## Motivation

[motivation] ### Physical Origins A vibrating string, a drumhead, a sound wave in air — each of these systems transmits disturbances at a finite speed. If we pluck a guitar string at one end, the displacement travels along the string as a pulse; it does not appear instantaneously everywhere. This is the defining physical characteristic of wave propagation, and the mathematical model that captures it is the **wave equation**: \begin{align*} \partial_t^2 u - \Delta u = 0. \end{align*} Here $u(x, t)$ represents the displacement (or pressure, or amplitude) at position $x \in \mathbb{R}^n$ and time $t > 0$. The equation states that the acceleration $\partial_t^2 u$ is proportional to the spatial curvature $\Delta u$, with the proportionality constant (the wave speed squared) normalised to $1$. ### How the Wave Equation Differs from the Heat Equation The [heat equation](/page/Heat%20Equation) $\partial_t u - \Delta u = 0$ is first-order in time: it requires only one initial condition $u(\cdot, 0) = g$, its solutions are instantaneously smooth (even if $g$ is rough), and disturbances propagate at infinite speed — a point source at the origin influences every point in $\mathbb{R}^n$ immediately. None of these features survive the passage to the wave equation. Because the wave equation is *second*-order in time, the initial-value problem requires two pieces of data: the initial displacement $g$ and the initial velocity $h$. Solutions propagate at finite speed — a compactly supported disturbance remains compactly supported for all time. And the wave equation does *not* smooth its data: singularities in $g$ or $h$ persist and travel along characteristic surfaces. These differences are not incidental; they reflect fundamentally different physics (dissipation versus propagation) and fundamentally different mathematics (parabolic versus hyperbolic). ### The Strategy: Explicit Formulas, Then Energy The plan for this page follows Evans §2.4. In one spatial dimension, the wave operator factors as a product of two first-order operators, and the general solution is an explicit superposition of two travelling waves — this is d'Alembert's formula. In higher dimensions, the factorisation breaks down, but the rotational symmetry of $\Delta$ provides a replacement: by averaging $u$ over spheres (the method of **spherical means**), the $n$-dimensional problem reduces to a one-dimensional one, yielding Kirchhoff's formula ($n = 3$) and Poisson's formula ($n = 2$). Finally, **energy methods** — which do not require explicit formulas — establish uniqueness and finite propagation speed in all dimensions, providing qualitative information that complements the quantitative representation formulas. [/motivation]

## Definition

The wave equation is the simplest second-order hyperbolic PDE: it is linear, has constant coefficients, and its principal symbol $\tau^2 - |\xi|^2$ vanishes on a cone rather than an ellipsoid. To state the initial-value problem precisely, we must specify both the initial displacement and the initial velocity, reflecting the second-order nature of the equation in time.

[definition:Wave Equation] The **wave equation** in $n$ spatial dimensions is the partial differential equation \begin{align*} \partial_t^2 u - \Delta u = 0, \end{align*} where $u: \mathbb{R}^n \times (0, \infty) \to \mathbb{R}$ and $\Delta = \sum_{i=1}^n \partial_{x_i}^2$ is the Laplacian in the spatial variables. [/definition]

The sign convention $\partial_t^2 u - \Delta u$ (rather than $\partial_t^2 u + \Delta u$) ensures that the equation is hyperbolic. With the opposite sign, the equation $\partial_t^2 u + \Delta u = 0$ would be the ultrahyperbolic equation in the variables $(x, t)$, which has entirely different character.

[definition:Initial-Value Problem For The Wave Equation] The **Cauchy problem** (initial-value problem) for the wave equation consists of finding $u \in C^2(\mathbb{R}^n \times [0, \infty); \mathbb{R})$ satisfying \begin{align*} \begin{cases} \partial_t^2 u - \Delta u = 0 & \text{in } \mathbb{R}^n \times (0, \infty), \\ u = g & \text{on } \mathbb{R}^n \times \{t = 0\}, \\ \partial_t u = h & \text{on } \mathbb{R}^n \times \{t = 0\}, \end{cases} \end{align*} where $g: \mathbb{R}^n \to \mathbb{R}$ is the **initial displacement** and $h: \mathbb{R}^n \to \mathbb{R}$ is the **initial velocity**. [/definition]

The requirement of two initial conditions — $g$ and $h$ — is the hallmark of a second-order-in-time evolution equation. Physically, specifying the position and velocity of every point on a vibrating membrane at time zero determines the motion for all future times.

