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Related Concept Videos

Shock Waves01:16

Shock Waves

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While deriving the Doppler formula for the observed frequency of a sound wave, it is assumed that the speed of sound in the medium is greater than the source's speed through it. When this condition is breached, a shock wave occurs.
When the source's speed approaches the speed of sound, constructive interference between successive wavefronts emitted by the source occurs immediately behind it. Initially, scientists believed that this constructive interference would result in such high...
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Sound as Pressure Waves01:17

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Sound waves, which are longitudinal waves, can be modeled as the displacement amplitude varying as a function of the spatial and temporal coordinates. As a column of the medium is displaced, its successive columns are also displaced. As the successive displacements differ relatively, a pressure difference with the surrounding pressure is created. The gauge pressure varies across the medium.
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Deriving the Speed of Sound in a Liquid01:09

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As with waves on a string, the speed of sound or a mechanical wave in a fluid depends on the fluid's elastic modulus and inertia. The two relevant physical quantities are the bulk modulus and the density of the material. Indeed, it turns out that the relationship between speed and the bulk modulus and density in fluids is the same as that between the speed and the Young's modulus and density in solids.
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Travelling Waves01:04

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A wave is a disturbance that propagates from its source, repeating itself periodically, and is typically associated with simple harmonic motion. Mechanical waves are governed by Newton's laws and require a medium to travel. A medium is a substance in which a mechanical wave propagates, and the medium produces an elastic restoring force when it is deformed.
Water waves, sound waves, and seismic waves are some examples of mechanical waves. For water waves, the wave propagation medium is...
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Damped Oscillations01:07

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In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
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Propagation of Waves01:07

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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.
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Related Experiment Video

Updated: Mar 19, 2026

Measurements of Waves in a Wind-wave Tank Under Steady and Time-varying Wind Forcing
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Expansion shock waves in regularized shallow-water theory.

Gennady A El1, Mark A Hoefer2, Michael Shearer3

  • 1Department of Mathematical Sciences , Loughborough University , Loughborough LE11 3TU, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|June 10, 2016
PubMed
Summary

Researchers discovered a new expansion shock wave in shallow-water equations. This novel shock defies classical conditions and persists due to non-local dispersion, showing robustness in simulations.

Keywords:
Benjamin–Bona–Mahony equationBoussinesq equationsLax entropy conditionnon-local dispersion

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Area of Science:

  • Fluid dynamics
  • Nonlinear wave phenomena
  • Mathematical physics

Background:

  • Shallow-water equations model fluid flow in shallow depths.
  • Classical shock waves adhere to entropy conditions.
  • Regularized equations incorporate dispersive effects.

Purpose of the Study:

  • To identify and characterize a new type of shock wave.
  • To analyze the behavior of expansion shocks in regularized shallow-water equations.
  • To investigate the conditions for the existence and persistence of expansion shocks.

Main Methods:

  • Constructing stationary expansion shock solutions.
  • Applying matched asymptotic expansions.
  • Performing numerical simulations.
  • Analyzing regularized shallow-water equations (Benjamin-Bona-Mahony, Boussinesq).

Main Results:

  • Identification of a novel expansion shock wave.
  • Demonstration that expansion shocks violate the Lax entropy condition.
  • Justification of expansion shock persistence in initial value problems.
  • Establishment of algebraic decay of the shock.
  • Observation of shock robustness with weak dissipation and asymmetric initial conditions.

Conclusions:

  • Expansion shocks are a new class of waves in certain shallow-water models.
  • The non-local dispersive term is crucial for expansion shock existence.
  • These shocks exhibit robust behavior under various conditions, including dissipation and solitary wave shedding.