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Modes of Standing Waves - I

Modes of Standing Waves - I

A close look at earthquakes provides evidence for the conditions appropriate for resonance, standing waves, and constructive and destructive interference. A building may vibrate for several seconds with a driving frequency matching the building's natural frequency of vibration; this produces a resonance that results in one building collapsing while the neighboring buildings do not. Often, buildings of a certain height are devastated, while other taller buildings remain intact. This phenomenon...

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Sometimes waves do not seem to move; rather, they just vibrate in place. Unmoving waves can be seen on the surface of a glass of milk kept in a refrigerator, which is one example of standing waves. Vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. These waves are formed or created by the superposition of two or more identical moving waves in opposite directions. The waves move through each other, with their...

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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...

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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.
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Simple harmonic motion is the name given to oscillatory motion for a system where the net force can be described by Hooke's law. If the net force can be described by Hooke's law and there is no damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position. To derive an equation for period and frequency, the equation of motion is used. The period of a simple harmonic oscillator is given...

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Related Experiment Video

Updated: May 18, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Simple waves in Hertzian chains.

B Edward McDonald¹, David Calvo

¹US Naval Research Laboratory, Washington, DC 20375, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 26, 2012

Summary

This study reveals novel simple waves in discrete sphere chains, offering a simplified model for shock wave dynamics. These findings advance understanding of wave propagation and energy dissipation in granular materials.

Area of Science:

Physics
Materials Science
Nonlinear Dynamics

Background:

Granular materials exhibit complex wave propagation phenomena.
Hertzian contact forces govern interactions in many granular systems.
Previous models often simplify or overlook specific wave behaviors.

Purpose of the Study:

To analyze wave dynamics in a discrete chain of spheres with Hertzian interactions.
To derive and investigate a reduced-order model for wave propagation.
To compare simulation results from discrete and continuum models.

Main Methods:

Examination of a discrete system of spheres under Hertz force (index 3/2) in the long wavelength limit.
Derivation of a continuum second-order equation of motion and a reduced first-order equation for simple waves.

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Last Updated: May 18, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

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Published on: December 4, 2017

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

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Numerical simulations of shock wave development and wave collisions using the discrete system and reduced equation.

Main Results:

Identification of novel simple waves governed by a first-order equation (reduced index 5/4).
Accurate comparison between discrete system simulations and the reduced first-order equation for shock wave development.
Observation of wave collision dynamics, including phase advance, and evolution towards a universal similarity solution.

Conclusions:

The reduced first-order equation effectively describes simple wave behavior in these granular chains.
The study provides a closed-form solution for waveform evolution, shock location, and amplitude.
Findings offer insights into energy dissipation and wave interactions in granular media.