02.13.04 · analysis / pde

Wave Equation, d'Alembert Solution, Spherical Means, and Huygens Principle

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Anchor (Master): Evans §2.4; John §5; Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale UP 1923); Courant-Hilbert, Methods of Mathematical Physics II (Interscience 1962), §VI; Strichartz, A Guide to Distribution Theory and Fourier Transforms (CRC 1994); Sogge, Lectures on Non-Linear Wave Equations, 2e (International Press 2008)

Intuition Beginner

The wave equation is the partial differential equation that describes how disturbances propagate at a finite speed through a medium without losing their shape. Pluck a taut string and a kink runs to the left and to the right at the same speed. Drop a stone in a still pond and a circular ripple grows outward, keeping the shape it started with, only larger. Strike a drumhead and a wavefront races out across the membrane. Snap a wire and a pulse travels down it.

All of these are governed by a single mathematical law: the rate at which the displacement accelerates equals a positive constant (the wave speed squared) times the Laplacian of the displacement.

The wave equation is the second-order time-derivative cousin of the heat equation. Where the heat equation has a first time derivative and describes irreversible diffusion toward equilibrium, the wave equation has a second time derivative and describes reversible oscillation around equilibrium. The doubling of the time derivative changes everything. The heat equation smooths out initial roughness; the wave equation preserves it. The heat equation has infinite propagation speed and is irreversible; the wave equation has finite propagation speed and is time-reversible (running a wave backward in time gives another valid wave). The heat equation has no oscillation; the wave equation is all about oscillation.

The same equation governs an enormous range of physical phenomena. Sound waves in air, light waves in vacuum, seismic waves in rock, gravitational waves in spacetime, vibrations in a guitar string, pulses along a transmission line, ripples on a pond, shock fronts in a gas: all obey a wave-equation-style law, with the wave speed determined by the physical properties of the medium. The wave equation is the canonical example of a hyperbolic PDE, the class of equations describing reversible time-evolution at a finite signal speed. The triad of elliptic, parabolic, and hyperbolic equations covers the three fundamental qualitative behaviours of linear second-order PDEs, with the Laplace equation, the heat equation, and the wave equation as the prototypes.

Three properties make the wave equation stand apart from its elliptic and parabolic relatives. First, finite propagation speed. A disturbance at one point cannot affect another point until enough time has passed for a signal travelling at the wave speed to cross the distance between them. This is the mathematical statement that the wave equation respects causality: cause precedes effect, and the speed of cause-to-effect transmission is bounded. The numerical value of the bound is the wave speed, which appears in the equation itself as the positive constant in front of the Laplacian.

Second, the qualitative behaviour of wave propagation depends sharply on the number of spatial dimensions. In one dimension a wave pulse travels at the prescribed speed without any distortion: the pulse at time $t$ is the original pulse shifted by a distance $c t$ . In three dimensions a localised disturbance produces a thin expanding spherical shell: the signal is sharp, and after the shell passes a given point the medium returns to rest.

In two dimensions, a disturbance produces a wake: the wavefront still travels at the wave speed, but after the wavefront passes there is a residual ripple that never quite dies away. This odd-versus-even dimension distinction is the heart of the sharp Huygens principle, named for Christiaan Huygens who explicated wave propagation in his 1690 Traité de la lumière.

Third, the wave equation in one spatial dimension has a remarkably simple solution, written down by Jean d'Alembert in 1747. The general solution is a sum of two travelling waves, one moving to the right and one moving to the left, each at the wave speed. Given any initial shape and any initial velocity, the d'Alembert formula provides the explicit displacement at every later time, in closed form, by evaluating the initial data at two specific past points and integrating the initial velocity over the interval between them. The formula is exact, requires no approximation, and works for any reasonable initial data. It is one of the cleanest and earliest closed-form solutions of any partial differential equation.

In higher dimensions the explicit solution is more elaborate but follows the same recipe: write the solution as an integral against the initial data, with the integration domain determined by the wave speed and the elapsed time. In three space dimensions the integral runs over a sphere (the Kirchhoff formula, due to Gustav Kirchhoff in 1882). In two space dimensions the integral runs over a disk (the Poisson formula, due to Simeon Denis Poisson in 1818, more cleanly derived a century later by Hadamard via his method of descent).

