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written by
Kyle Forinash and Wolfgang Christian
This is a collection of interactive tutorials on the fundamentals of waves. The tutorials emphasize concepts that are not usually easy to illustrate in textbooks. Java applets are used to illustrate the physics. The lessons begin with very simple wave properties and end with an examination of nonlinear wave behavior.
33 supplemental documents are available
Waves Tutorial ePub PreviewThis ePub document contains a preview of the Waves Tutorial. Use an ePub 3 reader that supports Math ML and JavaScript, such as the iBooks Reader on Apple devices or the Gitden on Android. The complete ePub tutorial is available in Apple iTunes. … download 1231kb .epub Last Modified: December 22, 2015 
Introduction to Waves TutorialWaves: An Interactive Tutorial is a set of 33 exercises designed to teach the fundamentals of wave dynamics. It starts with very simple wave properties and ends with an examination of nonlinear wave behavior. The emphasis here is on the properties of waves which … download 241kb .zip Last Modified: July 28, 2015
Sine WavesThis simulation shows a perfect, smooth wave out on the ocean far enough from shore so that it has not started to break (complications involved in describing real waves will be discussed later in this tutorial). download 247kb .zip Last Modified: May 28, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Speed of a WaveThere are three different velocities involved with describing a wave, one of which will be introduced in this simulation. download 265kb .zip Last Modified: May 28, 2018 previous versions
Transverse WavesTransverse waves are the kind of wave you usually think of when you think of a wave. The motion of the material constituting the wave is up and down so that as the wave moves forward the material moves perpendicular (or transverse) to the direction the wave moves. download 312kb .zip Last Modified: May 28, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Simple Harmonic MotionThe Simple Harmonic Motion simulation shows the motion a mass on a spring graphs its time dependence. download 274kb .zip Last Modified: May 28, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Simple Harmonic Motion and ResonanceThe Simple Harmonic Motion and Resonance simulation shows a driven damped harmonic oscillator. The user can select under damped, over damped, and critically damped conditions. download 276kb .zip Last Modified: May 28, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Longitudinal WavesThe Longitudinal Waves simulation shows waves where the motion of the material is back and forth in the same direction that the wave moves. Sound waves (in air and in solids) are examples of longitudinal waves. download 275kb .zip Last Modified: May 28, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Water WavesWater Waves, like many real physical waves, are combinations of three kinds of wave motion; transverse, longitudinal and torsional. download 267kb .zip Last Modified: May 28, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Two-Dimensional WavesThe Two-Dimensional Waves simulation shows a plane wave in two dimensions traveling in the x-y plane, in the x direction, viewed from above. In these simulations the amplitude (in the z direction, towards you) is represented in grey-scale. download 280kb .zip Last Modified: May 29, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Adding Linear Waves (Superposition)Linear waves have the property, called superposition, that their amplitudes add linearly if they arrive at the same point at the same time. This simulation shows the sum of two wave functions u(x,t) = f(x,t) + g(x,t). download 240kb .zip Last Modified: May 29, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
InterferenceThe Interference simulation shows a top view of a source making waves on the surface of a tank of water (imagine tapping the surface of a pond with the end of a stick at regular intervals). The white circles coming from the spot represents the wave crests with … download 285kb .zip Last Modified: May 29, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Group VelocityThe Group Velocity simulation shows how several waves add together to form a single wave shape (called the envelope) and how we can quantify the speed with two numbers, the group velocity of the combined wave and the phase velocity. download 292kb .zip Last Modified: May 29, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Other Wave FunctionsThe Other Wave Functions simulation explores how any function of x and t which has these variables in the form x - v t will be a traveling wave with speed v. download 228kb .zip Last Modified: May 29, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Fourier Analysis and SynthesisFourier analysis is the process of mathematically breaking down a complex wave into a sum of of sines and cosines. Fourier synthesis is the process of building a particular wave shape by adding sines and cosines. Fourier analysis and synthesis can be done for any … download 1088kb .zip Last Modified: May 29, 2018 previous versions
