Weird Science

Lissajous Curves – The Beauty of Oscillations

Polish ver­sion is here

The fol­lo­wing article was ori­gi­nally publi­shed in the jour­nal Młody Tech­nik (eng. Young Tech­ni­cian) (6/2015):

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Ples M., Krzywe Lis­sa­jous - piękno drgań (eng. Lis­sa­jous Curves – The Beauty of Oscil­la­tions), Młody Tech­nik, 6 (2015), Wydaw­nic­two AVT, pp. 76-77

Intro­duc­tion

I believe a few cla­ri­fi­ca­tions are in order before we begin. What exac­tly are these enig­ma­tic Lis­sa­jous curves (some­ti­mes also refer­red to as Lis­sa­jous figu­res or Bow­ditch curves)?

A mathe­ma­ti­cian or phy­si­cist would imme­dia­tely reply that they are para­me­tric curves descri­bed by the fol­lo­wing equ­a­tions:

x(t) = A sin(at + φ), y(t) = B sin(bt)

These curves repre­sent har­mo­nic oscil­la­tions. Altho­ugh first descri­bed by the Ame­ri­can mathe­ma­ti­cian Natha­niel Bow­ditch, they are more widely known under the name of Jules Anto­ine Lis­sa­jous, who stu­died them using a device of his own con­struc­tion.

Jules Anto­ine Lis­sa­jous (1822–1880) was a French mathe­ma­ti­cian and phy­si­cist. He stu­died at the École Nor­male Supérieure and, after ear­ning his doc­to­rate, served as rec­tor of the Aca­demy of Cham­béry and later the Aca­demy of Besa­nçon. Lis­sa­jous had a par­ti­cu­lar inte­rest in aco­u­stic waves. He deve­lo­ped an appa­ra­tus made from vibra­ting tuning forks with small mir­rors atta­ched. This setup allo­wed him to obse­rve the curves that would later bear his name.

I under­stand that the for­mula above may not appeal to eve­ry­one — espe­cially those who, for rea­sons unk­nown to me, feel an aver­sion to mathe­ma­tics. For­tu­na­tely, this can also be expla­i­ned gra­phi­cally (Fig. 1). The illu­stra­tion shows the three sim­plest cases of Lis­sa­jous curves, which occur when the fre­qu­en­cies of the two sinu­so­i­dal oscil­la­tions (in the x and y direc­tions, per­pen­di­cu­lar to each other) are equal. Cases A, B, and C dif­fer in their phase shift, which is the argu­ment φ in the for­mula above. Here's how it works:

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Fig. 1 – Lis­sa­jous curves; left column – resul­ting oscil­la­tion, right column – com­po­nent oscil­la­tions

By care­fully selec­ting the two fre­qu­en­cies and the phase shift, one can gene­rate distinc­tly dif­fe­rent curves, many of which have sur­pri­sin­gly ele­gant and aesthe­tic sha­pes.

But how can we actu­ally gene­rate such curves?

There are many ways. For exam­ple, we could (as Lis­sa­jous did) obse­rve a light beam reflec­ted off mir­rors vibra­ting in per­pen­di­cu­lar pla­nes. Ano­ther excel­lent method is using an oscil­lo­scope that allows for elec­tron beam deflec­tion in both the X and Y axes — just con­nect two elec­tri­cal signals with adju­sta­ble fre­qu­en­cies and pha­ses to the inputs.

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A Lis­sa­jous curve on an ana­log oscil­lo­scope
ani­ma­tion: sup­ple­men­tary mate­rial

Finally, you can also gene­rate Lis­sa­jous curves using a pen­du­lum.

The Expe­ri­ment

To pro­duce Lis­sa­jous curves at home, it's conve­nient to use a pen­du­lum with an appro­priate design (Fig. 2).

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Fig. 2 – Con­struc­tion of a pen­du­lum for obse­rving Lis­sa­jous curves

Let’s con­si­der how such a pen­du­lum works. We know that for small displa­ce­ments, the oscil­la­tion period T of a pen­du­lum is descri­bed by the equ­a­tion:

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where l is the length of the pen­du­lum and g is the stan­dard acce­le­ra­tion of gra­vity. A pen­du­lum with the pro­po­sed con­struc­tion will oscil­late with period T1 in the plane per­pen­di­cu­lar to the dia­gram, and with period T2 in the plane of the dia­gram. These periods will dif­fer due to the dif­fe­rent effec­tive leng­ths of the pen­du­lum in those pla­nes, L1 and L2 respec­ti­vely. If L2 is only sli­gh­tly shor­ter than L1, we won’t obse­rve a dra­ma­tic dif­fe­rence in oscil­la­tion fre­qu­ency. Howe­ver, we will see a gra­dual change in the rela­tive phase (ran­ging from 0 to 2π). This makes it pos­si­ble to trace inte­re­sting Lis­sa­jous curves.

In my expe­ri­ments, the pen­du­lum leng­ths were appro­xi­ma­tely:

The mass of the pen­du­lum was 0.05 kg (1.76 oz).

The best way to visu­a­lize Lis­sa­jous curves is by using long-expo­sure pho­to­gra­phy. You’ll need to attach a sim­ple light source to the pen­du­lum bob, made from an LED con­nec­ted in series with a resi­stor (R between 1–5 kΩ) and a minia­ture 3V bat­tery (CR2032 coin cell). The camera sho­uld be posi­tio­ned below the pen­du­lum and poin­ted upward, as shown in Fig. 2. Set the pen­du­lum in motion and take pho­tos in com­plete dark­ness with the light source tur­ned on. The expo­sure time sho­uld be deter­mi­ned expe­ri­men­tally — usu­ally seve­ral seconds.

Ima­ges of Lis­sa­jous curves cap­tu­red this way are shown in Photo. 1.

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Photo. 1 – Gene­ra­ted Lis­sa­jous curves

By sli­gh­tly adju­sting the values of L1 and L2, and using a lon­ger expo­sure time, we cap­tu­red the curve shown in Photo. 2.

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Photo. 2 – Ano­ther Lis­sa­jous curve

As we can see, the figu­res pro­du­ced — even with this very sim­ple method — pos­sess a uni­que geo­me­tric beauty.

Of course, the appe­a­rance of the curves can be modi­fied across a wide range. You sim­ply need to adjust the pen­du­lum leng­ths and the camera’s expo­sure time.

Refe­ren­ces:

All pho­to­gra­phs and illu­stra­tions were cre­a­ted by the author.

The above text inc­lu­des minor edi­to­rial modi­fi­ca­tions com­pa­red to the ver­sion publi­shed in the jour­nal, aimed at sup­ple­men­ting and adap­ting it for online pre­sen­ta­tion.

Marek Ples

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