3.3 d’Alembert’s solution

The 1-D wave equation $\partial_t^2 y = c^2 \partial_x^2 y$ has a remarkable property: its general solution can be written down explicitly, without separation of variables, without Fourier series, without any further input. This was Jean d’Alembert’s 1747 result, and it remains one of the cleanest exact solutions in mathematical physics.

⏳ The history — d'Alembert and the Vibrating-String controversy

In 1747 Jean le Rond d’Alembert, then 29 years old, published a paper Recherches sur la courbe que forme une corde tendue mise en vibration in the Berlin Academy’s proceedings (d’Alembert 1747). It contained the general solution $y(x, t) = F(x - ct) + G(x + ct)$ to the 1-D wave equation he had derived for a vibrating string. The result is the same formula we use today.

What followed was one of the great mathematical controversies of the 18th century. Euler argued that $F$ and $G$ could be any functions — including those with corners (e.g., the initial shape of a plucked string, which has a sharp peak). D’Alembert insisted they had to be analytic, drawn from the class of well-behaved functions Newton and Leibniz had developed calculus for. Daniel Bernoulli proposed yet a third view: any vibration is a sum of sinusoidal modes — what we now call a Fourier series.

The dispute lasted decades. It was only resolved in 1822 by Fourier (Fourier 1822), whose work on heat flow showed that arbitrary functions could be expanded as trigonometric series, vindicating Bernoulli and forcing a redefinition of what “function” even meant. The controversy is the origin of modern analysis.

Read the original: Recherches sur la courbe que forme une corde tendue mise en vibration (Jean le Rond d'Alembert, 1747) — French

The statement

Every solution of $\partial_t^2 y = c^2 \partial_x^2 y$ has the form

\boxed{\;\; y(x, t) \;=\; F(x - c t) \;+\; G(x + c t).\;\;}

$F$ and $G$ are arbitrary twice-differentiable functions, fixed by initial conditions.

$F(x - c t)$ is a right-going wave: at time $t = 0$ its shape is $F(x)$ ; at time $t$ the same shape has shifted to the right by $c t$ . $G(x + c t)$ is left-going: shifted left by $c t$ . The general solution is the superposition of one right-going and one left-going profile.

▶ Direct verification: any

F(x - ct)

solves the wave equation Derivation

Compute partial derivatives of $y = F(\xi)$ with $\xi = x - ct$ :

\partial_t y = F'(\xi) \cdot (-c) = -c F'(\xi),

\partial_t^2 y = -c \cdot F''(\xi) \cdot (-c) = c^2 F''(\xi).

\partial_x y = F'(\xi), \qquad \partial_x^2 y = F''(\xi).

So $\partial_t^2 y = c^2 \partial_x^2 y$ — verified. Identical computation for $G(x + ct)$ with $\eta = x + ct$ . By linearity, the sum $F(x - ct) + G(x + ct)$ also solves the equation.

▶ Why these are the *only* solutions: a change of variables Derivation

Introduce new coordinates $\xi = x - ct$ and $\eta = x + ct$ . Then

\partial_x = \partial_\xi + \partial_\eta, \qquad \partial_t = -c \partial_\xi + c \partial_\eta,

\partial_t^2 - c^2 \partial_x^2 = (c^2 \partial_\xi^2 - 2 c^2 \partial_\xi \partial_\eta + c^2 \partial_\eta^2) - c^2(\partial_\xi^2 + 2\partial_\xi\partial_\eta + \partial_\eta^2)

= -4 c^2 \partial_\xi \partial_\eta.

The wave equation becomes $\partial_\xi \partial_\eta y = 0$ . Integrating once in $\xi$ : $\partial_\eta y = G'(\eta)$ . Integrating again in $\eta$ : $y = G(\eta) + F(\xi)$ . Both $F$ and $G$ are arbitrary functions of one variable, fixed by initial conditions.

Initial conditions

Given initial position $y(x, 0) = f(x)$ and initial velocity $\partial_t y(x, 0) = g(x)$ , d’Alembert’s formula reads

y(x, t) \;=\; \tfrac12\big[\, f(x - c t) + f(x + c t)\, \big] \;+\; \frac{1}{2 c} \int_{x - c t}^{x + c t} g(\xi)\, d\xi.

A pulse with no initial velocity ( $g \equiv 0$ ) splits into two half-amplitude copies — one travelling left, one travelling right — and from then on each propagates independently.

What the formula tells us

Finite propagation speed. The value of $y$ at $(x, t)$ depends only on initial data inside the domain of dependence $[x - c t, x + c t]$ . Information cannot reach $(x, t)$ from outside that interval. The wave equation respects $c$ as a strict speed limit.
Two characteristics. The lines $x - c t = \text{const}$ (right-going) and $x + c t = \text{const}$ (left-going) are the characteristic curves of the equation. Information propagates along them.
No spreading. A 1-D wave pulse keeps its shape exactly as it travels. The 1-D wave equation is non-dispersive. (This will not be true in higher dimensions, where geometrical spreading attenuates a pulse, or in dispersive media where different frequency components travel at different speeds.)

Harmonic traveling waves

The general solution admits any profile $F$ , but one special case is the building block for everything that follows: a sinusoid. A harmonic traveling wave is

y(x, t) \;=\; A \cos\big(k(x - c t)\big) \;=\; A \cos(k x - \omega t), \qquad \omega = c k.

where

$y(x, t)$: transverse displacement of the string m
$A$: wave amplitude (peak displacement) m
$x$: position along the string m
$t$: time s
$k$: wavenumber, $k = 2\pi/\lambda$ rad/m
$\omega$: angular frequency, $\omega = 2\pi f$ rad/s
$c$: wave speed set by the medium m/s

It is doubly periodic. In space it repeats every wavelength $\lambda = 2\pi/k$ ; in time it repeats every period $T = 2\pi/\omega = 1/f$ , where $f$ is the frequency (cycles per second). In exactly one period the profile advances exactly one wavelength, so the propagation speed is

c \;=\; \frac{\lambda}{T} \;=\; \lambda f.

This is the fundamental kinematic relation of every harmonic wave. The medium fixes $c$ ; wavelength and frequency are then locked together — raise $f$ and $\lambda$ must shrink in proportion, their product pinned at $c$ .

frequency f = 500 Hz wave speed c = 343 m/s

wavelength λ = c/f = 69 cm period T = 1/f = 2.00 ms

The speed c and frequency f are set independently; the wavelength λ = c/f follows. Raise the frequency and the bands pack in (λ shrinks); raise the speed and they stretch out (λ grows). The product λf is always c. Default c = 343 m/s is air at room temperature.

Drag the two sliders independently. Frequency $f$ and speed $c$ are free; the wavelength $\lambda = c/f$ is whatever they leave behind, and the measured span at the top tracks it. The acoustic plane wave of 5.1 is exactly this relation instantiated for air ( $c \approx 343$ m/s). And sinusoids are not merely one example among many: because the equation is linear and any profile can be built from sinusoids by Fourier analysis (refresher: Fourier series →), understanding the harmonic wave is understanding all of them.

Why this matters

D’Alembert’s formula is the entire story of a 1-D wave in a uniform medium. Every other technique we use — separation of variables, Fourier series, Green’s functions — reproduces it in one form or another. It is the irreducible content of “the wave equation”.

The catch is that real wave problems include boundaries. A finite string is clamped at both ends. A tube is closed at one end. A room has six walls. The boundaries reflect waves, and the reflections build up into standing modes. Next lesson: how reflection works.