6.2 The 1-D wave equation: d’Alembert and characteristics

The 1-D wave equation,

\frac{\partial^2 u}{\partial t^2} \;=\; c^2\, \frac{\partial^2 u}{\partial x^2},

is the simplest PDE that supports propagation. Despite being a partial differential equation, it has an explicit, closed-form general solution that can be written down without any ansatz or integration of modes — d’Alembert’s formula, derived in 1747. The structure of that formula reveals everything that matters about hyperbolic PDEs: information travels at speed $c$ , along curves called characteristics, and the value of the field at any point depends on initial data over a precisely bounded interval.

This lesson develops the formula and the geometric picture together.

The general solution

We claim that any function of the form

u(x, t) \;=\; F(x - c t) \;+\; G(x + c t)

solves the wave equation, for arbitrary twice-differentiable $F$ and $G$ . To verify, compute the partial derivatives:

\frac{\partial u}{\partial t} = -c\, F'(x - c t) + c\, G'(x + c t), \qquad \frac{\partial^2 u}{\partial t^2} = c^2\, F''(x - c t) + c^2\, G''(x + c t),

\frac{\partial u}{\partial x} = F'(x - c t) + G'(x + c t), \qquad \frac{\partial^2 u}{\partial x^2} = F''(x - c t) + G''(x + c t).

Substitute into $u_{tt} - c^2 u_{xx}$ :

u_{tt} - c^2 u_{xx} \;=\; c^2 F'' + c^2 G'' - c^2 (F'' + G'') \;=\; 0. \;\checkmark

The verification is short, but the content is large. The function $F(x - c t)$ is a profile that moves to the right at speed $c$ — at time $t$ , the argument $x - ct$ takes the value it had at $x_0 - 0 = x_0$ when $t = 0$ , provided $x = x_0 + c t$ . So a shape sitting at $x_0$ at $t = 0$ has migrated to $x_0 + c t$ at time $t$ . Likewise $G(x + c t)$ travels left at speed $c$ . The general solution is the sum of an arbitrary right-going wave and an arbitrary left-going wave. Nothing else satisfies the equation.

input frequency f

1 kHz

characteristic place: 15.2 mm from stapes
Q (fixed): 8

The interactive above shows the right-going and left-going pieces in action. Drag the initial pulse; pause and step; watch the two halves split and recombine after reflection at the boundary. The shape is preserved exactly as it propagates — a hallmark of the linear wave equation in 1-D and a feature it does not share with the heat equation or with most wave equations in higher dimensions, both of which spread or smear the profile as it evolves.

The d’Alembert formula

The general solution above contains two arbitrary functions, $F$ and $G$ . To pin them down we need two pieces of initial data — the analogue of “two free constants for a second-order ODE” from Foundations 5.1. Specifically, give

u(x, 0) \;=\; f(x), \qquad \frac{\partial u}{\partial t}(x, 0) \;=\; g(x).

The initial shape of the wave is $f$ ; the initial velocity of every material point is $g$ . The d’Alembert formula expresses the solution at any later $(x, t)$ in terms of these two functions:

\boxed{\;u(x, t) \;=\; \frac{1}{2}\bigl[\,f(x - c t) + f(x + c t)\,\bigr] \;+\; \frac{1}{2 c} \int_{x - c t}^{x + c t} g(s) \, ds.\;}

▶ Deriving d'Alembert from the general solution

We have $u(x, t) = F(x - ct) + G(x + ct)$ and want to solve for $F$ and $G$ in terms of the initial data $f$ and $g$ .

At $t = 0$ :

f(x) \;=\; u(x, 0) \;=\; F(x) + G(x). \quad (*)

Differentiating $u$ in time and evaluating at $t = 0$ :

g(x) \;=\; u_t(x, 0) \;=\; -c\, F'(x) + c\, G'(x).

Divide by $c$ and integrate from a fixed reference point $a$ to $x$ :

\frac{1}{c} \int_a^x g(s)\, ds \;=\; -F(x) + F(a) + G(x) - G(a).

Rearrange:

-F(x) + G(x) \;=\; \frac{1}{c} \int_a^x g(s)\, ds \;+\; F(a) - G(a). \quad (**)

The two equations $(*)$ and $(**)$ are linear in $F(x), G(x)$ . Add them and divide by 2 to get $G(x)$ ; subtract and divide by 2 to get $F(x)$ . After substituting back and noting that the $F(a) - G(a)$ constants cancel, you get

F(x) = \frac{1}{2} f(x) - \frac{1}{2c} \int_a^x g(s)\, ds, \qquad G(x) = \frac{1}{2} f(x) + \frac{1}{2c} \int_a^x g(s)\, ds.

Substitute $x \to x - ct$ in the first and $x \to x + ct$ in the second, add them together, and use $\int_a^{x + ct} - \int_a^{x - ct} = \int_{x - ct}^{x + ct}$ . The boxed d’Alembert formula falls out.

Two terms, two contributions. The first term — the average of $f$ at the two points $x \pm c t$ — is the shape contribution: half the initial shape went left at speed $c$ and half went right, so what arrives at $(x, t)$ from the initial shape is half the value at $x - ct$ (the right-mover that started there) plus half the value at $x + ct$ (the left-mover). The second term — the integral of $g$ over the segment between $x \pm c t$ — is the velocity contribution: every material point in the segment from $x - ct$ to $x + ct$ was, at $t = 0$ , moving with some velocity, and exactly its share has had time to reach $(x, t)$ by time $t$ .

