Differential equations and dynamical systems

The wave equation

A plucked string, a dropped stone in a pond, and a beam of light all obey the same partial differential equation. Two families of solutions capture everything: waves that travel, and waves that stand still and merely oscillate in place.

IntuitionA plucked string, frozen in time

Pluck a guitar string and record its height at every point, at every instant: you get a function $u(x,t)$ , the displacement at position $x$ and time $t$ . Drop a stone in a still pond and the ripples spreading outward are governed by the same kind of law, just in two spatial dimensions instead of one. In every case, the key physical fact is local: how a point accelerates depends only on how curved the wave is right there — a taut string pulls a dip back up harder the sharper the dip is. That single idea, "acceleration proportional to curvature," is the wave equation.

A ripple spreading outward, like circular waves on a pond after a stone is dropped at the center — height decays with distance while the wave pattern expands outward at constant speed.

UndergraduateFrom Newton's second law to a partial differential equation

Model an idealized string under tension $T$ with mass density $\rho$ per unit length, vibrating with small vertical displacement $u(x,t)$ . Newton's second law applied to a short segment $[x, x+dx]$ , using the small-angle approximation for the tension's vertical components, gives $\rho\, u_{tt} = T\, u_{xx}$ . Writing $c^2 = T/\rho$ (a speed, since $T$ has units of force and $\rho$ mass per length) turns this into the standard form.

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

UndergraduateTraveling waves: d'Alembert's solution

On the whole line $\mathbb{R}$ , change to characteristic coordinates $\xi = x - ct$ , $\eta = x + ct$ . The chain rule turns $u_{tt} = c^2 u_{xx}$ into the far simpler $u_{\xi\eta} = 0$ , whose general solution is $u = F(\xi) + G(\eta) = F(x-ct) + G(x+ct)$ for arbitrary (twice-differentiable) functions $F, G$ — a right-moving wave plus a left-moving wave, both traveling at speed $c$ without changing shape. Matching this to initial position $u(x,0)=\varphi(x)$ and initial velocity $u_t(x,0)=\psi(x)$ pins down $F$ and $G$ exactly.

Theorem: d'Alembert's formula

For $u_{tt} = c^2 u_{xx}$ on $\mathbb{R}\times(0,\infty)$ with $u(x,0)=\varphi(x)$ , $u_t(x,0)=\psi(x)$ , the unique solution is $u(x,t) = \tfrac{1}{2}\big(\varphi(x-ct)+\varphi(x+ct)\big) + \tfrac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)\,ds$ .

Why is it true?

Anything happening at $(x,t)$ can only depend on initial data in $[x-ct, x+ct]$ : information cannot travel faster than $c$ . This finite propagation speed is the defining feature that separates the wave equation from, say, the heat equation, where an initial disturbance is felt everywhere instantly.

Proof

Change to characteristic coordinates as above to get $u=F(x-ct)+G(x+ct)$ ; imposing $u(x,0)=\varphi(x)$ and $u_t(x,0)=\psi(x)$ gives two equations for $F$ and $G$ in terms of $\varphi$ and an antiderivative of $\psi$ , which combine into the stated formula.

Example: A plucked pulse splits in two

An infinite string starts at rest ( $\psi(x)=0$ ) with a triangular pulse $\varphi(x) = 1-|x|$ for $|x|\le1$ and $\varphi(x)=0$ otherwise. Describe $u(x,t)$ for $t>0$ .

Solution

Since $\psi=0$ , the integral term vanishes and $u(x,t) = \tfrac12\big(\varphi(x-ct)+\varphi(x+ct)\big)$ : the original pulse, at half its height, splits into two identical copies moving apart at speed $c$ , one to the right and one to the left. Wherever the two copies still overlap (for small $t$ ), the heights simply add, momentarily reconstructing part of the original shape; once $ct>1$ they are fully separated.

Example: Seismic wave arrival time and epicenter distance

A seismograph station detects a P-wave (compressional, speed $c_P = 6\text{ km/s}$ ) arriving at time $t_P = 0$ and an S-wave (shear, speed $c_S = 3.5\text{ km/s}$ ) arriving $\Delta t = 45$ seconds later. Assuming both waves are generated simultaneously at the same source and travel in straight lines, find the distance $d$ to the earthquake epicenter.

Solution

Step 1 — Set up the timing equations. The P-wave travels distance $d$ in time $d/c_P$ and the S-wave takes time $d/c_S$ . The measured lag is $\Delta t = d/c_S - d/c_P$ .

**Step 2 — Solve for $d$ .** Factoring out $d$ : $\Delta t = d\left(\tfrac{1}{c_S} - \tfrac{1}{c_P}\right)$ , so $d = \dfrac{\Delta t}{\tfrac{1}{c_S}-\tfrac{1}{c_P}}$ .

Step 3 — Substitute values. $\tfrac{1}{c_S}-\tfrac{1}{c_P} = \tfrac{1}{3.5}-\tfrac{1}{6} \approx 0.2857 - 0.1667 = 0.1190\text{ s/km}$ , so $d = \dfrac{45}{0.1190} \approx 378\text{ km}$ . This is the standard seismological S-P method used to locate earthquakes with a single station.

