Lissajous Figures
When two simple waves are combined, but one is rotated out of phase with respect to the other, the result is no longer a sine wave, but a combination of the frequencies, the rotational phase difference, and any time offset between the two original waves. Such a combination of sine waves, when graphed or otherwise visually displayed is known as a Lissajous figure.
In mathematics, sine waves are the result of graphing such as y=sin(x) for enough values of x to provide a smooth oscillating curve about the x axis. The function x=sin(y) is 90 degrees out of phase with y=sin(x) because it creates a curve which oscillates about the y axis. In the physical world, perhaps the best known example of sine waves is sound. The frequency of a sine wave is the distance, or time span, between each "crest" in the curve. In the case of y=sin(x), this crest is the point at which y is greatest. Energy in the electromagnetic spectrum (EMS) exhibits sine waves oscillation with very low to very high frequencies. Not only does sound have a place at the low end of this spectrum, but many other forms of energy are found along it as well, including radio waves, light, ultraviolet rays and so forth.
Returning to the audio portion of the EMS, a perfect tone that remains steady at a single pitch can be written as a simple trigonometric function like those above with additional parameters appropriate to match the tone's frequency of oscillation. Two such functions combined into a graphic representation can show Lissajous patterns. But the easiest way to create Lissajous figures is to introduce separate alternating current voltages to the horizontal and vertical inputs of a cathode-ray tube oscilloscope. The interaction of the two signals can then be viewed on the screen. If the waves are rotationally 90 degrees out of phase, but the frequencies and time phase are the same, the figure resulting from the combination of the waves is a straight line. For all other values of time displacement, the result is an. When the time phase is off by exactly 1/4 the frequency of the wave, a special case of the ellipse, a circle is formed. If the frequencies of the two waves are not the same, the Lissajous figure becomes more complex. For example, if one wave has a frequency that is twice that of the other, a sort of "figure eight" pattern results. It could be said that there is one twist, or cross-over point in the pattern at this point. Each time the frequency of one wave becomes a multiple of the other a definite pattern results with more and more twists in it as the frequency difference increases. Such figures are the result of two simple oscillating motions at right angles to each other reaching a harmonic state. When harmonics are reached, the screen of the oscilloscope shows Lissajous figures.
A very common practical use of all of this is found in phase and frequency measurements and adjustments of electronic circuits. By comparing the visual information provided by Lissajous figures, with known or expected patterns, or performance parameters, technicians can repair electronic equipment and scientists can study important phenomena. In addition, now that it is so common to create music electronically, such sine wave comparisons and indicators can also aid in the arts by providing information about the relationships between sounds that musicians use in compositions or performances.
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