## Solution in One Dimension

### D'Alembert's Formula

In one spatial dimension, the wave operator $\partial_t^2 - \partial_x^2$ has a remarkable algebraic property that is unavailable in higher dimensions: it factors as $(\partial_t - \partial_x)(\partial_t + \partial_x)$. This factorisation reduces the wave equation to a pair of transport equations, each of which can be solved by the [method of characteristics](/page/Method%20of%20Characteristics). The general solution is therefore a superposition of a right-travelling wave $F(x - t)$ and a left-travelling wave $G(x + t)$, and imposing the initial data determines $F$ and $G$ uniquely.

[quotetheorem:665]

The formula reveals the mechanism of one-dimensional wave propagation with crystalline clarity. The term $\frac{1}{2}(g(x + t) + g(x - t))$ shows that the initial displacement splits into two copies, each travelling at unit speed in opposite directions. The integral term $\frac{1}{2}\int_{x-t}^{x+t} h \, d\mathcal{L}^1$ shows that the initial velocity contributes to $u(x, t)$ through its integral over the interval $[x - t, x + t]$ — the **domain of dependence** of the point $(x, t)$. Data outside this interval have no influence on $u(x, t)$, which is the first manifestation of finite propagation speed.

### Why the Factorisation Fails in Higher Dimensions

The factorisation $\partial_t^2 - \partial_x^2 = (\partial_t - \partial_x)(\partial_t + \partial_x)$ is specific to one dimension. In $\mathbb{R}^n$ with $n \geq 2$, the operator $\partial_t^2 - \Delta$ does not factor over the reals as a product of first-order operators: there is no pair of real vector fields $X, Y$ with $XY = \partial_t^2 - \Delta$. This is because $\Delta = \partial_{x_1}^2 + \cdots + \partial_{x_n}^2$ is a sum of squares with a positive sign, and the symbol $\tau^2 - |\xi|^2$ factors as $(\tau - |\xi|)(\tau + |\xi|)$ only at the level of symbols, not at the level of differential operators (since $|\xi| = \sqrt{\xi_1^2 + \cdots + \xi_n^2}$ is not a polynomial in $\xi$). A fundamentally different technique is needed.

[example:One Dimensional Wave With Gaussian Initial Displacement] Consider the Cauchy problem in one dimension with $g(x) = e^{-x^2}$ and $h(x) = 0$ (zero initial velocity). By [D'Alembert's Formula](/theorems/665): \begin{align*} u(x, t) = \frac{1}{2}\bigl(e^{-(x+t)^2} + e^{-(x-t)^2}\bigr). \end{align*} At $t = 0$, this is the single Gaussian $e^{-x^2}$. At time $t > 0$, the solution consists of two Gaussians of half the original amplitude, centred at $x = t$ and $x = -t$, moving apart at unit speed. The initial displacement has split cleanly into two travelling pulses. By $t = 5$, the pulses are centred at $\pm 5$ with negligible overlap — the wave has separated. This clean separation is a one-dimensional phenomenon. In higher dimensions, the geometry of wave fronts is more complex: Kirchhoff's formula for $n = 3$ involves surface [integrals](/page/Integral) over expanding spheres, not point evaluations. [/example]

[example:One Dimensional Wave With Piecewise Initial Velocity] Take $g(x) = 0$ and $h(x) = \mathbb{1}_{[-1, 1]}(x)$, the indicator [function](/page/Function) of $[-1, 1]$. Although $h$ is not $C^1$, d'Alembert's formula still gives a well-defined function: \begin{align*} u(x, t) = \frac{1}{2}\int_{x - t}^{x + t} \mathbb{1}_{[-1, 1]}(y) \, d\mathcal{L}^1(y) = \frac{1}{2}\mathcal{L}^1\bigl([x - t, x + t] \cap [-1, 1]\bigr). \end{align*} For $t < 1$ and $|x| < 1 - t$, the interval $[x - t, x + t]$ lies entirely inside $[-1, 1]$, so $u(x, t) = t$. For large $t$ and $|x|$ outside $[-1 - t, 1 + t]$, the intersection is empty and $u = 0$. The solution is a trapezoidal pulse that broadens linearly in time, its leading edges moving at speed $\pm 1$. The discontinuities of $h$ propagate along the characteristics $x \pm t = \pm 1$, illustrating that the wave equation does not smooth its data — unlike the heat equation, which would immediately regularise $h$ to a $C^\infty$ function. [/example]

## Spherical Means and Higher Dimensions

### The Euler–Poisson–Darboux Equation

Since the factorisation approach is unavailable in dimensions $n \geq 2$, we exploit instead the rotational invariance of $\Delta$. The idea, due to Poisson and systematised by Euler and Darboux, is to average a solution $u$ over spheres and study the resulting function of the radius. If $u$ solves the wave equation, then its spherical mean satisfies a one-dimensional PDE — the **Euler–Poisson–Darboux (EPD) equation** — which is close enough to the one-dimensional wave equation that explicit formulas are again available.