Part I: Oscillations | Chapter 3

Coupled Oscillators

From two masses on springs to the wave equation

3.1 Two Coupled Oscillators

Derivation 1: Normal Modes of Two Masses

Consider two identical masses $m$ connected by springs: each mass is attached to a wall by a spring of constant $k$, and the two masses are connected to each other by a coupling spring of constant $\kappa$. The equations of motion are:

$$m\ddot{x}_1 = -kx_1 - \kappa(x_1 - x_2)$$$$m\ddot{x}_2 = -kx_2 - \kappa(x_2 - x_1)$$

The key insight is to look for normal modes — patterns of motion where all parts of the system oscillate at the same frequency. Define symmetric and antisymmetric coordinates:

$$q_+ = x_1 + x_2 \quad (\text{symmetric mode})$$$$q_- = x_1 - x_2 \quad (\text{antisymmetric mode})$$

Adding and subtracting the equations of motion gives two decoupled equations:

Normal Mode Frequencies

$$\ddot{q}_+ + \omega_+^2 q_+ = 0 \quad \text{with } \omega_+ = \sqrt{k/m}$$$$\ddot{q}_- + \omega_-^2 q_- = 0 \quad \text{with } \omega_- = \sqrt{(k + 2\kappa)/m}$$

In the symmetric mode, both masses move together and the coupling spring is unstretched. In the antisymmetric mode, masses move in opposite directions and the coupling spring contributes an additional restoring force.

3.2 Beating Phenomenon

Derivation 2: Energy Transfer and Beats

If we start with only mass 1 displaced ($x_1(0) = A$, $x_2(0) = 0$), both normal modes are excited equally. The individual displacements are:

$$x_1(t) = \frac{A}{2}[\cos(\omega_+ t) + \cos(\omega_- t)] = A\cos\!\left(\frac{\omega_- - \omega_+}{2}t\right)\cos\!\left(\frac{\omega_- + \omega_+}{2}t\right)$$$$x_2(t) = \frac{A}{2}[\cos(\omega_+ t) - \cos(\omega_- t)] = A\sin\!\left(\frac{\omega_- - \omega_+}{2}t\right)\sin\!\left(\frac{\omega_- + \omega_+}{2}t\right)$$

The energy shuttles back and forth between the two masses at the beat frequency:

Beat Frequency

$$\omega_{\text{beat}} = |\omega_- - \omega_+|$$

For weak coupling ($\kappa \ll k$), the beat frequency is much smaller than the oscillation frequency, giving a slowly modulated oscillation — the hallmark of beating.

Analogy: Two tuning forks of slightly different pitch placed near each other will exchange energy through the air. The audible "wa-wa-wa" sound is beating. In quantum mechanics, a similar energy exchange occurs between coupled quantum states (Rabi oscillations).

3.3 N Coupled Oscillators

Derivation 3: Dispersion Relation

Extend to $N$ identical masses connected by identical springs of constant $\kappa$, with fixed endpoints. The equation of motion for the $j$-th mass is:

$$m\ddot{x}_j = \kappa(x_{j+1} - 2x_j + x_{j-1})$$

Try a normal mode solution $x_j(t) = A\sin(jka)e^{-i\omega t}$ where $a$is the equilibrium spacing and $k$ is the wavenumber. Substituting:

$$-m\omega^2 = \kappa(e^{ika} - 2 + e^{-ika}) = 2\kappa(\cos(ka) - 1)$$

Dispersion Relation

$$\omega(k) = 2\sqrt{\frac{\kappa}{m}}\left|\sin\!\left(\frac{ka}{2}\right)\right|$$

This is the dispersion relation for a monatomic chain. It shows that the relationship between frequency and wavenumber is nonlinear, meaning different wavelengths travel at different speeds — the system is dispersive.

Key Features

Maximum Frequency

$$\omega_{\text{max}} = 2\sqrt{\kappa/m}$$

No modes exist above this cutoff. In a crystal, this sets the Debye frequency.

Long-Wavelength Limit

$$\omega \approx v_s k \quad (ka \ll 1)$$

Linear dispersion: $v_s = a\sqrt{\kappa/m}$ is the sound speed.