Characteristics

The two combinations $x - c t$ and $x + c t$ that keep appearing have a name. They are the characteristic variables of the 1-D wave equation. The curves

x - c t = \text{const}, \qquad x + c t = \text{const}

are the characteristic curves — two families of straight lines in the $(x, t)$ plane with slopes $\pm 1/c$ . They are the trajectories along which information propagates: right-going wave packets travel along the $x - ct =$ const family, left-going packets travel along $x + ct =$ const.

view:

wave speed c: 1.00

Drag the black dot anywhere in the plot. Two characteristics — one left-going (red), one right-going (blue) — reach back from (x, t) to the t = 0 axis. The bold gold segment between them is the domain of dependence: the only stretch of initial data that can possibly affect the solution at (x, t). Everything outside that segment is causally disconnected.

The interactive above shows both families simultaneously. Drag the black target point to a chosen $(x, t)$ . The two characteristics that reach $(x, t)$ — one right-going, one left-going — strike the initial line $t = 0$ at the two endpoints $x \pm c t$ of the bolded gold segment. That segment is the domain of dependence of $(x, t)$ : the only portion of the initial data that can affect $u(x, t)$ , exactly the same interval the d’Alembert formula integrates $g$ over. Initial data outside that segment is causally disconnected from the solution at $(x, t)$ .

Switch to the region of influence view and drag the source point along the $t = 0$ axis. The shaded wedge is the set of future $(x, t)$ points that the initial data at $x_0$ can reach. Anything outside the wedge at time $t$ cannot have heard from $x_0$ yet, because the signal would have to travel faster than $c$ .

The slope of the characteristics is set by $c$ . Slide $c$ and watch the wedges widen (information travels faster) or narrow (slower). For relativistic wave equations the analogue of $c$ is the speed of light, and the characteristic cones become light cones — the rigid frame on which special relativity is built.

Why “hyperbolic” PDEs all behave like this

The wave equation in higher dimensions, the Maxwell equations, the equations of elasticity, the linearised equations of gas dynamics — all are hyperbolic, and all share the structure of this lesson. A disturbance at a point spreads at finite speed; the value of the field at any spacetime point depends only on a bounded region of the initial data; the boundaries of that region are characteristic surfaces (cones in higher dimensions).

For practical purposes this gives the Sound book three things:

A propagating-pulse picture that survives the transition to higher dimensions and curved geometries (with shape distortion appearing, but the characteristic structure intact). This is the source of every “ray”, “wavefront”, and “geometric acoustics” argument later.
A causality argument. The acoustic field at a microphone at time $t$ only depends on what happened inside a sphere of radius $c t$ around it. Anything farther away is, for the moment, irrelevant.
A reflection mechanism. When a characteristic meets a boundary, the reflection sends a new characteristic back. The choreography of incoming and outgoing characteristics is what reflection, refraction, and diffraction are made of.

The d’Alembert solution is special to 1-D — in 2-D and 3-D there is no closed-form analogue this clean, and waves spread along cones of characteristics rather than two single ones. But the geometric picture (characteristics, domain of dependence, region of influence) is universal.

For the full physical derivation of the 1-D wave equation from a string under tension, see Sound 3.2 — The continuum limit and the 1-D wave equation. For the application to reflection and standing waves, see Sound 3.4.

⏳ The history — d'Alembert, Euler, and the vibrating-string controversy

Jean le Rond d’Alembert derived the traveling-wave solution $u(x, t) = F(x - ct) + G(x + ct)$ in his 1747 Recherches sur la courbe que forme une corde tendue mise en vibration, the first solution of a partial differential equation in history. The setup was a vibrating string of length $L$ pinned at both ends; his solution combined right- and left-going waves to satisfy both the wave equation and the boundary conditions.

A controversy followed almost immediately. Leonhard Euler in 1748 pointed out that d’Alembert’s $F$ and $G$ — being functions of the spatial coordinate $x \pm ct$ — could in principle be arbitrary curves, not just analytic formulae. D’Alembert insisted on smooth analytic functions only; Euler insisted on admitting “geometric” curves like piecewise-linear shapes. Daniel Bernoulli in 1753 proposed a third approach: the solution should be a superposition of sinusoidal modes — exactly the Fourier-series picture — which led to a further dispute between Bernoulli, d’Alembert, and Euler over whether any function could be represented as such a sum.

The full reconciliation came only after Fourier’s 1822 Théorie analytique de la chaleur (and a century of subsequent foundational work in analysis): yes, the two pictures are equivalent and both admit arbitrary reasonable functions, but doing so required a more careful understanding of what “function” and “convergence” meant. The 75-year vibrating-string controversy ended up being the seed dispute that motivated modern analysis. See also the History block in 7.1 — the two stories are continuous.

The next lesson, 6.3 — Separation of variables, develops the other major solution technique for the wave equation — the one that works on bounded domains where characteristics keep bouncing off the boundaries and a “mode” is a more useful way to think than a “ray.”