lissajous pattern experiment

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Background: lissajous figures, create a lissajous pattern, subtracting two signals, activity: the lissajous pattern, a classic phase measurement.

All periodic signals can be described in terms of amplitude and phase. To perform amplitude, frequency, and phase measurements using an oscilloscope and to make use of Lissajous figures for phase and frequency measurements.

We all learn that in basic circuit theory classes. You surely have needed to calculate a signal's phase change when it passes through a circuit. Fortunately, you can measure phase on the lab bench with oscilloscope hardware such as the ADALM1000 and its accompanying ALICE desktop software using several methods.

As in all the ALM labs we use the following terminology when referring to the connections to the M1000 connector and configuring the hardware. The green shaded rectangles indicate connections to the M1000 analog I/O connector. The analog I/O channel pins are referred to as CA and CB. When configured to force voltage / measure current –V is added as in CA- V or when configured to force current / measure voltage –I is added as in CA-I. When a channel is configured in the high impedance mode to only measure voltage –H is added as CA-H. Scope traces are similarly referred to by channel and voltage / current. Such as CA- V , CB- V for the voltage waveforms and CA-I , CB-I for the current waveforms.

Lissajous (pronounced LEE-suh-zhoo) figures were discovered by the French physicist Jules Antoine Lissajous. He would use sounds of different frequencies to vibrate a mirror. A beam of light reflected from the mirror would trace patterns which depended on the frequencies of the sounds. Lissajous' setup was similar to the apparatus which is used today to project laser light shows.

Before the days of digital frequency meters and phase-locked loops, Lissajous figures were used to determine the frequencies of sounds or radio signals. A signal of known frequency was applied to the horizontal axis of an oscilloscope, and the signal to be measured was applied to the vertical axis. The resulting pattern was a function of the ratio of the two frequencies.

Lissajous figures often appeared as props in science fiction movies made during the 1950's. One of the best examples can be found in the opening sequence of The Outer Limits TV series. (“Do not attempt to adjust your picture--we are controlling the transmission.”) The pattern of criss-cross lines is actually a Lissajous figure.

The phase difference, or phase angle, is the difference in phase between the same points, say a zero crossing, in two different waveforms with the same frequency. A common example is the phase difference between the input signal and output signal after it passes through a circuit, cable, or PC board trace. A waveform with a leading phase has a specific point occurring earlier in time than the same point on the other signal. That would be the case of when a signal passes through, a capacitor: the current in the capacitor will lead the voltage across the capacitor by 90º. Conversely, a waveform with lagging phase has a specific point occurring later in time than the other paired signal waveform. Two signals are in opposition if they are 180º out of phase. Signals that differ in phase by ±90º are in phase quadrature (one quarter of 360º).

The time (phase) relationship between two sine waves can of course be measured from a time domain plot such as figure 1. The Time measurement capabilities of most any Oscilloscope, the ADALM1000 and the ALICE software included, can display the relative phase between channel A and channel B in degrees and/or the time delay between A and B. The software scans the waveforms looking for the time points where they cross their average value (zero crossing with DC offset removed). It then uses those time points to report frequency, period, phase, delay, duty-cycle etc. Noise and jitter will introduce errors in the results.

Figure 1, Time plot of two sine waves.

Old timers who have started out their careers using an analog oscilloscope probably remember using the classic Lissajous pattern to measure the phase difference of two sine waves. It can be measured by cross plotting the two sine waveforms on the X-Y display in ALICE as shown in Figure 2. In this figure, the voltage waveform on channel A provides the horizontal or X displacement. Channel B provides the vertical or Y deflection. The Lissajous pattern indicates the phase difference by the shape of the X-Y plot. A straight line indicates a 0º or 180º phase difference. The angle of the line depends on the difference in amplitude between the two signals, a line at 45º to the horizontal means the amplitudes are equal. While a circle indicates a 90º difference. It will only be a true circle if the amplitudes are equal. Phase differences between 0º and 90º appear as tilted ellipses and phase is determined by measuring the maximum vertical deflection (Ymax) and the vertical deflection at zero horizontal deflection (Yx=0). In Figure 2, cursors mark these two locations on the X-Y plot. Note that this is only valid if the X-Y plot is centered on 0,0. Any DC offset in the two waveforms must be removed first.

Figure 2 Using a classic Lissajous display lets you measure the phase difference between two sine waves.

Marker readouts in upper left of the plot show the required values for computing the phase difference.

