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Related Concept Videos

Equations of Wave Motion01:02

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.
Velocity and Acceleration of a Wave00:51

Velocity and Acceleration of a Wave

A wave propagates through a medium with a constant speed, known as a wave velocity. It is different from the speed of the particles of the medium, which is not constant. In addition, the velocity of the medium is perpendicular to the velocity of the wave. The variable speed of the particles of the medium implies that there must be acceleration associated with it. 
The velocity of the particles can be obtained by taking the partial derivative of the position equation with respect to time. We can...
Electromagnetic Wave Equation01:24

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...
Wave Parameters01:10

Wave Parameters

The simplest mechanical waves are associated with simple harmonic motion and repeat themselves for several cycles. These simple harmonic waves can be modeled using a combination of sine and cosine functions. Consider a simplified surface water wave that moves across the water's surface. Unlike complex ocean waves, in surface water waves, water moves vertically, oscillating up and down, whereas the disturbance of the wave moves horizontally through the medium. If a seagull is floating on the...
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...
Graphing the Wave Function01:13

Graphing the Wave Function

Consider the wave equation for a sinusoidal wave moving in the positive x-direction. The wave equation is a function of both position and time. From the wave equation, two different graphs can be plotted.

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Related Experiment Video

Updated: May 15, 2026

Measurements of Waves in a Wind-wave Tank Under Steady and Time-varying Wind Forcing
08:54

Measurements of Waves in a Wind-wave Tank Under Steady and Time-varying Wind Forcing

Published on: February 13, 2018

Evolution of basic equations for nearshore wave field.

Masahiko Isobe1

  • 1Graduate School of Frontier Sciences, The University of Tokyo, Chiba, Japan. isobe@k.u-tokyo.ac.jp

Proceedings of the Japan Academy. Series B, Physical and Biological Sciences
|January 16, 2013
PubMed
Summary

This paper reviews theories for periodic and random waves, detailing wave equations like the mild-slope and Boussinesq equations for predicting wave transformations in various conditions.

Area of Science:

  • Oceanography
  • Fluid Dynamics
  • Applied Mathematics

Background:

  • Understanding wave theories is crucial for coastal engineering and marine science.
  • Existing models often simplify complex wave behaviors like randomness and nonlinearity.

Purpose of the Study:

  • To provide a systematic overview of wave theories and their validity.
  • To introduce and classify various wave equations for different applications.
  • To present methods for analyzing random and nonlinear waves.

Main Methods:

  • Review of established theories for periodic waves (Stokes, cnoidal).
  • Development of methods for estimating directional spectra of random waves.
  • Derivation and discussion of wave equations: mild-slope, parabolic approximations, Boussinesq, and nonlinear mild-slope equations.

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Related Experiment Videos

Last Updated: May 15, 2026

Measurements of Waves in a Wind-wave Tank Under Steady and Time-varying Wind Forcing
08:54

Measurements of Waves in a Wind-wave Tank Under Steady and Time-varying Wind Forcing

Published on: February 13, 2018

Visualization of Flow Field Around a Vibrating Pipeline Within an Equilibrium Scour Hole
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Visualization of Flow Field Around a Vibrating Pipeline Within an Equilibrium Scour Hole

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Visualizing Hyporheic Flow Through Bedforms Using Dye Experiments and Simulation
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Main Results:

  • Established validity ranges for Stokes and cnoidal wave theories.
  • Introduction of methods to handle random wave spectra.
  • Classification of wave equations based on their assumptions and applications, including linear and nonlinear models.

Conclusions:

  • A systematic classification of wave equations aids in theoretical understanding and selection.
  • The presented methods and equations are applicable to predicting wave transformations in diverse marine environments.
  • This work provides a comprehensive framework for analyzing complex wave phenomena.