### 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

*will appear. It will stay visible until it is deactivated. It offers the following options:*

**Camera Inspector**

**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

__
__

__From function_3d, made to work by changing all cos to sin, sqrt to
pow( , 0.5) etc due to a programming fixed realtionship to replace all
a,b,c,t__

**cos(t)*(b*x+a*y)-c**: Plane in space

**a*cos(t)*(x^2+y^2)-c**: Rotational paraboloid

**cos(t)*((b*x)^2+(a*y)^2)-c**: General paraboloid

**cos(t)*((b*x)^2-(a*y)^2)-c**: Parabolic saddle

**4*sqrt((a^2)*abs(cos(t))-0.04*(x^2)-0.04*(y^2))**: Sphere

**sqrt((c^2)-((c/a)*x*cos(t))^2 - ((c/a)*y*cos(t))^2)**: Rotational
ellipsoid

**sqrt((c^2)-((c/a)*x)^2 - ((c/b)*y*cos(t))^2)**: General ellipsoid

**sqrt(a*(cos(t))^2+x^2+y^2)-c**: Rotational hyperboloid

**sqrt((a^2)+b*(x^2)+c*(y^2))-p**: General hyperboloid

**sqrt((a^2)-cos(t)*(b*(x^2)-c*(y^2))**: Elliptic-hyperbolic saddle

**cos(t)*x*y**: Hyperbolic saddle

**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

### For Teachers

### Translations

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support fullscreen? | Yes. Chrome/Opera No. Firefox/ Sumsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee

### end faq

### Sample Learning Goals

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### 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

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### Video

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### Version:

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

### Other Resources

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