The same domain-of-dependence picture that makes the d'Alembert formula transparent generalises: the displacement at a point at time $t$ is determined entirely by the initial data on the backward light cone, the cone of past points from which a signal could have reached the observation point.

The takeaway in a single sentence: the wave equation describes finite-speed propagation of disturbances, d'Alembert's 1747 formula solves it in one dimension, Kirchhoff's 1882 formula solves it in three dimensions via spherical means, Poisson's 1818 formula (Hadamard's method of descent) solves it in two dimensions, and the sharp Huygens principle distinguishes odd-dimensional clean propagation from even-dimensional propagation with a wake.

Visual Beginner

Picture an infinite taut string at rest along the horizontal axis. At time zero, lift the middle section into a triangular bump, then release the string from rest. A moment later, the triangular bump has split into two half-height triangular bumps, one travelling to the left at the wave speed and one travelling to the right at the wave speed. A moment later, the two bumps have separated farther; the string between them is flat. A while later still, the two bumps have moved well away from the origin, each carrying half the original displacement, both keeping the original triangular shape exactly.

This single picture, the splitting of an initial shape into a left-travelling copy and a right-travelling copy each at half the original height, is the d'Alembert solution in one space dimension. Every solution of the one-dimensional wave equation on the infinite line decomposes this way, with the initial velocity making a separate contribution that fills in the region between the two travelling shape-copies.

The picture in three dimensions is different and more striking. Imagine sending out a disturbance from a single point at a single instant. The disturbance does not stay put: it propagates outward as a thin expanding spherical shell at the wave speed. After the shell has expanded to a given radius, the medium inside the shell is once again at rest; the disturbance has cleanly passed through. This is the three-dimensional Huygens principle in action: a localised disturbance produces a sharp expanding wavefront with no after-effects.

Sound from a clap reaches your ears as a brief sharp pulse and then is gone, not as a sustained rumble. Light from a flash reaches your eye as a brief sharp flash and then is gone, not as a fading afterglow. The clean propagation is what allows speech and sight to convey discrete signals: a wave pulse arrives, is heard or seen, and is gone, ready for the next pulse to arrive cleanly.

The picture in two dimensions is intermediate and reveals the strangeness of the Huygens distinction. Drop a stone in a pond and a circular ripple expands outward. But unlike the three-dimensional case, the water inside the ripple is not at rest after the ripple has passed: it continues to oscillate with a slowly decaying amplitude. The two-dimensional wave equation has a tail.

A speaker confined to a two-dimensional membrane would not be able to send discrete signals cleanly, because every signal would be followed by a long lingering tail that would obscure the next signal. The fact that we live in three space dimensions, where the wave equation respects the sharp Huygens principle, is what makes acoustic and optical communication possible.

Worked example Beginner

We solve the one-dimensional wave equation on an infinite string with a simple triangular initial profile and zero initial velocity. The setup: at time zero, the string displacement is equal to one minus the absolute value of $x$ on the interval from minus one to one, and zero elsewhere (a triangular hat function). The initial velocity of the string is zero everywhere. The wave speed is one. We compute the displacement at the origin at time two and at the point $x = 1$ at time one.

Step 1. Set up the wave equation. The displacement $u (x, t)$ satisfies $u_{tt} = u_{xx}$ for every $x$ on the real line and every $t > 0$ , with initial position $u (x, 0) = f (x)$ where $f$ is the triangular hat function ( $f (x) = 1 - ∣ x ∣$ for $∣ x ∣ \leq 1$ and $f (x) = 0$ for $∣ x ∣ > 1$ ), and initial velocity $u_{t} (x, 0) = g (x) = 0$ for every $x$ .

Step 2. Apply the d'Alembert formula. With initial velocity zero, the d'Alembert formula reduces to the average of the two travelling-wave copies of the initial profile: $u (x, t) = \frac{1}{2} f (x - t) + \frac{1}{2} f (x + t) .$ This is the formal statement of the picture in the Visual panel: the initial bump splits into a half-height left-moving copy and a half-height right-moving copy.

