About

Function plotter z = f ( x , y )

3D Waves

This function plotter displays functions of typical 3D waves z = f(x , y) that may contain up to 4 continuously variable parameters a, b, c, p. In the predefined waves p controls the speed of the waves in the xy plane

At start of the simulation you will first see the projection of a plane wave in space traveling in the x direction, viewed under perspective distortion. It is embedded into an xyz tripod, and is accompanied by the xy-plane z = 0. This plane can be deactivated by its check box.

Other predefined waves can by selected in the ComboBox.

By phase modulation of one or more of the variables the periodic waves are animated, so that they appear to be moving in space as a function of time. As you see in the formula field, for example, the periodic function sin(x-t) is creating the movement in the x direction. The program calculates functions in time in steps of ∆t = p*0.1 milliseconds. Slider p thus controls the speed of animation. p = 0 freezes the graph.

Play starts the animation, with time t starting at 0, as indicated in the t number field. With (x - t) rising linearly, the wave progresses in space. Sliders a, b, c define amplitude and orientation. Pause freezes the animation at any spatial position. Reset leads back to the initial conditions.

Scaling of all three axes has a range of ∓1. The xy-plane cuts the z-axis at the center of the z-arrow. The minimum and maximum position of the z-axis is marked by a red and a green point.

The orientation of the tripod in space can be changed by drawing with the mouse.

Other ways of visualization are described on the next page.

Predefined wave forms are selected in the ComboBox with a mouse click.

Parameters a,b,c can be varied by sliders while the animation is running. By editing the fomulas you can change the parts that are animated. You can input new formulas to create your own waves. Do not forget to press the ENTER key after a change!

Touching a wave with the mouse pointer lets its color filling disappear; the wire mesh of calculation will be pronouncedly visible.

Visualization alternatives

Rotation:  Mark any point within the tripod by the mouse pointer and draw (while the left mouse key stays pressed).

Shift:  Draw while the Strg key is pressed.

Zoom:  Draw while the Shift key is pressed

Show coordinates:  Mark a point on the surface while pressing the Alt key. When you draw, a cutting plane will pass through the surface. Depending on orientation, different cuts can be evaluated.

Camera- Inspector: Press the right mouse key. A context menu will appear. Choose Elements option/ drawing 3D panel/ Camera. The Camera Inspector will appear. It will stay visible until it is deactivated. It offers the following options:

Perspective:  Distant lines appear shorter than near ones.

No perspective:  No perspective distortion.

Planar xy or yz or yx : One looks onto the respective planes.

Other options: Degree and angle of perspective can be defined.

Optimizing  parameters. The spatial impression can be optimized by adjusting parameters. The optimum will be different for different projections.

The context menu also offers programs for producing a picture or a video.

Predefined functions

a*sin(b*x-t): Plane wave x

a*sin(b*y-t): Plane wave y

0.3*sin(6*pi*a*(b*x+c*y)/sqrt(b*b+c*c)-t): Plane wave direction adjustable

a*(sin(b*y-t)+sin(b*y-t)): Interference concurrent f1

a*(sin(b*y-t)+sin(-b*y-t)): Interference countercurrent f1

a*(sin(b*y-t)+sin(c*y-t)): Interference concurrent f1 + f2

a*(sin(b*y-t)+sin(-c*y-t)): Interference countercurrent f1 + f2

a*(sin(b*x-t)+sin(c*y-t)): Interference perpendicular f1 + f2

a*(sin(b*(y-(c-pi)*x)-t)+sin(b*(y+(c-pi)*x)-t)): Interference angle < 90 adjustable (c)

a*(sin(b*(y-(c-pi)*x)-t)+sin(b*(-y+(c-pi)*x)-t)): Interference angle > 90 adjustable (c)

a*sin(b*(x*x+y*y)-t): Radial wave outward

a*sin(b*(x*x+y*y)+t): Radial wave inward

a*(sin(b*(x^2+y^2)-t)+sin(b*(x^2+y^2)+t)): Standing radial wave

0.4*a*sin(b*(x^2+y^2)-t)/sqrt(0.1+x^2+y^2): Surface wave outward

0.2*a*sin(b*(x^2+y^2)-t)/(0.1+x^2+y^2): Space wave outward

E1: Test the different waves without change of parameters. Rotate the frames and train your 3D perception of these functions.

E2: Study the formulas and develop a sense for the relation between formula and wave.

