Physics · Oscillations and waves · Grade 11

Coupled oscillations

Why do two pendulums joined by a spring swing in turn?

The situation

In 1665 the Dutch scientist Christiaan Huygens, watching two pendulum clocks hung from the same beam, noticed a strange thing: however they started, the pendulums ended up swinging in opposite directions. If you join two pendulums with a light spring and push only one of them, you will see something even more interesting: the first slowly stops, the second begins to swing — and then everything repeats in reverse. Atoms in a crystal are joined to one another by just such “springs”. How does energy pass from pendulum to pendulum — and from atom to atom?

Topic

Coupled oscillations and normal modes. Two pendulums coupled by a spring and a chain of atoms — phonons

Key ideas

Beats and normal modes
Phonon dispersion
Lissajous figures

Formulas

ω_{1} = \sqrt{\frac{g}{l}}, ω_{2} = \sqrt{\frac{g}{l} + \frac{2 k d^{2}}{m l^{2}}}

Normal frequencies of coupled pendulums: in phase (the spring is not stretched) and in antiphase. If the spring is attached to the bobs ( $d = l$ ): $ω_{2} = \sqrt{g / l + 2 k / m}$ .

x = 2 A \cos \frac{(ω_{2} - ω_{1}) t}{2} \cos \frac{(ω_{2} + ω_{1}) t}{2}, T_{t} = \frac{2 π}{ω_{2} - ω_{1}}

Beats: the sum of two oscillations of close frequencies. The amplitude varies with period $T_{t}$ ; in sound, the number of beats per second is $| f_{2} - f_{1} |$ .

A^{2} = A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos (φ_{2} - φ_{1}), Z = \sqrt{R^{2} + {(ω L - \frac{1}{ω C})}^{2}}

Adding oscillations of the same frequency with a phasor diagram; voltages in an RLC circuit add in the same way.

x = A \cos ω_{x} t, z = B \cos (ω_{z} t + δ), \frac{ω_{z}}{ω_{x}} = \sqrt{\frac{l_{x}}{l_{y}}}

Mutually perpendicular oscillations — Lissajous figures. In a Y-shaped sand pendulum the ratio is set by the lengths of the strings.

ω = 2 \sqrt{\frac{K}{m}} | \sin \frac{q a}{2} |, ω_{\mp}^{2} = \frac{K}{μ} \mp K \sqrt{\frac{1}{μ^{2}} - \frac{4 \sin^{2} (q a / 2)}{m_{1} m_{2}}}

The dispersion law of monatomic and diatomic chains: $q$ — wave number, $a$ — lattice constant, $1 / μ = 1 / m_{1} + 1 / m_{2}$ ; “−” is the acoustic branch, “+” the optical branch.

Theory

Simple harmonic motion

If, on displacement from equilibrium, the restoring force is proportional to the displacement, the motion is simple harmonic: $\ddot{x} + ω^{2} x = 0$ , with solution $x = A \cos (ω t + φ)$ . For a spring pendulum $ω = \sqrt{k / m}$ , for a simple pendulum $ω = \sqrt{g / l}$ , for a physical pendulum $ω = \sqrt{m g a / I}$ ( $a$ — distance from the pivot to the centre of mass). A single pendulum is covered in the “Swing” and “Spring pendulum” experiments. Model: the angles are small (≤ 10°), the rods are weightless, and the spring stays horizontal.

Normal modes and normal coordinates

A system with two degrees of freedom has two normal modes — motions in which all parts oscillate with one frequency. In phase the spring is not stretched: $ω_{1} = \sqrt{g / l}$ . In antiphase the spring pushes both bobs back: $ω_{2} = \sqrt{g / l + 2 κ}$ , $κ = k d^{2} / (m l^{2})$ . The normal coordinates $q_{1} = (θ_{1} + θ_{2}) / 2$ , $q_{2} = (θ_{1} - θ_{2}) / 2$ are independent simple oscillations; any motion is their sum. With $n$ degrees of freedom, the $n$ natural frequencies are found from the equation $\det (𝐊 - ω^{2} 𝐌) = 0$ — molecules and crystals vibrate in the same way.

