The Wave Equation

Part 14, Chapter 14: Differential Equations

Learning objectives

State the wave equation and identify the wave speed $c$
Write d'Alembert's solution on the infinite line as two travelling waves plus an integral over initial velocity
Solve the finite-string problem by separation of variables and recognize the resulting standing waves
Contrast wave propagation (finite speed, energy conservation) with heat diffusion (infinite speed, smoothing)

The wave equation governs everything from the vibrating guitar string to electromagnetic radiation to seismic shock waves. Its solutions split cleanly into travelling waves (on the line) or standing waves (on a finite string), and unlike the heat equation it preserves energy, nothing decays, nothing smooths. The two pillars are d'Alembert's formula (which gives the explicit solution on $\mathbb{R}$ as two waves travelling in opposite directions) and separation of variables (which on a finite string produces the normal modes that are the vibrating-string Fourier basis).

The equation

The one-dimensional wave equation:

$\dfrac{\partial^2 u}{\partial t^2} = c^2 \dfrac{\partial^2 u}{\partial x^2}$

$c > 0$ is the wave speed, the speed at which disturbances propagate. The PDE is hyperbolic, linear, homogeneous, and second order in both $x$ and $t$ . Because $t$ enters at second order, you need two initial conditions: $u(x, 0) = f(x)$ (initial displacement) and $u_t(x, 0) = g(x)$ t(x,0)=g(x) (initial velocity).

D'Alembert's solution on the infinite line

On all of $\mathbb{R}$ , the wave equation has a closed-form solution. Substituting $\xi = x - ct$ and $\eta = x + ct$ turns $u_{tt} - c^2 u_{xx} = 0$ 0 into $u_{\xi \eta} = 0$ xieta=0, whose general solution is

$u(x, t) = F(x - ct) + G(x + ct)$

for arbitrary functions $F, G$ . The first term $F(x - ct)$ is a wave travelling rightward at speed $c$ ; the second is a wave travelling leftward at speed $c$ . Matching the initial conditions $u(x, 0) = f(x)$ and $u_t(x, 0) = g(x)$ t(x,0)=g(x) gives d'Alembert's formula:

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

If the initial velocity $g \equiv 0$ , this reduces to two half-amplitude copies of the initial profile travelling in opposite directions, an exact bilateral split.

The finite-string problem: standing waves

For a string of length $L$ fixed at both ends, $u(0, t) = u(L, t) = 0$ , separation of variables produces the same spatial eigenfunctions as the heat equation, $\sin(n\pi x / L)$ , but the time ODE is now second order and oscillatory:

$T_n''(t) + (cn\pi/L)^2 T_n(t) = 0$ 0

so $T_n(t) = A_n \cos(\omega_n t) + B_n \sin(\omega_n t)$ n(t)=Ancos(omegant)+Bnsin(omegant) with natural frequencies $\omega_n = cn\pi/L$ n=cnpi/L. The general solution is

$u(x, t) = \displaystyle\sum_{n=1}^{\infty} \bigl[A_n \cos(\omega_n t) + B_n \sin(\omega_n t)\bigr] \sin(n\pi x / L)$

Each term $\sin(n\pi x / L) \cos(\omega_n t)$ nt) is a standing wave: a fixed spatial shape multiplied by a time-varying amplitude. The points where $\sin(n\pi x / L) = 0$ are nodes, they stay still. The maxima between nodes are antinodes, oscillating at the natural frequency.

Plot the first standing wave $u(x, t) = \sin(\pi x / L) \cos(c\pi t / L)$ for several values of $t$ . The shape stays $\sin(\pi x / L)$ ; only the overall amplitude oscillates. Now plot the second mode $\sin(2\pi x / L) \cos(2c\pi t / L)$ : notice the new node at $x = L/2$ , fixed forever in time.

