Understanding the Wave Equation and Normal Modes: From Coupled Oscillators to Continuous Media (MIT OCW AO3)

Overview

This MIT OpenCourseWare lecture examines how a wave equation arises from an infinite chain of coupled oscillators and how to analyze the resulting normal modes in the continuous limit. The instructor demonstrates the separation of variables, revealing that the time and space parts of the solution must separate into constants, yielding harmonic time dependence and sinusoidal spatial shapes. The talk then connects boundary conditions, wave numbers, and frequencies via a dispersion relation, and shows how finite systems are obtained by enforcing boundary conditions that quantize the modes. Fourier series is introduced as a practical tool to determine mode amplitudes from initial conditions, with visualizations of mode shapes on a string and discussions of resonance and superposition.

Introduction

The lecture begins by revisiting how a discrete chain of masses connected by springs can be described by an equivalent wave equation in the continuum limit. The separation of variables is introduced as a powerful method to solve the infinite set of coupled equations, with the goal of finding normal modes that describe collective motions of the system.

From Discrete to Continuous: The Wave Equation

The instructor explains the transition from a discrete system with spacing A to a continuous field si(x, t). In the continuous description, the matrix of couplings becomes a differential operator, specifically a second derivative with respect to position, leading to the familiar wave equation. The variables separate into a spatial function A(x) and a temporal function B(t).

Separation of Variables and Time-Space Factorization

By dividing the equation into a product AB, and equating the time-dependent and space-dependent parts to a constant, the course demonstrates that the left-hand side depends only on time while the right-hand side depends only on space. This constant is denoted in various ways, and its introduction yields two ordinary differential equations, one for the time part and one for the spatial part, each solvable by standard methods.

Time Dependence and Spatial Shape

The time component solves as a harmonic oscillator with angular frequency, while the spatial component yields sinusoidal shapes. The normal modes are thus products of a time-dependent harmonic factor and a spatial sine function, with the overall frequency determined by the dispersion relation that links frequency to wave number.

Boundary Conditions and Mode Quantization

Introducing a concrete boundary scenario, such as a string fixed at one end and attached to a movable ring at the other, shows how boundary conditions fix the phase and allowed wave numbers. The left boundary fixes the phase to yield alpha = 0 while the right boundary imposes a cosine condition at the end of the string, leading to a discrete set of allowable wave numbers K_N. This quantization explains how a continuous system produces a finite set of normal modes when boundaries are present.

Normal Modes, Dispersion, and Visualization

The normal modes are depicted as shapes that scale in time with a common frequency. The dispersion relation, omega_N = v_p K_N, connects the mode frequency to its spatial wave number, and demonstrates how higher modes oscillate more rapidly. The speaker emphasizes that while the whole string exhibits a traveling wave shape, individual points move up and down, illustrating a resonance-like, collective motion rather than back-and-forth motion for each particle.

Determining Amplitudes with Initial Conditions and Fourier Series

Determining the amplitudes A_N requires initial conditions. The boundary conditions set the form of K_N and alpha_N, while initial shape and velocity fix the amplitudes via Fourier series. The orthogonality of sine modes is used to project the initial shape onto the basis of normal modes, isolating each A_N. A concrete example shows how a piece of the string with a given initial displacement can be decomposed into a sum over N of A_N sin(K_N X + alpha_N) cos(omega_N t) or sin(omega_N t) depending on the choice of initial conditions and the phase, enabling practical calculation of the mode content.

General Solution and Physical Insights

The final form is a linear combination of all normal modes, each with its own amplitude and phase, with the time evolution governed by the dispersion relation. The lecture highlights how the continuum limit reveals a linear, wave-like propagation with a simple yet rich structure of normal modes, and how boundary conditions and initial conditions determine which modes participate. A remark on how the observed omega versus K relation aligns with a continuum model offers insight into the internal length scales that govern wave behavior.

Takeaways

Key ideas include the wave equation as a continuum limit of coupled oscillators, separation of variables leading to time and space decoupling, boundary conditions quantizing normal modes, the dispersion relation linking frequency and wave number, and the role of Fourier series in extracting mode amplitudes from initial data. The discussion ties together resonance, superposition, and the practical determination of normal modes in both infinite and finite systems.