LISSAJOUS (lil'sb,'zhiTa') FIGURES. A name given to certain phenomena designed to show optically the composition of vibratory mo tions. On April 6, 1857, Jules Antoine Lis sajous presented to the Academy of Sciences in Paris a memoir on the optical study of vibratory movements, wherein he set forth for the first time that series of peculiar curves which have since borne his name, and which are generated by the combination of two vibrations taking place in the same plane, but at right angles to each other.
Suppose a body is vibrating back and forth be tween E and IV. Fig. I, in simple harmonic mo tion, when an impulse is given to it which alone would set it into similar vibration be tween N and S. As sume that the time of vibration of the two motions is the same, then the result of their combination will depend upon the rela tive 'phase' of the two vibrations. Several characteristic c a se s may be considered as typical of the real in finity of possible variations. If the body tend to start from 0 toward E at the same instant that it tends to start from 0 toward N, then its real motion would be along the diagonal of the rectangle on OE and ON, and the result ing vibration would be as shown in Fig. 2 a. This is the case where both harmonic cycles start at the same instant from the position of equilibrium, 0; that is, the difference of phase is zero. if it tend to start from 0 toward N and IV simultaneously, then the resulting vibration will be shown at Fig. 2 e. In this case the EW motion would have executed a half cycle when the NS commences. that is, the phase dif ference is one-half.
For a phase difference of one-quarter the body would be at E when the NS motion starts and the result is shown in Fig. 2 c. A phase differ ence of three-quarters would put the body at \V when the other motion starts and the figure would be the same as c except that it would cir culate in the opposite direction.
The result of a phase difference of one-eighth is shown in b, and that of three-eighths in d. If the amplitude of the two motions he the same it is evident that Fig. 2 c would be a circle. A small weight or plumb-bob hanging upon a string serves very well to illustrate the above forms of compound vibration.
When the rates of vibration or periodic times of the two components are unequal much more complex results are produced, which, however, reduce to comparative simplicity when the rate of one bears a simple ratio to the rate of the other. When the two rates are in the ratio of 1 to 2 there results a series of curves some of which are shown in Fig. 3. for phase differences of one-eighth, one-fourth, and one half of the ENV motion. Similarly Fig. 4 shows the corresponding curves when the rates of vibra tion are in the ratio of 2 to 3.
Lissajous attaehed a small mirror to one prong of each of two tuning-forks of the same pitch. One fork is mounted vertically, Fig. 5. and the other horizontally. A ray of light from the lamp, L, is made to fall first on the mirror of the vertical fork, thence upon that of the horizontal fork, and finally on the screen S. if F, alone vibrates. then the spot of light on the screen will move np and down. owing to the tilting of the mirror on that fork. if F, is the only one in motion the spot will move hori zontally; and finally if both vibrate simultane ously, then a combination curve will be exhibited, similar to some one •of the forms of Fig. 2. It the two forks have differ ent. rates of vibra tion then complex will appear ; if one is the octavo of the other. it will be of the class of Fig. 3: and Fig. 4 would be given by two forks hay ing the interval of the fifth. In an other form of ap paratus for the same purpose the forks are replaced by two steel sprints the rates of which can be varied by their length and by a small adjustable weight. Fig. G illustrates another method of showing these forms of vibration.
The plumb-bob I) is free to swing perpendicular to the plane of the paper around the line AB with an effective length of the pendulum of (1): in the plane of the paper it can only swing around E with the effective length DE. means of the sliding clamp E it is possible to adjust the two rates of vibration to definite ratios and then the bob 1) NVii I execute some one or all of the above curves Many other methods have been de vised for showing or drawing these curves.