Φ2 - Φ1 = ± sin−1 (Yx=0/Ymax) for when the top of the ellipse is located in quadrant 1 (Q1)

Φ2 - Φ1 = ± 180-sin−1 (Yx=0/Ymax) for when the top of the ellipse is located in quadrant 2 (Q2)

The sign of the phase difference is determined by inspecting the channel time traces.

In the figure 2 example, the Ymax value is 1.538, YX=0 is 1.064, and the top of the ellipse is in Q1: Φ2 - Φ1 = ± sin−1 (1.064/1.538) = ± sin−1 (0.692) = 44º

The accuracy of this method is dependent on the placement of the cursors but it produces reasonable results with certain artistic panache.

Hardware like the ADALM1000 and ALICE desktop software offer multiple techniques to measure phase. Direct measurement in the time domain supports both static and dynamic measurements of phase. Frequency domain based calculation provides somewhat more accurate results for static phase measurements but requires you to take the difference of the FFT phase data at the fundamental frequency.

Materials: ADALM1000 hardware module

The default function of an oscilloscope is to display voltage signals on the Y axis vs time on the X axis. The ALICE software has a special function that allows the user to plot one signal on the X axis and the other on the Y axis.

Begin with both AWG generators set for 1.0 Min and 4.0 Max values and a frequency of 1 kHz . Use the Time display to ensure that both waveform generators are producing the same signal.

You should see two nearly identical sine waves, both in phase with the other. Remember that the sine wave is defined by three parameters – amplitude, frequency and phase. Check to see that the vertical range and position settings for both channels are the same. (Use the V /Div to adjust this if needed.) If the amplitudes are not the same, the sine waves will not be the same amplitude. If the phases are not the same, the sine waves will not line up horizontally. If the frequencies are not the same, the wave the oscilloscope is triggered on will be stationary, while the other wave will move (slowly we hope) to the left or right. Figure 3 shows signals that have different amplitude and frequency (and DC offset).

Figure 3, Two sine waves that vary in amplitude and frequency.

We can now generate a Lissajous pattern by opening the X-Y Plotter tool. Select CA- V for the X Axis and CB- V for the Y Axis. Compare what you see to the examples of Lissajous figures. Use your favorite screen capture software tool to take a picture of the image and save it. Your TA or instructor can help you with this. Include this picture with your report. [Hint: Use the STOP button to freeze your figure at the point you want to take a picture.]

You should now play with the AWG settings and produce several other representative patterns. Create one pattern that you find particularly interesting. Take a picture of it with the screen capture software.

Next, we will do a different kind of comparison of the two sine waves, one that will prove to be very important in the development of measurement techniques. In this measurement, we will compare two signals to see how close they are to one another by subtracting one from the other.

Go back to the Time display. This should the two sine waves plotted vs time. Adjust them again so that they are as identical as possible. (Try to get them displayed on top of one another by using the Min, Max, Freq and Phase controls in the AWG generator controls.)

Click on the Math Button to open the Math function controls. Click on the Built In Expression list and select CAV-CBV, which will produce a third trace that is the difference between the two channels. You will need to increase the V /Div to 1.0 on both channels to see the difference signal. The two AWG channel outputs have DC offset which should also sum to zero if they are the same. Also adjust the vertical position such that the traces do not go off the grid.

If you have adjusted the two sine waves to be identical, their difference should be zero. How well did you do? Note that the amplitude, the frequency, and the phase must be identical to make the difference zero.

Now make the two signals as identical as possible by adjusting the difference signal away. What did you have to do?

From this activity, you should see the value of comparing one signal against some kind of known reference signal. It is possible to tune a guitar, for example, by comparing the tone a string makes with an electronic reference tone. This results in a perfectly tuned instrument, even when the player has less than perfect pitch. You have also seen how we make what are called differential measurements. There are many advantages to making differential over absolute measurements. You have seen one of the key reasons since differential measurements allow you to focus on smaller quantities since you are working with the difference between two signals.

For Further Reading:

Lissajou Curves Lissajou Curves

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Instructional Resources and Lecture Demonstrations

3a80.20 - lissajous figures - oscilloscope.

Plug one wave generator into each input of a dual trace scope. Set the oscilloscope to X - Y plotting. If each generator is at the same frequency the figure will be in the form of a circle. More complicated patterns will be obtained with higher harmonic frequencies. The Wavetek generators work much better for this because they are more stable and they also have a 'stabilization mode' so that you can lock onto a desired frequency.

