Waves & Oscillations · Upper Division

The Wave Equation

The wave equation is the PDE that governs sound, light, vibrating strings, water waves, and quantum probability amplitudes. Its solutions — d'Alembert's formula, standing waves, and dispersive packets — appear across all of physics.

Prerequisites

Wave properties (Ch. 8) \cdot Fourier analysis (Ch. F) \cdot Partial derivatives

Learning Goals

Derive the 1D wave equation from Newton's second law applied to a vibrating string.
Apply d'Alembert's formula to find the motion of a plucked string from initial conditions.
Use separation of variables to find normal modes and their frequencies for a fixed string.
Distinguish phase velocity and group velocity and compute them from a dispersion relation.
Calculate the power transmitted by a traveling wave from amplitude and wave parameters.

WE.1 Derivation from a Vibrating String

Consider a taut string with tension T and linear mass density μ. For small transverse displacements y(x, t), Newton's second law applied to a small element gives:

\frac{\partial^2y}{\partial t^2}=v^2\frac{\partial^2y}{\partial x^2} \qquad v=\sqrt{\frac{T}{\mu}} \qquad \text{(1D wave equation)}

(WE.1)

This is the canonical form of the wave equation with wave speed v. The same equation governs longitudinal sound waves (v = √(B/ρ), B = bulk modulus), EM waves (v = 1/√(με)), and quantum mechanical free particles (v → operator, giving Schrödinger).

WE.2 D'Alembert's Solution

Theorem WE.1 — D'Alembert's Formula (1747)

The general solution to the 1D wave equation with initial conditions y(x,0) = f(x) and \partialy/

\partialt(x,0) = g(x) is

y(x, t) = \frac{1}{2}[f(x-vt) + f(x+vt)] + (1/2v) \int(x-vt to x+vt) g(s) ds

The first term is the superposition of two copies of the initial shape — one traveling right at speed v, one left at speed v. Every solution is a sum of rightward and leftward traveling waves.

Example WE.1 — Plucked String

$A string of length L is plucked at its center to height h and released from rest (g = 0$ . Describe the subsequent motion.

Initial shape:

f(x) = triangular peak: 2hx/L for 0 \le x \le L/2, then 2h(L-x)/L for L/2 \le x \le L

D'Alembert:

y(x,t) = \frac{1}{2}[f(x-vt) + f(x+vt)], extended as odd periodic function

Motion:The triangle splits into two half-height triangles

that travel in opposite directions. They bounce off the fixed ends (boundary condition y=0

T = 2L/v

Fourier view:The triangular shape has Fourier series with only odd

harmonics. Each harmonic has frequency nv/(2L) — the harmonics of a guitar string. The n=1

/(2L).

WE.3 Separation of Variables and Normal Modes

For a string fixed at x=0 and x=L, try y(x,t) = X(x)T(t). Substituting:

\frac{\ddot T}{T}=v^2\frac{X''}{X}=-\omega^2 \qquad \text{(separation constant)}

(WE.2)

The spatial equation X'' + (ω/v)²X = 0 with boundary conditions X(0)=X(L)=0 gives:

X_n(x)=\sin\left(\frac{n\pi x}{L}\right) \qquad \omega_n=\frac{n\pi v}{L} \qquad n=1,2,3,\ldots

(WE.3)

These are the normal modes (standing waves). The general solution is a superposition:

y(x,t)=\sum_{n=1}^{\infty}\sin\left(\frac{n\pi x}{L}\right)\left[A_n\cos(\omega_nt)+B_n\sin(\omega_nt)\right]

(WE.4)

The coefficients Aₙ and Bₙ are determined by the initial conditions via Fourier sine series — Aₙ = (2/L)∫(0 to L) f(x) sin(nπx/L) dx.

WE.4 Dispersion Relations

Definition WE.1 — Dispersion Relation

The dispersion relation

\omega(k) relates angular frequency \omega to wavenumber k for waves in a medium. For the standard w

ve equation: \omega = vk (linear, non-dispersive). Two velocities characterize dispersive waves

:Phase velocity:

v_{p} = \omega/k — speed of a single-frequency wave crest

Group velocity:

v_{g} = d\omega/dk — speed of a wave packet (carries energy and information

When v_{p} ≠ v_{g}, the medium is

dispersive: different frequencies travel at different speeds, and pulses spread out over time.