Step 3. Compute $u (0, 2)$ . Substitute $x = 0$ and $t = 2$ : $u (0, 2) = \frac{1}{2} f (0 - 2) + \frac{1}{2} f (0 + 2) = \frac{1}{2} f (- 2) + \frac{1}{2} f (2) .$ Both $f (- 2)$ and $f (2)$ equal zero (since $f$ is supported on $[- 1, 1]$ ). So $u (0, 2) = 0$ . The string at the origin is at rest at time two: both halves of the original bump have passed through and moved away.

Step 4. Compute $u (1, 1)$ . Substitute $x = 1$ and $t = 1$ : $u (1, 1) = \frac{1}{2} f (1 - 1) + \frac{1}{2} f (1 + 1) = \frac{1}{2} f (0) + \frac{1}{2} f (2) = \frac{1}{2} \cdot 1 + \frac{1}{2} \cdot 0 = \frac{1}{2} .$ The point $x = 1$ at time $t = 1$ is exactly the location of the peak of the right-moving half-bump (since the right-moving bump has shifted by the wave speed times the elapsed time, putting its peak at $x = 0 + 1 = 1$ ). The displacement there is half the original peak height, which is one-half.

Step 5. Interpret. The d'Alembert formula has a transparent geometric interpretation: at any time $t$ , the displacement at any point $x$ is the average of the initial profile at the two points $x - t$ and $x + t$ . These two points are exactly the points from which a signal travelling at the wave speed could have reached $(x, t)$ in the elapsed time. They are the domain of dependence of the space-time point $(x, t)$ . The wave equation respects causality: only initial data at points that can be reached by signals travelling at the wave speed can influence the displacement.

What this tells us: the d'Alembert formula converts an initial-value problem for the wave equation into two function evaluations. The same recipe handles any reasonable initial data, with the integral of the initial velocity between $x - t$ and $x + t$ appearing as a separate additive term whenever the initial velocity is nonzero. The wave equation in one space dimension is one of the few PDEs with a fully explicit closed-form solution for every initial-value problem on the infinite line.

Check your understanding Beginner

Exercise (easy, multiple choice).

Which equation is the wave equation in one space dimension, with $u (x, t)$ the displacement, with the wave speed set to one?

A. $u_{xx} = 0$ B. $u_{t} = u_{xx}$ C. $u_{tt} = u_{xx}$ D. $u_{t} = u_{x}$

Hint

The wave equation relates the second time derivative of the displacement to the second spatial derivative of the displacement.

Answer

C. $u_{tt} = u_{xx}$ . The wave equation in one space dimension sets the second time derivative of the displacement equal to the second spatial derivative (with the wave speed absorbed into the units; the dimensional version is $u_{tt} = c^{2} u_{xx}$ ). Feedback-correct: this is the canonical hyperbolic equation. Feedback-wrong: A is the steady-state Laplace equation (no time dependence); B is the heat equation (first time derivative on the left); D is a first-order transport equation moving a signal to the right at unit speed.

Exercise (easy, true-false).

According to the d'Alembert formula, the displacement at $(x, t)$ of a one-dimensional wave equation with zero initial velocity and initial profile $f$ equals $\frac{1}{2} [f (x - t) + f (x + t)]$ when the wave speed is one.

Hint

The d'Alembert formula gives the general solution as the average of two travelling copies of the initial profile, with a separate term for the initial velocity that vanishes when the initial velocity is zero.

Answer

True. The d'Alembert formula in one space dimension with wave speed one gives the displacement as the average of the two travelling copies of the initial position $f$ , plus an extra term that is one-half times the area under the initial velocity $g$ between the two backward characteristic points $x - t$ and $x + t$ . When $g \equiv 0$ , the extra term vanishes and the displacement reduces to the average of the two travelling copies of $f$ .