• Tiandong Li, Ruotong Zhu, Huilin Jin, Hongchun Yang, Minghe Wu, Baohua Teng, "Further Understanding for Lissajous Figures", TPT, Vol. 59, #1, Jan. 2021, p. 62.
• C. Criado and N. Alamo, "A Simple Construction to Illustrate Lissajous Figures", TPT, Vol. 42, #4, April 2004, p. 248.
• E.Y.C. Tong, "Lissajous Figures", TPT, Vol.  35, # 8, p. 491- 493, Nov. 1997.
• Thomas B. Greenslade, "All About Lissajous Figures", TPT, Vol. 31, # 6, Sept. 1993, p. 364.
• Fred B. Otto, "Three-Dimensional Lissajous Figures", TPT, Vol. 29, # 4, Apr. 1991, p. 197.
• Arthur Quinton, "The Generator of Lissajous Figures", TPT, Vol. 16, # 3, Mar. 1978, p. 178.
• Frank G. Karioris, "Projection Sine-Sine Grid and Lissajous Figures", TPT, Vol. 13, # 5, May 1975, p. 294.
• Thomas B. Greenslade, Jr., "Lissajous Figure Demonstrator (Photo)", AJP, Vol. 73, # 9, Sept. 2005, p. 892.
• P. Jasselette and J. Vandermeulen,  "More on Lissajous Figures",  AJP, 52, # 2, Feb. 1986.
• Mu-Shiang Wu, W. H. Tsai,  "Corrections for Lissajous Figures in Books",  AJP, 52, # 7, July 1984, p. 657.
• Sn- 2, 3:  Freier and Anderson,  A Demonstration Handbook for Physics.
• M-930:  "Lissajous Patterns on Scope",  DICK and RAE Physics Demo Notebook.
• Yaakov Kraftmakher, "1.5, Lissajous Patterns", Experiments and Demonstrations in Physics, ISBN 981-256-602-3, p. 20.
• 1.25:  Charles Taylor,  The Art and Science of Lecture Demonstration, p. 50- 52.
• Gerard L'E Turner, Nineteenth-Century Scientific Instruments, p. 145-146.
• The Project Physics Course - Teachers Resource Book,  "Concepts of Motion",  "Frequency Measurements,"  p. 120.

Disclaimer: These demonstrations are provided only for illustrative use by persons affiliated with The University of Iowa and only under the direction of a trained instructor or physicist.  The University of Iowa is not responsible for demonstrations performed by those using their own equipment or who choose to use this reference material for their own purpose.  The demonstrations included here are within the public domain and can be found in materials contained in libraries, bookstores, and through electronic sources.  Performing all or any portion of any of these demonstrations, with or without revisions not depicted here entails inherent risks.  These risks include, without limitation, bodily injury (and possibly death), including risks to health that may be temporary or permanent and that may exacerbate a pre-existing medical condition; and property loss or damage.  Anyone performing any part of these demonstrations, even with revisions, knowingly and voluntarily assumes all risks associated with them.

Lissajous Patterns of CRO or Cathode Ray Oscilloscope

When the horizontal and vertical deflection plates of a CRO ( Cathode Ray Oscilloscope ) are connected to two sinusoidal voltages, the patterns that appear on the screen are called Lissajous pattern . Shape of these Lissajous pattern changes with changes of phase difference between signal and ration of frequencies applied to the deflection plates (traces) of CRO . Which makes these Lissajous patterns very useful to analysis the signals applied to deflection plated of CRO. These lissajous patterns have two Applications to analysis the signals. To calculate the phase difference between two sinusoidal signals having same frequency. To determine the ratio frequencies of sinusoidal signals applied to the vertical and horizontal deflecting plates.

Calculation of the phase difference between two Sinusoidal Signals having same frequency

