of Thin-Walled Members: Problem Solving

The shear center of a channel section with uniform thickness, height, and width, is determined by computing the shear force in the member and calculating the moments of inertia of the sections.
To compute the shear forces, find the shear flow at a specific distance from the endpoint using the vertical shear and the moment of inertia values. The total shear force on the flange is calculated by integrating the shear flow from one end of the flange to the other.
Next, calculate the moments of...

Propagation of Waves

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...

Standing Waves in a Cavity

Standing Waves in a Cavity

A household microwave and lasers are examples of standing electromagnetic waves in a cavity. When two conducting metal plates are placed parallel at the nodal planes, it creates a cavity where standing waves are formed. The cavity between the two planes is analogous to a stretched string held at the points x = 0 and x = L. Here, the distance 'L' between the two planes must be an integer multiple of half of the wavelength. The wavelengths that satisfy this condition are given by:

Electromagnetic Wave Equation

Electromagnetic Wave Equation

Maxwell's equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose trajectory can thus be determined using suitable boundary conditions. The objective of electromagnetism is thus theoretically complete.
However, although electric and magnetic fields were first introduced as mathematical constructs to simplify the description of mutual forces between charges, a natural question emerges from Maxwell's equations: What...

Modes of Standing Waves: II

Modes of Standing Waves: II

The starting point for expressing the modes of standing waves is understanding the boundary conditions that the waves must follow. The boundary conditions are derived from the physical understanding of how the standing waves are sustained, that is, how the vibrating particles of the medium behave at the boundaries imposed on them.
For a tube open at one end and closed at the other filled with air, the modes are such that there is always an antinode at the open end and a node at the closed end.

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Related Experiment Video

Updated: Jun 13, 2026

Fabrication of Zero Mode Waveguides for High Concentration Single Molecule Microscopy

Fabrication of Zero Mode Waveguides for High Concentration Single Molecule Microscopy

Published on: May 12, 2020

Novel solution of 2-D waveguides using the finite element method.

|April 17, 2010

Summary

A new finite element method solves 2-D waveguide equations, handling medium inhomogeneities. This technique accurately calculates transverse field components for far-field radiation patterns.

Area of Science:

Electromagnetics and Wave Propagation
Computational Physics

Background:

Waveguide analysis is crucial for electromagnetic devices.
Solving complex waveguide equations, especially with inhomogeneities, presents significant challenges.

Purpose of the Study:

To introduce a novel finite element method (FEM) for solving two-dimensional (2-D) waveguide equations.
To demonstrate the method's capability in handling various medium inhomogeneities automatically.

Main Methods:

The study employs the finite element method (FEM) to solve the full-wave equation for 2-D waveguides.
The technique directly computes transverse electric and magnetic field components.

Main Results:

The developed FEM approach successfully solves 2-D waveguide equations, incorporating medium inhomogeneities.

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Fabrication And Characterization Of Photonic Crystal Slow Light Waveguides And Cavities

Fabrication And Characterization Of Photonic Crystal Slow Light Waveguides And Cavities

Published on: November 30, 2012

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Last Updated: Jun 13, 2026

Fabrication of Zero Mode Waveguides for High Concentration Single Molecule Microscopy

Fabrication of Zero Mode Waveguides for High Concentration Single Molecule Microscopy

Published on: May 12, 2020

Fabrication And Characterization Of Photonic Crystal Slow Light Waveguides And Cavities

Fabrication And Characterization Of Photonic Crystal Slow Light Waveguides And Cavities

Published on: November 30, 2012

The method provides transverse field components essential for subsequent analysis.

Conclusions:

The novel FEM technique offers an effective solution for analyzing 2-D waveguides with complex media.
The computed transverse field components facilitate accurate far-field radiation pattern calculations.