Examples of dispersion relations:

\omega^2=v^2k^2+\omega_p^2 \qquad \text{(plasma, where }\omega_p\text{ is the plasma frequency)}

(WE.5)

\omega^2=gk+\frac{\gamma k^3}{\rho} \qquad \text{(deep water: gravity + surface tension)}

(WE.6)

\omega=\frac{\hbar k^2}{2m} \qquad \text{(non-relativistic de Broglie waves)}

(WE.7)

Example WE.2 — Group Velocity of a Wave Packet

$Two cosine waves with nearly equal frequencies: k_{1} = k-\deltak, k_{2} = k+\deltak, \omega_{1} = \omega-\delta\omega, \omega_{2} = \omega+\delta\omega$ . Find the envelope velocity.

Superpose:

y = cos(k_{1}x - \omega_{1}t) + cos(k_{2}x - \omega_{2}t

Sum-to-product:=

2 cos((k_{1}-k_{2})x/2 - (\omega_{1}-\omega_{2})t/2) \cdot cos((k_{1}+k_{2})x/2 - (\omega_{1}+\omega_{2})t/2

Result:=

2 cos(\deltak \cdot x - \delta\omega \cdot t) \cdot cos(kx - \omegat

Interpretation:

Fast oscillation at (k, \omega) — the carrier wave. Slow envelope at (\deltak, \delta\omega) — the beat. The e

nvelope travels at v_{envelope} = \delta\omega/\deltak \to d\omega/dk in the limit

Conclusion:

v_{group} = d\omega/dk — the group velocity carries the packet and, in QM, the probability densit

WE.5 Energy and Power in Waves

For a traveling wave on a string y = A sin(kx − ωt), the power transmitted past any point is:

P=\frac{1}{2}\mu\omega^2A^2v=\frac{1}{2}TkA^2\omega

(WE.8)

More usefully: P = ½μvω²A². The energy per wavelength is E_λ = P · (λ/v) = ½μωA² · λ. The energy density (energy per unit length) is u = ½μω²A² — equally divided between kinetic (½μẏ²) and potential (½T(∂y/∂x)²) at each instant, averaged over time.

Definition WE.2 — Common Traps

Boundary conditions choose the modes: the PDE alone does not decide whether sine, cosine, or mixed modes appear.
Phase velocity is not always signal speed: in dispersive media, energy and information travel with group velocity.
Fourier coefficients are set by initial conditions: normal modes are the basis, not the full answer by themselves.
Average power is cycle-averaged: instantaneous energy density oscillates between kinetic and potential forms.

Exercises — WE.1–WE.5 The Wave Equation

A guitar string is 65 cm long, under 70 N tension, with linear density \mu = 5\times10^{-4} kg/m. Fi

nd the wave speed, fundamental frequency, and first three harmonics.

m/s

Straightforward

A Gaussian pulse travels in a medium with cubic dispersion \omega = ck + \alphak^{3}. Describe qualitat

ively how the pulse shape changes over time. What is this called in fiber optics?

Intermediate

3.Find the normal modes of a 2D square membrane (fixed boundary, side L). Which modes are degenerate? Sketch the nodal patterns for the lowest modes.

Intermediate

Show that the Schrödinger equation i\hbar \partial\psi/\partialt = (-\hbar^{2}/2m)\partial^{2}\psi/\partialx^{2} has a plane wave solution. F

ind the phase and group velocities. Which matches the classical particle velocity?

Challenging

Key Takeaways

$Wave equation: \partial^{2}y/\partialt^{2} = v^{2}\partial^{2}y/\partialx^{2}. Applies to strings (v=\sqrt(T/\mu)), sound, EM, QM$ .
$D'Alembert: general solution = f(x-vt) + g(x+vt) — rightward plus leftward waves$ .
$Normal modes: sin(n\pix/L) with \omega_{n} = n\piv/L. Superpose with Fourier coefficients$ .
$Phase velocity v_{p} = \omega/k (crest speed); group velocity v_{g} = d\omega/dk (packet speed, carries e$ nergy).
$Non-dispersive: \omega = vk \to v_{p} = v_{g}. Dispersive: v_{p} ≠ v_{g}, pulses spread$ .
$Energy density: u = \frac{1}{2}\mu\omega^{2}A^{2}, equally kinetic and potential (averaged over cycle$ .