SL No.	Phase angle difference ‘ø’	Lissajous Pattern appeared at CRO Screen
1	0 & 360	or 330	or 315	or 300	or 270	or 240	or 210		There are two cases to determine the phase difference ø between two signals applied to the horizontal and vertical plates, Examples Lissajous Pattern Sl No. Lissajous Pattern Ratio of Frequencies f /f 1 /f = 4/2 = 2 2 /f = 3/1 = 3 3 /f = 6/4 = 3/2 4 /f = 6/8 = 3/4 5 /f = 4/3 Leave a Comment Cancel reply History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture Games & Quizzes On This Day One Good Fact New Articles Lifestyles & Social Issues Philosophy & Religion Politics, Law & Government World History Health & Medicine Browse Biographies Birds, Reptiles & Other Vertebrates Bugs, Mollusks & Other Invertebrates Environment Fossils & Geologic Time Entertainment & Pop Culture Sports & Recreation Visual Arts Demystified Image Galleries Infographics Top Questions Britannica Kids Saving Earth Space Next 50 Student Center Lissajous figure Our editors will review what you’ve submitted and determine whether to revise the article. Maths.com - Lissajous Figures Lissajous figure , also called Bowditch Curve , pattern produced by the intersection of two sinusoidal curves the axes of which are at right angles to each other. First studied by the American mathematician Nathaniel Bowditch in 1815, the curves were investigated independently by the French mathematician Jules-Antoine Lissajous in 1857–58. Lissajous used a narrow stream of sand pouring from the base of a compound pendulum to produce the curves. If the frequency and phase angle of the two curves are identical, the resultant is a straight line lying at 45° (and 225°) to the coordinate axes. If one of the curves is 180° out of phase with respect to the other, another straight line is produced lying 90° away from the line produced where the curves are in phase ( i.e., at 135° and 315°). Otherwise, with identical amplitude and frequency but a varying phase relation, ellipses are formed with varying angular positions, except that a phase difference of 90° (or 270°) produces a circle around the origin. If the curves are out of phase and differing in frequency, intricate meshing figures are formed. Of particular value in electronics, the curves can be made to appear on an oscilloscope, the shape of the curve serving to identify the characteristics of an unknown electric signal. For this purpose, one of the two curves is a signal of known characteristics. In general, the curves can be used to analyze the properties of any pair of simple harmonic motions that are at right angles to each other. Experiments Lissajous curves Create magnificent patterns using only light! Carefully review the general safety advice on the back of the box cover before starting the experiment. Be very careful in a dark room – turn the lights off only when necessary; be sure to prepare everything you’ll need in the darkness in advance; clear the path from the light switch to the table or, even better, ask someone else to turn the lights off. Never eat or drink any of the substances provided. Do not use for culinary purposes. Disassemble the setup after the experiment. Step-by-step instructions Flip the sheet of phosphorescent paper to shield the glossy side from light. Putting your UV light on a string will let it swing freely as it draws with light! The clamp divides the string into two parts that can move independently of each other. The time it takes for the UV light to swing back and forth will depend on the string length from the UV light to the clamp. Try different lengths and see! Adjust the length of the string so the UV light is not touching the table. The glow of the phosphorescent paper shows how the UV light moves. You can observe special figures called Lissajous curves. Adjusting the clamps will create different figures. Which do you find the most interesting? Pour liquids down the sink. Wash with an excess of water. Dispose of solid waste together with household garbage. Scientific description Dozens of experiments you can do at home Kids are now able to engage with science in a way that they simply wouldn’t have been able to in the past as they shrink themselves down to see the world at a molecular level ToyNews Access denied under U.S. Export Administration Regulations. Electronic Measuring Instruments Introduction Performance Characteristics Measurement Errors Measuring Instruments DC Voltmeters AC Voltmeters Other AC Voltmeters DC Ammeters Signal Generators Wave Analyzers Spectrum Analyzers Basics of Oscilloscopes Special Purpose Oscilloscopes Lissajous Figures Other AC Bridges Transducers Active Transducers Passive Transducers Measurement Of Displacement Data Acquisition Systems Useful Resources Quick Guide Selected Reading UPSC IAS Exams Notes Developer's Best Practices Questions and Answers Effective Resume Writing HR Interview Questions Computer Glossary Lissajous figure is the pattern which is displayed on the screen, when sinusoidal signals are applied to both horizontal & vertical deflection plates of CRO. These patterns will vary based on the amplitudes, frequencies and phase differences of the sinusoidal signals, which are applied to both horizontal & vertical deflection plates of CRO. The following figure shows an example of Lissajous figure. The above Lissajous figure is in elliptical shape and its major axis has some inclination angle with positive x-axis. Measurements using Lissajous Figures We can do the following two measurements from a Lissajous figure. Frequency of the sinusoidal signal Phase difference between two sinusoidal signals Now, let us discuss about these two measurements one by one. Measurement of Frequency Lissajous figure will be displayed on the screen, when the sinusoidal signals are applied to both horizontal & vertical deflection plates of CRO. Hence, apply the sinusoidal signal, which has standard known frequency to the horizontal deflection plates of CRO. Similarly, apply the sinusoidal signal, whose frequency is unknown to the vertical deflection plates of CRO Let, $f_{H}$ and $f_{V}$ are the frequencies of sinusoidal signals, which are applied to the horizontal & vertical deflection plates of CRO respectively. The relationship between $f_{H}$ and $f_{V}$ can be mathematically represented as below. $$\frac{f_{V}}{f_{H}}=\frac{n_{H}}{n_{V}}$$ From above relation, we will get the frequency of sinusoidal signal, which is applied to the vertical deflection plates of CRO as $f_{V}=\left ( \frac{n_{H}}{n_{V}} \right )f_{H}$ (Equation 1) $n_{H}$ is the number of horizontal tangencies $n_{V}$ is the number of vertical tangencies We can find the values of $n_{H}$ and $n_{V}$ from Lissajous figure. So, by substituting the values of $n_{H}$, $n_{V}$ and $f_{H}$ in Equation 1, we will get the value of $f_{V}$ , i.e. the frequency of sinusoidal signal that is applied to the vertical deflection plates of CRO. Measurement of Phase Difference A Lissajous figure is displayed on the screen when sinusoidal signals are applied to both horizontal & vertical deflection plates of CRO. Hence, apply the sinusoidal signals, which have same amplitude and frequency to both horizontal and vertical deflection plates of CRO. For few Lissajous figures based on their shape, we can directly tell the phase difference between the two sinusoidal signals. If the Lissajous figure is a straight line with an inclination of $45^{\circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be $0^{\circ}$. That means, there is no phase difference between those two sinusoidal signals. If the Lissajous figure is a straight line with an inclination of $135^{\circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be $180^{\circ}$. That means, those two sinusoidal signals are out of phase. If the Lissajous figure is in circular shape , then the phase difference between the two sinusoidal signals will be $90^{\circ}$ or $270^{\circ}$. We can calculate the phase difference between the two sinusoidal signals by using formulae, when the Lissajous figures are of elliptical shape . If the major axis of an elliptical shape Lissajous figure having an inclination angle lies between $0^{\circ}$ and $90^{\circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be. $$\phi =\sin ^{-1}\left ( \frac{x_{1}}{x_{2}} \right )=\sin ^{-1}\left ( \frac{y_{1}}{y_{2}} \right )$$ If the major axis of an elliptical shape Lissajous figure having an inclination angle lies between $90^{\circ}$ and $180^{\circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be. $$\phi =180 - \sin ^{-1}\left ( \frac{x_{1}}{x_{2}} \right )=180 - \sin ^{-1}\left ( \frac{y_{1}}{y_{2}} \right )$$ $x_{1}$ is the distance from the origin to the point on x-axis, where the elliptical shape Lissajous figure intersects $x_{2}$ is the distance from the origin to the vertical tangent of elliptical shape Lissajous figure $y_{1}$ is the distance from the origin to the point on y-axis, where the elliptical shape Lissajous figure intersects $y_{2}$ is the distance from the origin to the horizontal tangent of elliptical shape Lissajous figure In this chapter, welearnt how to find the frequency of unknown sinusoidal signal and the phase difference between two sinusoidal signals from Lissajous figures by using formulae. Return to PocketLab Home Page PocketLab/Phyphox Damped Lissajous Figures Lissajous Introduction Lissajous patterns have fascinated physics students for decades.  They are commonly observed on oscilloscopes by applying simple harmonic functions with different frequencies to the vertical and horizontal inputs.  Three examples are shown in Figure 1.  From left to right, the frequency ratios are 1:2, 2:3, and 3:4.  These Lissajous patterns were created by use of the parametric equation section of  The Grapher software written by the author of this lesson.  You are welcome to use this software for graphing prepared or custom functions, polar equations, and parametric equations.  The graphs of Figure 1 can only be obtained in driven oscillations, meaning that there is no damping of the oscillations as time goes on.  The parametric equations for Lissajous patterns of this type take on the form: x = A sin(at + delta) y = B sin(bt). In these equations, t is time,  A and B are the amplitudes of the oscillations, a and b are the frequencies, and delta is the phase difference between them.  