3.4 The Continuum Limit: The Wave Equation

Derivation 4: From Discrete to Continuous

Take the limit $N \to \infty$, $a \to 0$, with $m/a = \mu$(linear mass density) and $\kappa a = T$ (tension) held fixed. The discrete equation of motion:

$$\mu a \ddot{x}_j = \frac{T}{a}(x_{j+1} - 2x_j + x_{j-1})$$

Recognizing that $(x_{j+1} - 2x_j + x_{j-1})/a^2$ is the finite-difference approximation to $\partial^2 u/\partial x^2$:

The Wave Equation

$$\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial x^2} \quad \text{where } v = \sqrt{T/\mu}$$

This is one of the most important equations in physics. The discrete chain of coupled oscillators, in the continuum limit, gives rise to wave propagation. The wave speed$v$ depends on the tension and density of the medium.

The general solution is $u(x,t) = f(x - vt) + g(x + vt)$ — arbitrary shapes traveling to the right and left with speed $v$. This is d'Alembert's solution (1747).

3.5 Matrix Formulation

Derivation 5: Eigenvalue Problem

The coupled oscillator problem can be cast as a matrix eigenvalue problem. For $N$coupled oscillators, we write $\ddot{\mathbf{x}} = -\mathbf{K}\mathbf{x}$where $\mathbf{K}$ is the dynamical matrix:

$$\mathbf{K} = \frac{\kappa}{m}\begin{pmatrix} 2 & -1 & 0 & \cdots \\ -1 & 2 & -1 & \cdots \\ 0 & -1 & 2 & \cdots \\ \vdots & & & \ddots \end{pmatrix}$$

The normal modes are the eigenvectors of $\mathbf{K}$, and the squared normal mode frequencies are the eigenvalues. For $N$ masses with fixed endpoints, the$n$-th eigenvalue and eigenvector are:

Normal Modes (Fixed Endpoints)

$$\omega_n^2 = \frac{4\kappa}{m}\sin^2\!\left(\frac{n\pi}{2(N+1)}\right) \quad n = 1, 2, \ldots, N$$$$x_j^{(n)} = \sin\!\left(\frac{nj\pi}{N+1}\right)$$

These standing-wave patterns are the same as the modes of a vibrating string. The eigenvalue approach generalizes naturally to 2D and 3D lattices.

3.5b Diatomic Chain: Acoustic and Optical Branches

When the chain has alternating masses $m_1$ and $m_2$(or alternating spring constants), the dispersion relation splits into two branches:

Diatomic Chain Dispersion

$$\omega_\pm^2 = \kappa\!\left(\frac{1}{m_1} + \frac{1}{m_2}\right) \pm \kappa\sqrt{\left(\frac{1}{m_1} + \frac{1}{m_2}\right)^2 - \frac{4\sin^2(ka/2)}{m_1 m_2}}$$

Acoustic Branch ($\omega_-$)

Adjacent masses move in the same direction (in-phase). At long wavelengths,$\omega \approx v_s k$ (sound waves). This branch starts at zero and rises to a maximum at the zone boundary.

Optical Branch ($\omega_+$)

Adjacent masses move in opposite directions (out-of-phase). Called "optical" because in ionic crystals these modes can couple to electromagnetic radiation. There is a band gap between the two branches where no propagating modes exist.

The band gap has frequency range $\sqrt{2\kappa/m_1} < \omega < \sqrt{2\kappa/m_2}$(assuming $m_1 > m_2$). Waves at frequencies in the gap are evanescent. This is the mechanical analog of electronic band gaps in semiconductors and photonic band gaps in photonic crystals.

