The Wave Equation — Mathematics Knowledge System

Overview

The wave equation, uₜₜ = c²u_xx, is the canonical hyperbolic PDE and describes phenomena where disturbances propagate at a finite speed c without dissipating — vibrating strings, sound waves, electromagnetic waves, and (in higher dimensions) light and seismic waves. Unlike the heat equation, solutions of the wave equation do not smooth out or decay: an initial disturbance splits and travels outward, preserving its shape (in one dimension) as it moves. The general solution on an infinite domain is d'Alembert's formula, a sum of two traveling waves moving in opposite directions; on a finite domain with boundary conditions, solutions instead decompose into a Fourier series of standing-wave normal modes with frequencies fixed by the boundary conditions.

Intuition

Pluck a guitar string in the middle: the displacement you created doesn't sit still or smear out like heat — it splits into two identical, smaller pulses that race off toward each end at a fixed speed c, reflect, and keep bouncing back and forth, forever (with no friction). D'Alembert's formula makes this rigorous for an infinite string: the initial shape f(x) literally decomposes into a left-moving copy and a right-moving copy, each just riding along at speed c unchanged. On a finite string pinned at both ends, only wave shapes that fit an integer number of half-wavelengths between the pins can persist — these are the standing-wave normal modes, the same math that gives a guitar string its fundamental pitch and overtones.

Formal Definition

Definition

The one-dimensional wave equation, its d'Alembert traveling-wave solution on the infinite line, and its standing-wave (separated) solution on a finite string of length L:

u_{tt} = c^2 u_{xx}

One-dimensional wave equation

u(x,t) = \tfrac{1}{2}\big[f(x-ct)+f(x+ct)\big] + \tfrac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds

d'Alembert's solution, u(x,0)=f(x), u_t(x,0)=g(x)

u_n(x,t) = \sin\!\left(\frac{n\pi x}{L}\right)\cos\!\left(\frac{n\pi c t}{L}\right), \quad n=1,2,3,\ldots

Standing-wave normal modes, fixed ends u(0,t)=u(L,t)=0

Worked Examples

With g=0, the integral term vanishes and only the two traveling copies of f remain.
$u(x,t) = \tfrac{1}{2}[f(x-2t) + f(x+2t)], \quad f(x)=\sin x$
Substitute f = sin.
$u(x,t) = \tfrac{1}{2}[\sin(x-2t) + \sin(x+2t)]$
Apply the sine addition formulas to simplify into a single product form.
$u(x,t) = \sin x \cos 2t$

Answer: u(x,t) = sin x cos 2t (verify: u_tt = -4 sin x cos 2t; u_xx = -sin x cos 2t; c²u_xx = 4(-sin x cos 2t) = -4 sin x cos 2t = u_tt ✓; u(x,0)=sin x·1=sin x ✓; u_t(x,0) = sin x·(-2sin0)=0 ✓).

Practice Problems

Difficulty 8/10

Use d'Alembert's formula to solve u_tt = 9u_xx with u(x,0)=cos x, u_t(x,0)=0.

Difficulty 7/10

For a string fixed at x=0 and x=L=2π released from rest at u(x,0)=5sin(3x/2)... actually use u(x,0)=5sin(x) with c=1, L=π. Find u(x,t).

Difficulty 8/10

A seismic P-wave travels through rock at c=6 km/s and is modeled 1-D as u_tt=36u_xx. If a disturbance starts as a localized pulse f(x) at t=0 with zero initial velocity, how far has each half of the pulse traveled after 10 seconds?

Quiz

Summary

The wave equation u_tt = c²u_xx is hyperbolic: disturbances propagate at finite speed c without decaying.
On an infinite domain, d'Alembert's formula writes the solution as two traveling waves moving in opposite directions.
On a finite domain with fixed ends, solutions decompose into standing-wave normal modes sin(nπx/L)cos(nπct/L), the basis of vibrating-string harmonics.

References

WebsiteWikipedia — Wave equation