E3: Orient the tripod to optimize the spatial impression of the animated wave.

E4: Vary parameters and study the influence on the appearance of the wave.

E5: Change signs (+/-) in the formula and study the effect.

E6: Introduce power of periodics in the formula. What happens?

E7: Edit the formula arbitrarily and consider in advance how that should influence the wave.

E8: Superimpose waves of different speed (e.g. one with cox(bx-pt), the other one with cos(bx-cpt). Do that for different frequencies of the waves.

E9: Delete the animation term and use p as a free fourth parameter in your own formula.

E10: Consider which forms of waves you have observed at the beach, and try to reduplicate some as a formula.

This file was created by Dieter Roess in Aug. 2008

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

 

Translations

Code	Language		Translator	Run

Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee; Félix J. García Clemente

Briefing Document: Plotter z = f(x,y,t) 3D Waves JavaScript Simulation Applet

1. Overview:

This document describes a JavaScript simulation applet designed for visualizing 3D waves represented by the function z = f(x, y, t). It's an open educational resource intended for learning and teaching mathematics, particularly in the context of wave phenomena. The simulation allows users to explore various predefined wave functions, manipulate parameters, and observe the resulting 3D wave behavior in real-time.

2. Main Themes and Ideas:

• Interactive 3D Visualization: The core concept is providing a tool to visually represent complex mathematical functions as 3D waves. This allows users to develop a better intuitive understanding of wave behavior that is difficult to achieve through equations alone. "This function plotter displays functions of typical 3D waves z = f(x , y) that may contain up to 4 continuously variable parameters a, b, c, p."
• Parameter Manipulation: A key feature is the ability to control parameters a, b, c, and p, enabling users to see how changes in amplitude, frequency, direction, and speed affect the wave. "Sliders a, b, c define amplitude and orientation. Pause freezes the animation at any spatial position."
• Real-time Animation: The simulation animates the waves, showing their movement as a function of time. This brings the concept of wave propagation to life. "By phase modulation of one or more of the variables the periodic waves are animated, so that they appear to be moving in space as a function of time. As you see in the formula field, for example, the periodic function sin(x-t) is creating the movement in the x direction."
• User-Defined Functions: The applet allows users to input their own wave equations, fostering experimentation and deeper understanding of the relationship between mathematical formulas and visual representations. "By editing the formulas you can change the parts that are animated. You can input new formulas to create your own waves. Do not forget to press the ENTER key after a change!"
• Multiple Visualization Options: Beyond basic animation, the simulation offers tools for rotation, shifting, zooming, and creating cutting planes to analyze the 3D wave from different perspectives. "The orientation of the tripod in space can be changed by drawing with the mouse."
• Educational Experiments: The resource suggests a series of experiments designed to guide users through exploration of the applet's features and wave properties. "E1: Test the different waves without change of parameters. Rotate the frames and train your 3D perception of these functions."
• Open Educational Resource: The resource is freely available and embeddable, promoting accessibility and collaboration in education. "Learning and Teaching Mathematics using Simulations"

3. Key Features and Functionality:

• Predefined Wave Functions: The simulation provides a range of predefined wave functions (e.g., plane waves, radial waves, interference patterns) as starting points for exploration. "Predefined wave forms are selected in the ComboBox with a mouse click."
• Parameter Controls: Sliders control the a, b, c, and p parameters, allowing real-time adjustment and observation of the effects.
• xy- plane toggling: The xy- plane can be deactivated by its check box.
• Animation Controls: "Play," "Pause," and "Reset" buttons provide control over the animation.
• Formula Input: A field for entering custom wave equations.
• Visualization Tools: Mouse-based controls for rotation, shifting, and zooming. Context menu options for perspective and other camera settings.

4. Intended Audience:

• Mathematics students.
• Physics students.
• Educators teaching wave phenomena.

5. Learning Goals (from the perspective of teachers):

• Visualize 3D sinusoidal functions.
• Understand the effects of amplitude, frequency, and direction on wave behavior.
• Explore the relationship between mathematical equations and their graphical representations.
• Experiment with user-defined wave functions.

6. Important Notes:

• The simulation requires a web browser that supports HTML5 and JavaScript.
• The applet is part of a larger collection of simulations and resources for mathematics and physics education.
• The simulation is originally based on Easy Java Simulations (EJS), but has been converted to JavaScript/HTML5 for broader accessibility.

