Schwingende Saiten, Intervalle der Musik and Lissajoussche Kurven

Auf dieser Seite werden zwei gleichzeitige Schwingungen mittels verschiedener Betrachtungsarten visualisiert: Als zwei parallel schwingende Saiten, als eine Saite die gleichzeitig Grund und Oberschwingung ausführt oder als Lissajoussche Kurve. Durch Ändern der Wellenzahlen = Schwingungsknoten und des Phasenwinkels zwischen den beiden Schwingungen können die Graphen verschiedener Intervalle der Musik erzeugt werden, ebenso die Erscheinungen von Schwebung und Interferenz. Weitere Deteils s.u..

m und n sind die jeweiligen Wellenzahlen der ersten und zweiten Saite.
Die Wellenzahl m=1 steht für eine komplette Sinuskurve einer Saite.
φ ist der Phasenwinkel zwischen den beiden Schwingungen, d.h. ob beide Schwingungen im Takt oder gegenläufig etc. schwingen.
"⇐ slow fast ⇒" reguliert die Geschwindigkeit der Animation
"octave up / down" = "⇑ / ⇓ " verdoppelt oder halbiert die Frequenzen
START - startet die Animation, STOP - beendet sie


  ⇐     slow      fast     ⇒  

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An ideal oscillation. is described mathematically by a sine curve. A musical interval corresponds to two independent oscillations. The time development of these two independent oscillators, for example two vibrating strings or a pendulum with two independent joints, are described by two standing sine waves. (The case of running waves like for instance radiation or water waves is omitted here since the fixed plucked string serves as the most important example).
These two waves can be represented graphically by
1) the movement of the two strings
2) the movement of one string where the two oscillations are superposed
3) a Lissajous curve

Lissajous curves

Lissajous curves are named after the french physicist Jules Antoine Lissajous. Their graphs are parametric plots. One oscillation follows the horizontal axis, the other one the vertical axis. Both are sine curves with different frequencies which are given by the corresponding two wavenumbers m and n. The relative phase between the two oscillations determines the starting point of the curve.


The addition of the two strings into a one-string-graph is called a superposition: Both waves are added without influencing each other. As a consequence the following phenomena arise:
Beats: Two waves of almost equal frequencies superpose. Try for example m=20 and n=19 or m=64 n=63 in the one-string-graph. While running press "⇑" to get the full picture. Now regulate phi slowly with the slider.
Interference: Start with m=n=3 in the two-string-graph. While running change slowly the phase with the slider. Now switch to the one-string-graph and do the same. A first surprise may be, that two sine curves which are shifted in phase do exactly add up to a third sine curve. As the phase approaches 1.0 pi complete extinction of the signal occurs. A phase of 0.0 or 2pi will result in maximum amplification. Switching between one-string-graph and two-string-graph explains this easily.

Lissajous curves in vivo

To get a better understanding of the Lissajous curves the choice "move Lissajous curve" was introduced. This allows you to see a continuous change of a Lissajous curve, while you alter m,n or the phase. For example if you choose m=5 n=6 and change the phase (preferably by using the arrows of the slider) while running "move Lissajous curve". The result looks like a three dimensional rotation.
In electrical engineering Lissajous curves are used in oscilloscope measurements to detect very small frequency deviations or phase differences. If you define m=1 n=1 and change the phase via the slider you will see a slowly moving elliptical shape. Try with m=50 n=49.99 . In the one-string-graph you notice the beats in time. In the "move Lissajous curve" you notice a slowly moving elliptical shape.

Large oscillations of the planets around the sun in our solar system are shown in Curves of planetary motion in geocentric perspective: Epitrochoids.
For a quantum mechanical treatment of oscillations see A Gallery of Quantum States. This site gives an introduction to the quantum mechanics of the light field explaining notions like the wave packet or the uncertainty relation.



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Last modified: Jun. 2013