Energy conservation

The total mechanical energy of the vibrating string is

$E(t) = \tfrac{1}{2} \displaystyle\int_0^L \bigl[(u_t)^2 + c^2 (u_x)^2\bigr] \, dx$

Differentiating $E$ in time and using $u_{tt} = c^2 u_{xx}$ with integration by parts shows $dE/dt = 0$ : energy is conserved exactly. This is in sharp contrast to the heat equation, where energy (in the sense of $L^2$ norm) decays exponentially to zero. The wave equation is time-reversible; the heat equation is not.

Where this shows up

Acoustics and musical instruments: A vibrating guitar string, organ pipe, or drum head satisfies the wave equation. The natural frequencies $\omega_n = cn\pi/L$ n=cnpi/L are exactly the harmonics you hear, tuning a guitar string changes $c$ (tension over linear density), which scales the entire harmonic series.
Electromagnetism: Maxwell's equations in vacuum reduce to the 3D wave equation $\nabla^2 E = (1/c^2) E_{tt}$ tt with $c = 1/\sqrt{\varepsilon_0 \mu_0}$ , the speed of light. Every light wave, radio wave, and microwave you have ever experienced satisfies this PDE.
Seismic waves: P-waves and S-waves through the Earth satisfy wave equations with different speeds. The difference in arrival times between P and S at a seismograph is what lets seismologists triangulate earthquake epicenters.
Quantum field theory: Free particles in QFT (Klein-Gordon, Dirac) satisfy wave equations or their relativistic generalisations. The mass term adds $m^2 u$ to the right side, but the propagation-at-finite-speed structure is identical.

Pause and think: Where does the factor $1/(2c)$ in d'Alembert's formula come from? Hint: differentiate $u(x, t) = F(x - ct) + G(x + ct)$ in $t$ and set $t = 0$ to recover the initial-velocity condition.

Try it

Identify the wave speed in $u_{tt} = 9 u_{xx}$ .
A string of length $L = 1$ has wave speed $c = 2$ . Compute the first three natural frequencies $\omega_n$ n.
Use d'Alembert with $f(x) = e^{-x^2}$ and $g \equiv 0$ , $c = 1$ . Sketch the solution at $t = 0, t = 1, t = 2$ .
For the $n$ -th standing-wave mode $u_n(x, t) = \sin(n\pi x / L) \cos(\omega_n t)$ , list the positions of the nodes inside $[0, L]$ .
Show that the energy $E(t)$ is conserved: differentiate, substitute $u_{tt} = c^2 u_{xx}$ , and integrate by parts using the fixed boundary conditions.

A trap to watch for

The wave equation has finite propagation speed $c$ : an initial disturbance at $x_0$ can only affect points $x$ with $|x - x_0| \leq ct$ by time $t$ . The triangular region $\{(x, t) : t \geq 0, |x - x_0| \leq ct\}$ is the future light cone of $(x_0, 0)$ ; outside this cone $u$ is unaffected by the initial datum at $x_0$ . Students who pattern-match the heat equation's "anything affects everything instantly" intuition onto the wave equation will get the qualitative behavior wrong. Heat: instant smearing. Wave: travelling pulse.

What you now know

You can write down d'Alembert's solution to the wave equation on $\mathbb{R}$ , solve the finite-string problem as a Fourier series of standing waves with natural frequencies $cn\pi/L$ , identify nodes and antinodes, and prove that total mechanical energy is exactly conserved. With this chapter complete, you have the basic Fourier-and-PDE toolkit that underlies all of mathematical physics, sufficient to read advanced texts on quantum mechanics, fluid dynamics, signal processing, or numerical methods.

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References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 14.

Evans, L. C. (2010). Partial Differential Equations (2nd ed.). AMS, ch. 2 (wave equation, d'Alembert).

Strauss, W. A. (2007). Partial Differential Equations: An Introduction (2nd ed.). Wiley, ch. 2 (wave equation).

John, F. (1991). Partial Differential Equations (4th ed.). Springer, ch. 2-3.

Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). Wiley, ch. 6-7 (electromagnetic waves).

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