3D Wave Simulation Study Guide

Quiz

Answer the following questions in 2-3 sentences each.

1. What is the primary function of the simulation?
2. How do the sliders a, b, c, and p affect the wave display?
3. Explain how the user can change the orientation of the tripod in the simulation.
4. Describe the purpose of the "Reset" button.
5. How does the simulation animate the waves?
6. What does the color change (disappearance of color filling) when touching the wave with the mouse pointer signify?
7. List three predefined functions available in the simulation.
8. How can the "Camera Inspector" be accessed, and what perspective options does it provide?
9. Explain the purpose of the xy-plane in the simulation.
10. Describe how you can input your own wave formula into the simulation.

Quiz Answer Key

1. The simulation displays functions of typical 3D waves, z = f(x, y), allowing users to visualize and manipulate wave parameters. It provides a graphical representation of waves, making it easier to understand their properties and behavior.
2. Sliders a, b, c define the amplitude and orientation of the wave, while slider p controls the speed of the animation. Changing these parameters allows users to observe how they influence the wave's shape and movement.
3. The orientation of the tripod in space can be changed by clicking and dragging the mouse within the tripod area. This allows users to view the wave from different angles and perspectives.
4. The "Reset" button returns the simulation to its initial conditions, resetting the parameters and the time to their starting values. It is useful for starting over after experimenting with different settings.
5. The simulation animates the waves by phase modulation of the variables in the wave function, such as sin(x-t), creating the illusion of movement in space as a function of time. The animation's speed is controlled by the p slider.
6. The color change signifies that the wire mesh calculation of the wave is becoming visible and pronounced. This allows the user to see the underlying structure and sampling of the wave function.
7. Three predefined functions are asin(bx-t), asin(by-t), and 0.3sin(6pia(bx+cy)/sqrt(bb+cc)-t). These represent a plane wave in the x-direction, a plane wave in the y-direction, and an adjustable plane wave direction, respectively.
8. The "Camera Inspector" can be accessed by pressing the right mouse key and selecting "Elements option/ drawing 3D panel/ Camera" from the context menu. It offers perspective options such as perspective, no perspective, and planar views (xy, yz, yx).
9. The xy-plane in the simulation serves as a reference plane at z = 0, helping users visualize the positive and negative parts of the wave relative to a flat surface. It can also be deactivated using its checkbox if desired.
10. By editing the formulas in the formula field, you can input your own wave formulas. After changing the formula, you must press the "ENTER" key for the simulation to recognize and display the new wave function.

Essay Questions

1. Discuss the educational value of the 3D wave simulation. How can it be used to enhance learning and understanding of wave phenomena?
2. Explain how the parameters a, b, c, and p influence the characteristics of the waves displayed in the simulation. Provide specific examples of how changing these parameters affects the wave's appearance and behavior.
3. Describe the different visualization alternatives provided by the simulation, such as rotation, shift, zoom, and show coordinates. How do these options contribute to a better understanding of the 3D wave functions?
4. Choose three of the predefined wave functions and compare and contrast their mathematical forms and resulting visual representations. How do the different formulas create different wave patterns?
5. Design an experiment using the simulation to investigate the superposition principle. Describe the steps you would take and the observations you would expect to make.

Glossary of Key Terms

• Amplitude (a): The maximum displacement of a wave from its equilibrium position. In the simulation, the 'a' parameter controls the amplitude of the wave.
• Frequency (b, c): A measure of how many cycles of a wave occur in a given unit of time or space. Parameters 'b' and 'c' influence the frequency and orientation of the wave.
• Phase Modulation: A technique used to animate the waves by changing the phase of the wave function over time, creating the illusion of movement.
• Parameter (p): A variable that can be adjusted to control the behavior and appearance of the wave. In the simulation, parameter 'p' primarily adjusts the speed of the animation.
• Perspective Distortion: The effect of making distant lines appear shorter than near ones, creating a sense of depth. The Camera Inspector allows toggling of perspective.
• Tripod: The XYZ coordinate axes displayed in the simulation, used as a visual reference for the wave's position and orientation in 3D space.
• Superposition: The principle that when two or more waves overlap in space, the resulting wave is the sum of the individual waves. The simulation includes several interference wave examples illustrating superposition.
• Interference: The phenomenon that occurs when two or more waves overlap, resulting in either constructive (increased amplitude) or destructive (decreased amplitude) effects.
• Wave Function: A mathematical function that describes the shape of a wave at a given point in space and time. In the simulation, the formula field allows users to input and modify wave functions.
• HTML5 Applet: A self-contained program written using HTML5, JavaScript, and CSS, designed to run within a web browser without requiring additional plugins.