MirrorsThe Mirrors simulation exploration of specular reflection fro plane, concave, and convex surfaces. download 271kb .zip Last Modified: May 29, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Collisions with BoundariesThe Collisions with Boundaries simulation shows how the phase of the wave may be different after reflection, depending on the surface from which they reflect. download 269kb .zip Last Modified: May 29, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Standing WavesThis simulation shows how a standing wave is formed from two identical waves moving in opposite directions. For standing waves on a string the ends are fixed and there are nodes at the ends of the string. This limits the wavelengths that are possible which in turn determines the frequencies download 260kb .zip Last Modified: May 29, 2018 previous versions
RefractionThis simulation shows how a wave that changes speed as it crosses the boundary of between two materials will also change direction if it crosses the boundary at an angle other than perpendicular. download 236kb .zip Last Modified: May 30, 2018 previous versions
LensesThis simulation shows how light rays are bent using the thin lens approximation which assumes the lens thickness is small compared to the curvature of the glass. download 257kb .zip Last Modified: May 30, 2018 previous versions
Path Difference and InterferenceThis simulation shows two identical waves that start at different locations. A third graph shows the sum of these two waves. download 247kb .zip Last Modified: May 30, 2018 previous versions
ImpedanceThis simulations represents a string as a row of individual masses connected by invisible springs. Waves are reflected in the middle of this string because the mass of the string is different on the left as compared with the right. download 256kb .zip Last Modified: May 30, 2018 previous versions
Dispersion of LightThis simulation shows visible light passing through a prism. You can choose the color and see what the index is for that wavelength. download 239kb .zip Last Modified: May 30, 2018 previous versions
Dispersion of Fourier ComponentsThis simulation starts with the first four components of the Fourier series for a traveling square wave with no dispersion. Changing the angular frequency of a component causes the initial wave function to distort due to dispersion. download 243kb .zip Last Modified: May 30, 2018 previous versions
DiffractionThis simulation shows what happens to a plane-wave light source (below the simulation, not shown) as it passes through an opening. The wavelength of the waves and the size of the opening can be adjusted. download 260kb .zip Last Modified: May 30, 2018 previous versions
Doppler EffectThis simulation models at the Doppler effect for sound; the black circle is the source and the red circle is the receiver. If either the source or the receiver of a wave are in motion the apparent wavelength and frequency of the received wave change. This is apparent … download 263kb .zip Last Modified: June 5, 2018 previous versions
AntennaThis simulation shows the effect of a wave traveling in the x-direction on a second charge inside a receiving antenna. Only the y-component of the change in the electric field is shown (so an oscillation frequency of zero will show nothing, because there is only a constant electric field). download 236kb .zip Last Modified: June 1, 2018 previous versions
Electromagnetic Plane WavesThis simulation shows a plane electromagnetic wave traveling in the y-direction. Both electric and magnetic fields are shown in the 3D representation. download 260kb .zip Last Modified: June 1, 2018 previous versions
PolarizationThis simulation shows the electric field component[s] for a wave traveling straight towards the observer in the +y direction. A polarized wave was previously defined to be an electromagnetic wave that has its electric field confined to change in only one direction. … download 253kb .zip Last Modified: June 1, 2018 previous versions
Wave EquationIn this simulation we look at the dynamics of waves; the physical situations and laws give rise to waves. We start with a string that has a standing wave on it and look at the forces acting on each end of a small segment of the string due to the neighboring sections. … download 252kb .zip Last Modified: June 1, 2018 previous versions