3.5c Standing Waves and Boundary Conditions

The boundary conditions determine which wavelengths are allowed. For a string of length$L$:

Fixed-Fixed (Dirichlet)

$u(0,t) = u(L,t) = 0$

$$\lambda_n = \frac{2L}{n}, \quad f_n = \frac{nv}{2L}$$

Guitar string, organ pipe (closed-closed)

Fixed-Free (Mixed)

$u(0,t) = 0$, $\partial_x u(L,t) = 0$

$$\lambda_n = \frac{4L}{2n-1}, \quad f_n = \frac{(2n-1)v}{4L}$$

Clarinet (odd harmonics only)

The standing wave is a superposition of left- and right-traveling waves:

$$u_n(x,t) = A_n \sin\!\left(\frac{n\pi x}{L}\right)\cos(\omega_n t + \phi_n)$$

The general motion of the string is a Fourier series superposition of all normal modes. This is the foundation of Fourier analysis: any function satisfying the boundary conditions can be expanded in terms of the normal modes (eigenfunctions) of the system.

Musical acoustics: The timbre (tone quality) of a musical instrument depends on which harmonics are present and their relative amplitudes. A clarinet produces only odd harmonics, giving it a distinctive hollow sound. A guitar string produces all harmonics, but the plucking position determines which harmonics are strongest (plucking at the center suppresses even harmonics).

3.5d Energy and Momentum in Waves

For a wave on a string with tension $T$ and linear mass density $\mu$, the energy densities are:

$$\text{Kinetic: } \frac{1}{2}\mu\left(\frac{\partial u}{\partial t}\right)^2 \quad \text{Potential: } \frac{1}{2}T\left(\frac{\partial u}{\partial x}\right)^2$$

For a traveling wave $u = A\cos(kx - \omega t)$, the kinetic and potential energy densities are equal at every point (equipartition), and the energy flows at the wave speed:

Energy Transport

$$\langle u_{\text{total}} \rangle = \frac{1}{2}\mu\omega^2 A^2$$$$\text{Power} = \langle u_{\text{total}} \rangle \cdot v = \frac{1}{2}\mu\omega^2 A^2 v$$

The intensity (power per unit area for 3D waves) scales as the square of the amplitude. This fundamental relationship holds for all types of waves: mechanical, electromagnetic, and quantum.

3.6 Applications

Phonons in Crystals

Atoms in a crystal lattice interact through interatomic forces, forming a system of coupled oscillators. The quantized normal modes are called phonons. The dispersion relation determines thermal conductivity, specific heat (Debye model), and electron-phonon interactions in superconductors.

Musical Instruments

A guitar string is the continuum limit of coupled oscillators. The normal modes are harmonics at frequencies $f_n = n f_1$. The timbre of an instrument depends on which harmonics are excited and their relative amplitudes.

Coupled Pendulums

Huygens observed in 1665 that two pendulum clocks mounted on the same beam would synchronize their oscillations. This is one of the earliest documented examples of coupled oscillator synchronization, now studied in nonlinear dynamics.

Neutrino Oscillations

The mass eigenstates of neutrinos differ from the flavor eigenstates, creating a coupled system. As neutrinos propagate, they oscillate between flavors (electron, muon, tau) in exact analogy with beating in coupled oscillators.

3.6b Fourier Analysis and Normal Mode Decomposition

The decomposition of motion into normal modes is the dynamical version of Fourier analysis. Any initial configuration of the coupled system can be expressed as a superposition of normal modes:

$$x_j(t) = \sum_{n=1}^{N} A_n \sin\!\left(\frac{nj\pi}{N+1}\right)\cos(\omega_n t + \phi_n)$$

The coefficients $A_n$ and $\phi_n$ are determined by the initial conditions through the orthogonality of the eigenvectors. This decomposition is exact for linear systems and is the foundation of:

Fourier Series

In the continuum limit, the sum becomes a Fourier series. Any periodic function can be expressed as a sum of sines and cosines — the eigenfunctions of the wave equation with periodic boundary conditions.

Spectral Analysis

Measuring the amplitude of each normal mode in the response gives the frequency spectrum. This is the physical basis of Fourier spectroscopy, signal processing, and quantum mechanics (energy eigenstates).

The power of normal mode analysis extends far beyond mechanics: in quantum mechanics, the energy eigenstates play the same role, and the time evolution of any state is determined by its decomposition into energy eigenstates. In quantum field theory, normal modes of the field correspond to particle states.