Sample Learning Goals

[text]

For Teachers

This simulation allows for graphical visualisation of a 3D sinusoidal function as a wave.

There are 15 predefined functions that can be inputted from the drop-down menu.

You can also adjust different parameters of the function to see how they affect the wave:

a: Amplitude

b: Frequency

c: Direction (only applicable to functions which allow for change in direction)

p: Simply adjusts the speed of the simulation.

In addition, the x-y plane can be toggled to more easily visualise the positive and negative parts of the wave.

 

Research

[text]

Video

[text]

 Version:

1. http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
2. http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

Other Resources

 

 

1. Linear Wavefront https://www.geogebra.org/m/kmzaends by Tan Seng Kwang
2. Circular Wavefront https://www.geogebra.org/m/twtnrtuh by Tan Seng Kwang

FAQ: 3D Wave Simulation Applet

• What is the purpose of the 3D Waves JavaScript Simulation Applet?
• The applet is a function plotter designed to visualize 3D wave functions, represented as z = f(x, y), which can include parameters a, b, c, and p. It allows users to observe and interact with these wave functions, exploring how different parameters influence their shape and movement. The primary goal is to aid in the learning and teaching of mathematics and physics concepts related to wave phenomena.
• How does the 'p' parameter affect the wave animation?
• The 'p' parameter controls the speed of animation of the waves. The program calculates the functions in time steps of ∆t = p*0.1 milliseconds. Therefore, increasing the value of 'p' will increase the animation speed, while setting 'p = 0' will freeze the graph.
• How can I interact with the 3D wave visualization?
• You can interact with the visualization in several ways:
• Rotation: Click and drag within the tripod to rotate the entire 3D scene.
• Shift: Hold the 'Strg' (Ctrl) key while clicking and dragging to shift the viewpoint.
• Zoom: Hold the 'Shift' key while clicking and dragging to zoom in or out.
• Show coordinates: Hold the 'Alt' key while clicking on the surface of the wave. A cutting plane will pass through the surface. Depending on orientation, different cuts can be evaluated.
• Camera Inspector: Right-click to access a context menu and choose "Elements option/ drawing 3D panel/ Camera" to open the Camera Inspector, allowing adjustments to perspective and viewing angles.
• What predefined wave functions are available in the applet?
• The applet provides several predefined wave functions, including:
• Plane waves in the x and y directions.
• Adjustable plane waves.
• Various interference patterns (concurrent, countercurrent, perpendicular, and adjustable angle).
• Radial waves (outward, inward, and standing).
• Surface and space waves.
• How can I modify the wave functions?
• You can modify the wave functions by editing the formulas directly in the formula field. After making changes, you must press the 'ENTER' key to apply them. You can adjust parameters a, b, c, and p using the sliders while the animation is running to observe their effects on the wave. You can also delete the animation term and use p as a free fourth parameter in the formula.
• What experiments can be performed using this simulation?
• The applet suggests several experiments to enhance understanding of wave behavior:
• Observing different wave types without changing parameters.
• Studying the relationship between formulas and wave shapes.
• Optimizing the tripod orientation for the best 3D impression.
• Varying parameters to see their impact on the wave's appearance.
• Changing signs in formulas to observe the effects.
• Introducing powers of periodics in the formula.
• Creating custom wave functions by editing the formula.
• Superimposing waves of different speeds and frequencies.
• Using p as a free fourth parameter in your own formula.
• Trying to reduplicate real-world waves you have observed.
• What does it mean when the color filling disappears when I touch a wave with the mouse pointer?
• Touching a wave with the mouse pointer makes its color filling disappear, causing the wire mesh of the calculation to be more pronouncedly visible. This allows you to see the underlying structure of the wave as it is being calculated.
• Where can I find more resources or related simulations?
• The documentation lists "Other Resources" such as:
• Linear Wavefront using GeoGebra by Tan Seng Kwang.
• Circular Wavefront using GeoGebra by Tan Seng Kwang. The documentation also lists related simulation applets including:
• Fourier Transform Wave Generator JavaScript Simulation Applet HTML5
• Free Particle Wavepacket JavaScript Simulation Applet HTML5