UndergraduateStanding waves and Fourier series on a finite string

Now pin the string down at both ends: $u(0,t)=u(L,t)=0$ for a string of length $L$ . Separating variables, $u(x,t)=X(x)T(t)$ , forces $X$ to satisfy $X''+\lambda X=0$ with $X(0)=X(L)=0$ — an eigenvalue problem whose solutions are exactly $X_n(x)=\sin(n\pi x/L)$ , $\lambda_n=(n\pi/L)^2$ , for $n=1,2,3,\dots$ . Each such $X_n$ pairs with $T_n(t)=\cos(n\pi ct/L+\text{phase})$ to give a normal mode: the whole string oscillating in place, all points crossing zero at the same instants. These are exactly the shapes seen in [Fourier series](/chuoi-fourier) — the wave equation is where that topic's sine basis was born, in Daniel Bernoulli's 1753 analysis of exactly this problem.

u(x,t) = \sum_{n=1}^{\infty} b_n \sin\!\left(\frac{n\pi x}{L}\right)\cos\!\left(\frac{n\pi c t}{L}\right), \qquad b_n = \frac{2}{L}\int_0^L \varphi(x)\sin\!\left(\frac{n\pi x}{L}\right)dx

A triangle wave (the shape of a string plucked at its midpoint) built from

5

sine harmonics — exactly the normal modes

\sin(n\pi x/L)

that the plucked string decomposes into.

AdvancedEnergy conservation and higher dimensions

The total energy $E(t) = \frac12\int_0^L \left(u_t^2 + c^2 u_x^2\right) dx$ (kinetic plus elastic potential) is conserved: differentiating under the integral, integrating by parts, and using $u_{tt}=c^2u_{xx}$ together with the fixed-end boundary conditions makes $E'(t)=0$ . This both confirms the physics (a frictionless string never runs out of energy) and gives a clean proof that d'Alembert's and the normal-mode solutions are the only solutions: if two solutions shared the same initial data, their difference would have zero energy, hence be identically zero.

Theorem: Conservation of wave energy

Let $u(x,t)$ solve the wave equation $u_{tt}=c^2 u_{xx}$ on $[0,L]$ with fixed-end boundary conditions $u(0,t)=u(L,t)=0$ . The total energy $E(t) = \tfrac{1}{2}\int_0^L \left(u_t^2 + c^2 u_x^2\right)dx$ is constant: $E'(t)=0$ for all $t>0$ .

Why is it true?

Energy has two parts: kinetic $\tfrac12\int u_t^2\,dx$ (mass moving) and elastic potential $\tfrac12 c^2\int u_x^2\,dx$ (string stretched). The wave equation turns kinetic energy into elastic energy and back, at the same rate, keeping the total constant — exactly as a pendulum trades kinetic and potential energy without loss.

Proof

Differentiate $E(t)$ under the integral sign: $E'(t) = \int_0^L \left(u_t u_{tt} + c^2 u_x u_{xt}\right)dx$ .

Substitute $u_{tt}=c^2 u_{xx}$ from the wave equation to get $E'(t) = \int_0^L \left(c^2 u_t u_{xx} + c^2 u_x u_{xt}\right)dx = c^2\int_0^L \left(u_t u_{xx} + u_x u_{xt}\right)dx$ . Recognize the integrand as the derivative $\tfrac{\partial}{\partial x}(u_t u_x)$ , so $E'(t) = c^2\Big[u_t u_x\Big]_0^L$ .

Apply the fixed-end boundary conditions $u(0,t)=u(L,t)=0$ , which force $u_t(0,t)=u_t(L,t)=0$ as well (differentiating the boundary conditions with respect to $t$ ). Therefore $E'(t) = c^2\big(u_t(L,t)u_x(L,t) - u_t(0,t)u_x(0,t)\big) = 0$ , so $E(t)=E(0)$ is constant for all $t\ge 0$ .

In two spatial dimensions, a rectangular drum membrane pinned at its edges has normal modes $\sin(mx)\sin(ny)$ ; the picture below shows the related pattern $\sin(x)\cos(y)$ , a standing wave rippling across a surface rather than along a line. Higher dimensions also bring a striking qualitative difference known as Huygens' principle: in odd spatial dimensions $3, 5, 7,\dots$ , a localized disturbance produces a sharp wavefront with silence afterward (this is why sound is intelligible), while in even dimensions $2, 4, \dots$ (and in dimension $1$ ) a disturbance leaves a persistent "wake" that never fully dies out.

A standing-wave mode

z=\sin x\cos y

on a 2D membrane: a snapshot at one instant, with the whole checkerboard pattern of peaks and troughs rising and falling in place rather than traveling sideways.

Historical note

The wave equation was the subject of one of mathematics' great 18th-century controversies. Jean le Rond d'Alembert derived the traveling-wave solution in 1747. Leonhard Euler agreed with the mathematics in 1748 but allowed more general (even non-differentiable) initial shapes than d'Alembert was willing to accept, since a physically plucked string has a corner at the pluck point. Daniel Bernoulli then claimed in 1753 that every solution could instead be written as an infinite sum of the sine normal modes $\sin(n\pi x/L)$ — a claim both d'Alembert and Euler found hard to believe, since it seemed to imply that a shape with corners could be a sum of perfectly smooth sine curves. The dispute was only resolved decades later, once Joseph Fourier's work on heat (1807, 1822) and the 19th-century theory of convergence made precise what "equals" should mean for such an infinite sum.

Leonhard Euler

The wave equation $u_{tt} = c^2 u_{xx}$ needs how many pieces of initial data to determine a unique solution?

In d'Alembert's formula, the value at $(x,t)$ depends only on the initial data on which interval?

For a string of length $L$ fixed at both ends, the normal modes (standing waves) have spatial shape

d'Alembert's formula $u(x,t)=\tfrac12[\varphi(x-ct)+\varphi(x+ct)]+\tfrac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)\,ds$ reduces to just $u(x,t)=\tfrac12[\varphi(x-ct)+\varphi(x+ct)]$ precisely when

References

Lawrence C. Evans (2010). Partial Differential Equations · DOI:10.1090/gsm/019
Walter A. Strauss (2008). Partial Differential Equations: An Introduction
Jean le Rond d'Alembert (1747). Recherches sur la courbe que forme une corde tendue mise en vibration