Beats: how energy is transferred

If we displace only the first pendulum, that is an equal mixture of these two modes. Because the mode frequencies differ slightly, they sometimes reinforce and sometimes cancel each other — beats: the energy passes completely from the first pendulum to the second and returns with period $T_{t} = 2 π / (ω_{2} - ω_{1})$ . The weaker the coupling, the slower the exchange. Musicians also tune instruments by beats: two notes of 440 and 442 Hz swell and fade 2 times a second; when the beats vanish, the frequencies are equal.

Phasor diagram and Lissajous figures

A simple harmonic oscillation can be regarded as the projection of a vector (a phasor) rotating with angular velocity $ω$ . Adding oscillations in the same direction is adding vectors. In an alternating-current circuit $U_{R}$ is in phase with the current, $U_{L}$ leads the current by 90° and $U_{C}$ lags by 90°; the amplitude of the sum is $U_{m} = I_{m} Z$ , $\tan φ = (X_{L} - X_{C}) / R$ . In complex form $Z = R + i (ω L - 1 / ω C)$ ; at $X_{L} = X_{C}$ — resonance. Mutually perpendicular oscillations trace out a curve: if the ratio of the frequencies is a simple fraction, the figure is closed. These curves were studied in detail by N. Bowditch in 1815 and J. Lissajous in 1857.

A chain of atoms: dispersion, acoustic and optical branches

In a closed chain of $N$ cells (Born–von Kármán condition) the wave number can take only the values $q = 2 π j / (N a)$ — there are $N$ modes in each branch. For long waves $ω \approx v_{s} q$ , where $v_{s} = a \sqrt{K / m}$ is the speed of sound in the crystal. At $q = π / a$ the wave is standing and the group velocity $d ω / d q = 0$ . In a diatomic chain, in the acoustic branch the cell moves as a whole, while in the optical branch neighbouring atoms vibrate in antiphase (in ionic crystals this creates a dipole and absorbs infrared light). If $m_{1} \neq m_{2}$ , there is a forbidden gap of frequencies between the branches. Electrical analogy: a chain of $L$ – $C$ sections ( $m \leftrightarrow L$ , $1 / K \leftrightarrow C$ ) is a low-pass filter that does not transmit $ω > 2 / \sqrt{L C}$ . The CO₂ molecule: a linear O–C–O model gives two stretching vibrations; the model is about 12 % below the real symmetric frequency — it ignores the interaction between the two bonds.

Phonons and the density of states

In quantum mechanics the energy of each normal mode is $E = (n + \frac{1}{2}) ℏ ω$ — it changes only in portions of $ℏ ω$ . This quantum of a lattice vibration is called a phonon (the name was proposed by Ya. Frenkel; from the Greek “phōnē” — sound). In our chain $ℏ ω$ is tens of meV, comparable with $k T \approx 26$ meV at room temperature: that is why phonons determine heat capacity and thermal conductivity. The density of states $g (f)$ is the number of modes per frequency interval; where the dispersion curve flattens ( $d ω / d q = 0$ ) it rises sharply. More in the “Bands in solids” and “Quantum statistics” experiments.

Did you know?

In 1665 Christiaan Huygens noticed that the pendulums of two pendulum clocks hung from the same beam always ended up swinging in opposite directions however they were started, and explained this by a weak coupling transmitted through the beam. This is one of the first recorded observations of synchronisation.
The CO₂ molecule absorbs infrared light in the bending vibration (about 15 µm) and the antisymmetric stretch (about 4.26 µm). The symmetric stretch does not change the dipole moment of the molecule, so this vibration does not absorb IR light. It is exactly this absorption that makes CO₂ a greenhouse gas.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Waves on a rope and a spring — Grade 10
Sound: speed, pitch, ultrasound — Grade 10
The dutar string and sound — Grade 11
Waves and the Doppler effect — Grade 11
Bridge resonance — Grade 11
Nonlinear oscillations — Year 2
Fourier: the spectrum of sound — Year 3