3.6c Worked Example: Three Coupled Pendulums

Problem: Three identical pendulums of length $\ell$ and mass$m$ are coupled by springs of constant $\kappa$ between adjacent pairs. The walls are connected to the end pendulums by the same springs. Find the normal mode frequencies and mode shapes.

Solution

Using the formula for fixed-endpoint normal modes with $N = 3$:

$$\omega_n^2 = \omega_0^2 + \frac{4\kappa}{m}\sin^2\!\left(\frac{n\pi}{8}\right) \quad n = 1, 2, 3$$

where $\omega_0 = \sqrt{g/\ell}$ is the pendulum frequency. The three mode shapes are:

Mode 1 ($n=1$): All three move in the same direction with the middle pendulum having the largest amplitude:$(1, \sqrt{2}, 1)/2$

Mode 2 ($n=2$): The middle pendulum stays still while the outer two move in opposite directions:$(1, 0, -1)/\sqrt{2}$

Mode 3 ($n=3$): The outer pendulums move together while the middle one moves in the opposite direction:$(1, -\sqrt{2}, 1)/2$

3.7 Historical Context

Daniel Bernoulli (1753): Was the first to propose that any vibration of a string can be decomposed into a sum of normal modes (sinusoidal standing waves). This "superposition principle" was initially controversial but was later validated by Fourier's work.

Jean le Rond d'Alembert (1747): Derived the one-dimensional wave equation and found its general solution in terms of traveling waves. His solution $u = f(x-vt) + g(x+vt)$ showed that any wave shape can propagate without distortion on a non-dispersive string.

Joseph Fourier (1822): Proved that any periodic function can be expressed as a sum of sines and cosines. This Fourier analysis provides the mathematical foundation for decomposing arbitrary oscillations into normal modes.

Max Born and Theodore von Karman (1912): Applied the coupled oscillator model to crystal lattices, deriving the dispersion relation for phonons. Their work laid the foundation for solid-state physics and our understanding of thermal properties of materials.

3.8 Python Simulation

This simulation visualizes beating in two coupled oscillators, the normal modes of N coupled oscillators, the dispersion relation, and the continuum limit.

Coupled Oscillators: Beating, Normal Modes, Dispersion, and Wave Equation

Python

script.py124 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from scipy.integrate import odeint

# ============================================================
# Coupled Oscillators: Normal Modes, Beating, Dispersion
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Coupled Oscillators', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: Beating in two coupled oscillators ---
ax1 = axes[0, 0]
omega0 = 2 * np.pi
kappa_over_m = 0.3 * omega0**2
omega_plus = omega0
omega_minus = np.sqrt(omega0**2 + 2 * kappa_over_m)
t = np.linspace(0, 10, 3000)

A = 1.0
x1 = A * np.cos((omega_minus - omega_plus) / 2 * t) * np.cos((omega_minus + omega_plus) / 2 * t)
x2 = A * np.sin((omega_minus - omega_plus) / 2 * t) * np.sin((omega_minus + omega_plus) / 2 * t)

ax1.plot(t, x1, color='#22d3ee', linewidth=1.5, label='Mass 1', alpha=0.9)
ax1.plot(t, x2, color='#f97316', linewidth=1.5, label='Mass 2', alpha=0.9)
ax1.set_xlabel('Time (s)')
ax1.set_ylabel('Displacement')
ax1.set_title('Beating: Energy Transfer Between Masses')
ax1.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Normal modes of N=10 chain ---
ax2 = axes[0, 1]
N = 10
j_vals = np.arange(1, N + 1)

for n, color in [(1, '#22d3ee'), (2, '#f97316'), (3, '#a855f7'), (5, '#4ade80')]:
    mode = np.sin(n * j_vals * np.pi / (N + 1))
    mode = mode / np.max(np.abs(mode))
    ax2.plot(j_vals, mode, 'o-', color=color, linewidth=1.5, markersize=5, label='Mode n=' + str(n))

ax2.axhline(y=0, color='white', linewidth=0.3, alpha=0.3)
ax2.set_xlabel('Mass index j')
ax2.set_ylabel('Amplitude')
ax2.set_title('Normal Modes of 10-Mass Chain')
ax2.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Dispersion relation ---
ax3 = axes[1, 0]
ka = np.linspace(-np.pi, np.pi, 500)
omega_max = 2.0  # 2*sqrt(kappa/m) normalized

