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.

Although this material deals strictly with classical wave properties, it can be used as a remedial tutorial or to introduce concepts such as group velocity and interference in a quantum mechanics class.

Waves Tutorial ePub Preview This 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. …
This 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.

Introduction to Waves Tutorial Waves: 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 …Waves: 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 are difficult to illustrate in a static textbook figure. Simulations are not a substitute for laboratory work. However they allow for visualization of processes that cannot normally be seen (for example electric and magnetic fields). They allow for visualization of process that are too fast (for example waves) to follow in real time or too small to see (for example thermodynamics at the molecular scale). They allow manipulation of processes which might be dangerous (collisions) or hard to experiment with (waves). They also allow for easy repetition. For all of these reasons, simulations are an excellent way to introduce students to the complex phenomena of waves.

Sine Waves
This 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.
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Speed of a Wave
There 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
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Transverse Waves
Transverse 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.
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Simple Harmonic Motion and Resonance
The 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.
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Longitudinal Waves
The 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.
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Water Waves
Water 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.
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Two-Dimensional Waves
The 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.
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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.
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Interference The 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 …
The 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 troughs in between. Two sources can be seen at the same time and the separation between them and the wavelength of both can be adjusted

Group Velocity
The 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.
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Other Wave Functions
The 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.
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Fourier Analysis and Synthesis Fourier 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 …
Fourier 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 type of wave, not just the sound waves shown in this simulation.

Collisions with Boundaries
The 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.
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Standing Waves
This 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
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Refraction
This 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
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Lenses
This 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
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Path Difference and Interference
This 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
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Impedance
This 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
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Dispersion of Light
This 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
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Dispersion of Fourier Components
This 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
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Diffraction
This 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
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Doppler Effect This 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 …
This 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 shift in frequency of a moving source or observer is called the Doppler Effect. The speed of the wave is not affected by the motion of the source or receiver and neither is the amplitude.

EM Waves from an Accelerating Charge
This 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: May 30, 2018
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Antenna
This 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
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Electromagnetic Plane Waves
This 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
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Polarization This 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. …
This 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. In this simulation we further investigate polarized waves.

Wave Equation In 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. …
In 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. For visualization purposes the string is shown as a series of masses but the physical system is a continuous string. Although the derivation is for a string, similar results occur in many other systems. The ends of the section of string we are interested in are marked with red dots in the simulation. The tension acting on each end is shown with a vector (in red) and its components (green and blue).

Oscillator Chain In 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 …
In 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 that the stretching depends only on the difference in the y locations of the masses.

Non-Linear Waves
This 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
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Solitons This 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 …
This 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 steepening of a wave). In this case there may be a special wave pulse shape that can travel and maintain its shape called a soliton.

Sine Waves Source Code
This 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
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Speed of a Wave Source Code
This 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

6-8: 4F/M4. Vibrations in materials set up wavelike disturbances that spread away from the source. Sound and earthquake waves are examples. These and other waves move at different speeds in different materials.

6-8: 4F/M6. Light acts like a wave in many ways. And waves can explain how light behaves.

6-8: 4F/M7. Wave behavior can be described in terms of how fast the disturbance spreads, and in terms of the distance between successive peaks of the disturbance (the wavelength).

9-12: 4F/H5ab. The observed wavelength of a wave depends upon the relative motion of the source and the observer. If either is moving toward the other, the observed wavelength is shorter; if either is moving away, the wavelength is longer.

9-12: 4F/H6ab. Waves can superpose on one another, bend around corners, reflect off surfaces, be absorbed by materials they enter, and change direction when entering a new material. All these effects vary with wavelength.

9-12: 4F/H6c. The energy of waves (like any form of energy) can be changed into other forms of energy.

11. Common Themes

11B. Models

6-8: 11B/M4. Simulations are often useful in modeling events and processes.

Common Core State Standards for Mathematics Alignments

High School — Algebra (9-12)

Creating Equations^{?} (9-12)

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

High School — Functions (9-12)

Interpreting Functions (9-12)

F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.^{?}

F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.^{?}

F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Trigonometric Functions (9-12)

F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.^{?}

Common Core State Reading Standards for Literacy in Science and Technical Subjects 6—12

Craft and Structure (6-12)

RST.11-12.4 Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 11—12 texts and topics.

Range of Reading and Level of Text Complexity (6-12)

RST.11-12.10 By the end of grade 12, read and comprehend science/technical texts in the grades 11—CCR text complexity band independently and proficiently.

<a href="https://www.compadre.org/quantum/items/detail.cfm?ID=3146">Forinash, Kyle, and Wolfgang Christian. Waves: An Interactive Tutorial. August 9, 2005.</a>

Forinash, K., & Christian, W. (2005, August 9). Waves: An Interactive Tutorial. Retrieved March 4, 2021, from https://www.compadre.org/books/WavesIntTut

Forinash, Kyle, and Wolfgang Christian. Waves: An Interactive Tutorial. August 9, 2005. https://www.compadre.org/books/WavesIntTut (accessed 4 March 2021).

@misc{
Author = "Kyle Forinash and Wolfgang Christian",
Title = {Waves: An Interactive Tutorial},
Volume = {2021},
Number = {4 March 2021},
Month = {August 9, 2005},
Year = {2002}
}

%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

%0 Electronic Source %A Forinash, Kyle %A Christian, Wolfgang %D August 9, 2005 %T Waves: An Interactive Tutorial %V 2021 %N 4 March 2021 %8 August 9, 2005 %9 application/java %U https://www.compadre.org/books/WavesIntTut

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