In many real-world situations, however, the amplitudes decrease with time due to a variety of non-conservative forces such as friction.  For example, the periodic motion of physical pendulums experiences damping with the passage of time.  The above pair of parametric equations can easily be modified to include damping.  If we assume that the damping is negative exponential, then the equations take on the form: x = A sin(at + delta) exp(-dt) y = B sin(bt) exp(-dt). In these parametric equations, the exponential damping factor d has been included, with this factor assumed the same for both the x and y oscillations.  This lesson will involve the study of damped Lissajous patterns, a subject that is generally not discussed much in physics classes.  Never-the-less, such a study is quite interesting and well worth the time spent. The Damped Lissajous Experiment Setup Figure 2 shows the setup for our damped Lissajous lab.  The damped oscillations are provided by a pair of pendulums, each approximately two meters in length.  In fact, the pendulums can be a couple of inches different in length, as we want the periods to be slightly different .  It is best to have pendulums that are at least two meters in length, as this provides for longer periods that plot better with the PocketLab/Phyphox combination.  The pendulums are constructed from four meter sticks and are suspended from the ceiling in such a way that their planes of oscillation are perpendicular to one another.  A piece of  6" x 8"  card stock is taped to the bottom of each pendulum.  The purpose of the card stock is to provide a white surface from which the Voyager IR rangefinders can bounce their beams.  A pair of perpendicular Voyagers are arranged with their rangefinders facing the card stock on the pendulums.  The Voyagers are about 18" from the pendulums when the pendulums are at rest.   The Voyagers are connected via BLE (Bluetooth Low Energy) to a device running Phyphox software.  With the Lissajous Activity experiment running in Phyphox, the pendulums are released from rest about 4" from their respective Voyagers.  Phyphox records and displays the resultant Lissajous figure while the pendulums swing back and forth with damping. The 0.5 minute video below shows a typical data collection run of this experiment. Results and Comparison to Theory Figure 3 shows a typical Lissajous pattern plotted in Phyphox for this experiment.  Note the concentration of points near the "center" of the pattern.  Does this result agree with that predicted from theory? Let's consider the following set of parametric equations: x = sin(1.07t + π /2) exp(-0.03t);   y = sin(t) exp(-0.03t). A phase difference of π  is used.  Note that changing the phase difference can result in significant differences in the resultant pattern.  This equation implies a 7% difference between the x and y frequencies.  This was found by measuring the x and y periods with a stopwatch.  The damping factor was set to 0.03.  When this equation is graphed with the author's The Grapher software, the resultant graph appears as shown in Figure 4.  There are clear similarities between our experimental results and theory, including the  concentration of data points near the center of the figure.  This concentration of points is obtained when the oscillations  have been damped significantly.  Here is exactly how you would key in these parametric equations using The Grapher software: x=sin(1.07*t+PI)*exp(-0.03*t);|y=sin(t)*exp(-0.03*t) Phyphox Software Phyphox ( phy sical  pho ne e x periments) is a free app developed at the 2nd Institute of Physics of the RWTH Aachen University in Germany.  The author of this lesson has been working with a pre-release Android version of this app. It supports BLE (Bluetooth Low Energy) technology to transfer data from multiple Voyagers to the Phyphox app.  Since then, a public Android version of the Phyphox app has been made available. The experiment of this lesson is in a file named Lissajous.phyphox  and accompanies this lesson.  This file can then be opened in Phyphox and will appear in the  PocketLab Voyager  category of the main screen, similar to that in Figure 5. Both Voyagers should initially be turned off.  The first screen that you will see after selecting the  Lissajous  experiment from the menu contains the message shown in Figure 6.  In response to this message you should turn on the Voyager labeled X.  "PL Voyager" will appear in the message.  Click on "PL Voyager" and a message will tell you that Bluetooth is connecting to the device.  Another message similar to that of Figure 6 will then appear, and your response to this message should be to turn on the Voyager labeled Y.  When both connections are complete, you can start data collection whenever you are ready with the pulsating triangle in the upper right corner of the screen.  If you want to export the collected data , all you need to do is click the ellipsis in the upper right corner of the screen and select  Export Data  from the drop-down menu.  You can then choose the desired data format (Excel, CSV) and pick a method for sharing the data (Google Drive, Email, etc.).  See Figure 7. To access this free lesson, please sign up to receive communications from us:

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H1-27: SPEED OF SOUND - LISSAJOUS FIGURES

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Additional Info

• ID Code: H1-27
• Purpose: Measurement of the speed of sound in air using Lissajous figures.

As the microphone is moved away from the loudspeaker the vertical signal falls 90 degrees behind in phase, causing the Lissajous figure to form an ellipse. When the two signals are out of phase (180 degrees phase difference) the pattern is a line along the opposite diagonal. As the microphone is withdrawn further, the microphone signal becomes 270 degrees behind in phase and the pattern again becomes an ellipse. One important difference between the two ellipses is that they are rotating in opposite directions, but this is not observable on the oscilloscope. Withdrawal of one full wavelength, when the signal from the microphone lags a full period (360 degrees) behind the original condition, creates a pattern similar to the original pattern. In this case the signal picked up by the microphone is reduced in amplitude due to the inverse square law, reducing the slope of the line.

For the most accurate measurement a frequency meter is connected to the trigger output of the oscillator. In the case shown below:

The photographs above show the Lissajous patterns at 90 degree intervals as the microphone is withdrawn.

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H1-01 BELL IN VACUUM

H1-02 SPEAKER AND CANDLE

H1-03: BELLS

H1-04: BELL IN VACUUM - PORTABLE

H1-11: MICROPHONE AND OSCILLOSCOPE

H1-12: VISIBLE WAVEFORMS ON LARGE SPEAKER

H1-13 WAVEFORM GENERATOR, SPEAKER AND OSCILLOSCOPE

H1-21: SPEED OF SOUND - PHASE CHANGE

H1-22: SPEED OF SOUND - USING PULSES

H1-24: SPEED OF SOUND IN HELIUM

H1-25: SPEED OF SOUND BETWEEN TWO MICROPHONES

H1-26: SPEED OF SOUND IN GARDEN HOSE

H1-31: SOUND LEVEL METER

H1-32: WAVETEK AND AUDIO CART - EQUAL SOUND LEVEL STEPS

H1-41: ULTRASONIC MOTION DETECTOR

H1-43: ULTRASONICS AND HEARING

H1-44: ULTRASONIC MOTION DETECTOR WAVE FORM

H1-51: AUDIOTAPE 42 MIN - SCIENCE OF SOUND - SHORT VERSION

H1-52: AUDIOTAPE 82 MIN - SCIENCE OF SOUND - LONG VERSION

H1-53: AUDIOTAPE 8 MIN - WOMB SOUNDS

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Lissajous Curves

An interactive demonstration of Lissajous curves.

Maths Geometry Polar plot parametric

A Lissajous curve, named after Jules Antoine Lissajous is a graph of the following two parametric equations:

\( A \) and \( B \) represent amplitudes in the \( x \) and \( y \) directions, \( a \) and \( b \) are constants, and \( \phi \) is an phase angle. The user interface above allows you to modify each of these five parameters and see how the graph is affected. The values of \( a \) and \( b \) have a particularly strong effect on the shape of the curve.

\( a \) determines the number of horizontally aligned "lobes" and \( b \) determines the number of vertically aligned lobes. For example for \( a = 3\) and \( b = 2 \) you should see 3 horizontal lobes and 2 vertical. You may need to adjust \( \phi \) to clearly see the lobes as they sometimes overlap for certain values of \( \phi \). Also note, that the graph with, say, \( a = 6 \) and \( b = 4 \) will be identical to the graph of \( a = 3 \) and \( b = 2 \) as it is specifically the ratio which affects the shape of the graph.