# Monatomic chain
omega_mono = omega_max * np.abs(np.sin(ka / 2))
ax3.plot(ka / np.pi, omega_mono, color='#22d3ee', linewidth=2.5, label='Monatomic chain')

# Linear (continuum) approximation
omega_linear = omega_max / 2 * np.abs(ka)
mask = np.abs(ka) < 1.5
ax3.plot(ka[mask] / np.pi, omega_linear[mask], color='#f97316', linewidth=2, linestyle='--',
         label='Continuum limit (linear)')

# Diatomic chain (for comparison)
m1, m2 = 1.0, 2.0
kappa_d = 1.0
det_term = np.sqrt((1/m1 + 1/m2)**2 - 4 * np.sin(ka/2)**2 / (m1 * m2))
omega_optical = np.sqrt(kappa_d * ((1/m1 + 1/m2) + det_term))
omega_acoustic = np.sqrt(kappa_d * ((1/m1 + 1/m2) - det_term))
ax3.plot(ka / np.pi, omega_acoustic, color='#a855f7', linewidth=1.5, alpha=0.7, label='Diatomic: acoustic')
ax3.plot(ka / np.pi, omega_optical, color='#4ade80', linewidth=1.5, alpha=0.7, label='Diatomic: optical')

ax3.set_xlabel('ka / pi')
ax3.set_ylabel('omega (normalized)')
ax3.set_title('Dispersion Relations')
ax3.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Wave equation solution (traveling pulse) ---
ax4 = axes[1, 1]
x_arr = np.linspace(0, 10, 500)
v = 1.0

for t_val, alpha_val in [(0, 1.0), (1.5, 0.7), (3.0, 0.5), (4.5, 0.3)]:
    pulse = np.exp(-2 * (x_arr - 2 - v * t_val)**2)
    ax4.plot(x_arr, pulse + t_val * 0.3, color='#22d3ee', linewidth=2, alpha=alpha_val)
    ax4.text(9, t_val * 0.3 + 0.05, 't = ' + str(t_val) + 's', color='white', fontsize=8)

ax4.set_xlabel('Position x')
ax4.set_ylabel('Displacement (offset)')
ax4.set_title('Traveling Pulse: u = f(x - vt)')
ax4.set_ylim(-0.2, 2.5)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("COUPLED OSCILLATORS: RESULTS")
print("=" * 60)
print("\n--- Two Coupled Oscillators ---")
print("  omega_+ = " + str(round(omega_plus, 4)) + " rad/s (symmetric mode)")
print("  omega_- = " + str(round(omega_minus, 4)) + " rad/s (antisymmetric mode)")
print("  Beat frequency = " + str(round(omega_minus - omega_plus, 4)) + " rad/s")
print("  Beat period = " + str(round(2 * np.pi / (omega_minus - omega_plus), 4)) + " s")
print("\n--- N-Mass Chain Normal Modes ---")
for n in range(1, 6):
    omega_n = 2 * np.sin(n * np.pi / (2 * (N + 1)))
    print("  Mode " + str(n) + ": omega/omega_max = " + str(round(omega_n, 4)))
print("\n--- Continuum Limit ---")
print("  Wave speed v_s = a * sqrt(kappa/m)")
print("  Wave equation: d2u/dt2 = v^2 d2u/dx2")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary

Core Equations

Normal modes: $\omega_\pm = \sqrt{(k + \kappa \pm \kappa)/m}$
Beat frequency: $\omega_{\text{beat}} = |\omega_- - \omega_+|$
Dispersion: $\omega = 2\sqrt{\kappa/m}|\sin(ka/2)|$
Wave equation: $\partial_t^2 u = v^2 \partial_x^2 u$

Key Insights

Normal modes decouple coupled equations
Beating = superposition of close frequencies
Dispersion: $\omega(k)$ is generally nonlinear
Continuum limit: coupled oscillators become waves
Matrix eigenvalue problem for N oscillators

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