Oscillator ChainIn this simulation we examine waves that occur on chains of masses with mass M coupled together with elastic, Hooke's law forces (F = -?x where ? is the spring constant and x is the amount the spring stretches). The masses are constrained to only move up and down so … download 242kb .zip Last Modified: June 1, 2018 Released under a CC Noncommercial-Share Alike license. previous versions
Non-Linear WavesThis simulations shows what happens if forces other than tension act on a string. Some additional forces cause the dispersion we saw in simulations 22 and 23 but friction, dissipation and nonlinearity can cause other behavior as we will see here. download 235kb .zip Last Modified: June 1, 2018 previous versions
SolitonsThis simulation explores a special solution of the non-linear wave equation where the effects of dispersion and dissipation (which tend to make a wave pulse spread out) are exactly compensated for by a nonlinear force (which, as we have seen, tends to cause … download 453kb .zip Last Modified: June 1, 2018 previous versions
3 source code documents are available
Sine Waves Source CodeThis source code zip archive contains an XML representation of the Sine Wave JavaScript Model. Unzip this archive in your Ejs workspace to compile and run this model using EjsS 5. Although EjsS is a Java program, EjsS 5 creates a stand alone JavaScript program from this source code. download 37kb .zip Last Modified: March 20, 2015 previous versions
Speed of a Wave Source CodeThis source code zip archive contains an XML representation of the Speed of a Wave JavaScript Model. Unzip this archive in your Ejs workspace to compile and run this model using EjsS 5. Although EjsS is a Java program, EjsS 5 creates a stand alone JavaScript program from this source code. download 46kb .zip Last Modified: March 20, 2015 previous versions
EM Waves from an Accelerating ChargeThis simulation shows an accelerating positive charge and the electric field around it in two dimensions. Because the charge is accelerated there will be a disturbance in the field. The energy carried by the disturbance comes from the input energy needed to accelerate the charge. download 283kb .zip Last Modified: April 27, 2024 previous versions
AAAS Benchmark Alignments (2008 Version)4. The Physical Setting
4F. Motion
11. Common Themes
11B. Models
Common Core State Standards for Mathematics AlignmentsHigh School — Algebra (9-12)
Creating Equations? (9-12)
High School — Functions (9-12)
Interpreting Functions (9-12)
Trigonometric Functions (9-12)
Common Core State Reading Standards for Literacy in Science and Technical Subjects 6—12
Craft and Structure (6-12)
Range of Reading and Level of Text Complexity (6-12)
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<a href="https://psrc.aapt.org/items/detail.cfm?ID=3146">Forinash, Kyle, and Wolfgang Christian. Waves: An Interactive Tutorial. August 9, 2005.</a>
AIP Format
K. Forinash and W. Christian, (2002), WWW Document, (https://www.compadre.org/books/WavesIntTut).
AJP/PRST-PER
K. Forinash and W. Christian, Waves: An Interactive Tutorial (2002), <https://www.compadre.org/books/WavesIntTut>.
APA Format
Forinash, K., & Christian, W. (2005, August 9). Waves: An Interactive Tutorial. Retrieved March 16, 2025, from https://www.compadre.org/books/WavesIntTut
Chicago Format
Forinash, Kyle, and Wolfgang Christian. Waves: An Interactive Tutorial. August 9, 2005. https://www.compadre.org/books/WavesIntTut (accessed 16 March 2025).
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Forinash, Kyle, and Wolfgang Christian. Waves: An Interactive Tutorial. 2002. 9 Aug. 2005. 16 Mar. 2025 <https://www.compadre.org/books/WavesIntTut>.
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@misc{
Author = "Kyle Forinash and Wolfgang Christian",
Title = {Waves: An Interactive Tutorial},
Volume = {2025},
Number = {16 March 2025},
Month = {August 9, 2005},
Year = {2002}
}
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%A Kyle Forinash %A Wolfgang Christian %T Waves: An Interactive Tutorial %D August 9, 2005 %U https://www.compadre.org/books/WavesIntTut %O application/java
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%0 Electronic Source %A Forinash, Kyle %A Christian, Wolfgang %D August 9, 2005 %T Waves: An Interactive Tutorial %V 2025 %N 16 March 2025 %8 August 9, 2005 %9 application/java %U https://www.compadre.org/books/WavesIntTut Disclaimer: ComPADRE offers citation styles as a guide only. We cannot offer interpretations about citations as this is an automated procedure. Please refer to the style manuals in the Citation Source Information area for clarifications.
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