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. 206 COMMENTS Activity: The Lissajous pattern, A Classic phase measurement The pattern of criss-cross lines is actually a Lissajous figure. The phase difference, or phase angle, is the difference in phase between the same points, say a zero crossing, in two different waveforms with the same frequency. Lissajous curve A Lissajous figure, made by releasing sand from a container at the end of a Blackburn pendulum. A Lissajous curve / ˈlɪsəʒuː /, also known as Lissajous figure or Bowditch curve / ˈbaʊdɪtʃ /, is the graph of a system of parametric equations. which describe the superposition of two perpendicular oscillations in x and y directions of ... Discover the Mesmerizing Le Joujou Figures Lissajous figures are examples of complex harmonic motion, and can be easily demonstrated on an oscilloscope. I examine the history, the mathematics behind them, demonstrate how to generate them ... Lissajous figures The aim of this experiment is to show how to trace the Lissajous figures on the Cathode Ray Oscilloscope using two function generators connected to X and Y plates of CRO. The different figures ... 3. Draw Lissajous Pattern, Measurement of unknown frequency and In this video we have demonstrated the experiment of Lissajous Pattern at different Phase and Frequencies and also show how different shapes are coming in DSO with variation of Phase and ... 3A80.20 Sn- 2, 3: Freier and Anderson, A Demonstration Handbook for Physics. M-930: "Lissajous Patterns on Scope", DICK and RAE Physics Demo Notebook. Yaakov Kraftmakher, "1.5, Lissajous Patterns", Experiments and Demonstrations in Physics, ISBN 981-256-602-3, p. 20. 1.25: Charles Taylor, The Art and Science of Lecture Demonstration, p. 50- 52. Lissajous Figures The optical production of the curves was first demonstrated in 1857 by Jules Antoine Lissajous (1833-1880), using apparatus similar to that at the left. Today we can do the same experiment more easily with a laser beam that reflects from the two mirrors vibrating at right angles to each other and then traces the Lissajous figure on the wall. 3.5: Lissajous Figures Lissajous figures are figures that are turned out on the face of an oscilloscope when sinusoidal signals with different amplitudes and different phases are applied to the time base (real axis) and deflection plate (imaginary axis) of the scope. The electron beam that strikes the phosphorous face then had position. Lissajous Patterns of CRO or Cathode Ray Oscilloscope Ellipse Patterns: Lissajous patterns form ellipses that change shape depending on the phase difference and the quadrant through which the major axis passes. Cathode Ray Oscilloscope (CRO) is very important electronic device. CRO is very useful to analyze the voltage wave form of different signals. The main part of CRO is CRT ( Cathode Ray Tube ). Lissajous figure Lissajous figure, also called Bowditch Curve, pattern produced by the intersection of two sinusoidal curves the axes of which are at right angles to each other. First studied by the American mathematician Nathaniel Bowditch in 1815, the curves were investigated independently by the French. Lissajous curves Just like the stars and Earth, the UV light attached to the string moves in a complex pattern. The clamp in the middle causes the UV light to move in a combination of swings in two directions (back-and-forth and sideways ). Therefore, the UV light travels along a line called a Lissajous figure . A Classic phase measurement, The Lissajous pattern Figure 2 Using a classic Lissajous display lets you measure the phase difference between two sine waves. Marker readouts in upper left of the plot show the required values for computing the phase difference. Measurements using Lissajous Figures Lissajous Figures - Lissajous figure is the pattern which is displayed on the screen, when sinusoidal signals are applied to both horizontal & vertical deflection plates of CRO. These patterns will vary based on the amplitudes, frequencies and phase differences of the sinusoidal signals, which are applied to both horiz. PocketLab/Phyphox Damped Lissajous Figures Results and Comparison to Theory Figure 3 shows a typical Lissajous pattern plotted in Phyphox for this experiment. Note the concentration of points near the "center" of the pattern. Does this result agree with that predicted from theory? Figure 3 Let's consider the following set of parametric equations: Lissajous Lab The Lissajous Lab provides you with a virtual oscilloscope which you can use to generate these patterns. ( You will control the horizontal. You will control the vertical.) The applet also allows you to apply a signal to modulate the hue of the trace, so you can create colorful designs. The Lissajous Lab and description were created by Ed Hobbs. H1-27: Speed of Sound ID Code:H1-27. Purpose:Measurement of the speed of sound in air using Lissajous figures. Description:The signal to the loudspeaker is used as the horizontal input of an oscilloscope, and the signal picked up by the microphone is used as the vertical input, forming Lissajous figures. When they are in phase a diagonal line is produced, running ... Lissajous Curves A Lissajous curve, named after Jules Antoine Lissajous is a graph of the following two parametric equations: (1) x = A s i n ( a t + ϕ) (2) y = B s i n ( b t) A and B represent amplitudes in the x and y directions, a and b are constants, and ϕ is an phase angle. The user interface above allows you to modify each of these five parameters and ... #48: Basics of Lissajous Patterns on an Oscilloscope Another "Back to Basics" video: This video takes a fairly detailed look at the basics of Lissajous patterns on an oscilloscope. There are a LOT of videos that show Lissajous patterns on YouTube ... Lissajous figures: from Physclips Lissajous figures (or Lissajous curves) are produced in two dimensions when the x and y coordinates are given by two sine waves, which may have any amplitude, frequency and phase. This is a support page to the multimedia chapter Interference and Consonance in the volume Waves and Sound, which introduces interactions between sine waves. PDF Lissajous Figures Lissajous Figures. In the old days, whenever they showed an engineer working, there was usually an oscilloscope nearby with a pattern on the screen. Most often, the pattern was a Lissajous Figure. Jules Antoine Lissajous (1822-1880) was a French physicist who was interested in waves, and around 1855 developed a method for displaying them ... PDF Lissajous Figures Lissajous figure. His setup was similar to the modern device used to project. aser light shows. Lissajous figures were used in the old days to determine the frequencies of sounds. or radio signals. They found their way into popular culture in many sci-fi movies and TV shows, including the opening footage for The Outer. A simple Lissajous curves experimental setup The aim of this study is to develop an experimental setup to produce Lissajous curves. The setup was made using a smartphone, a powered speaker (computer speaker), a balloon, a laser pointer and a piece of mirror. Lissajous curves are formed as follows: a piece of mirror is attached to a balloon. The balloon is vibrated with the sound signal ... essay homework paper phd project resume review study thesis