Formal definition Intermediate+

Let $U \subseteq R^{n}$ be an open set and let $T > 0$ . Write $U_{T} = U \times (0, T]$ . The wave equation with wave speed $c > 0$ is the second-order linear PDE $u_{tt} - c^{2} Δ u = 0 on U_{T},$ where $u_{tt} = \partial^{2} u / \partial t^{2}$ and $Δ u = \sum_{i = 1}^{n} \partial^{2} u / \partial x_{i}^{2}$ is the spatial Laplacian. Setting $c = 1$ by rescaling time gives the canonical form $u_{tt} - Δ u = 0$ . The inhomogeneous wave equation has a forcing term $f : U_{T} \to R$ : $u_{tt} - c^{2} Δ u = f on U_{T} .$ A classical solution is a function $u \in C^{2} (U_{T})$ satisfying the equation pointwise ^{[Evans 2010 §2.4]}.

Cauchy problem. The Cauchy problem (initial-value problem) on the whole space $R^{n}$ asks for $u : R^{n} \times [0, T] \to R$ satisfying $⎩ ⎨ ⎧ u_{tt} - Δ u = 0 u (\cdot, 0) = f u_{t} (\cdot, 0) = g on R^{n} \times (0, T], on R^{n}, on R^{n},$ where $f : R^{n} \to R$ is the initial position and $g : R^{n} \to R$ is the initial velocity. The wave equation is second-order in time, so well-posedness requires two pieces of initial data, one for the position and one for the velocity, in contrast to the first-order-in-time heat equation, which requires only the initial position.

Theorem (d'Alembert formula, $n = 1$ ). Let $f \in C^{2} (R)$ and $g \in C^{1} (R)$ . The function $u (x, t) = \frac{1}{2} [f (x - c t) + f (x + c t)] + \frac{1}{2 c} \int_{x - c t}^{x + c t} g (y) d y$ is in $C^{2} (R \times R)$ , satisfies $u_{tt} - c^{2} u_{xx} = 0$ everywhere, $u (x, 0) = f (x)$ , and $u_{t} (x, 0) = g (x)$ for every $x \in R$ . The formula gives the unique classical solution of the one-dimensional Cauchy problem ^{[d'Alembert 1747]}. The two right-hand-side terms each have a clean physical interpretation. The first is the average of the two travelling-wave copies of the initial position. The second is half the average of the initial velocity over the interval between the two backward characteristics through $(x, t)$ .

Definition (domain of dependence and range of influence). The domain of dependence of a space-time point $(x_{0}, t_{0})$ for the wave equation with wave speed $c$ is the set of points $x \in R^{n}$ from which a signal travelling at speed $c$ could have reached $(x_{0}, t_{0})$ during the elapsed time $t_{0}$ , namely $Dom (x_{0}, t_{0}) = {x \in R^{n} : ∣ x - x_{0} ∣ \leq c t_{0}} .$ The range of influence of a space-time region is the set of space-time points reachable from it by a signal travelling at speed $c$ . The wave equation has the property that the value of the solution at $(x_{0}, t_{0})$ depends only on the initial data within $Dom (x_{0}, t_{0})$ , not on initial data outside.

Theorem (Kirchhoff formula, $n = 3$ ). Let $f \in C^{3} (R^{3})$ and $g \in C^{2} (R^{3})$ . The function $u (x, t) = \partial_{t} [\frac{t}{4 π ( c t ) ^{2}} \int_{\partial B_{c t} (x)} f (y) d S (y)] + \frac{t}{4 π ( c t ) ^{2}} \int_{\partial B_{c t} (x)} g (y) d S (y)$ solves the three-dimensional Cauchy problem $u_{tt} - c^{2} Δ u = 0$ with $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ ^{[Kirchhoff 1882]}. The integrals are surface integrals over the sphere of radius $c t$ centred at $x$ . The displacement at $(x, t)$ depends only on the values of $f$ and $g$ on the spherical wavefront at distance $c t$ from $x$ , not on values inside or outside the sphere. This is the analytic statement of the sharp Huygens principle in three space dimensions: a signal travels precisely on the wavefront, with no support inside or outside.

Theorem (Poisson formula via descent, $n = 2$ ; Hadamard 1923). Let $f \in C^{3} (R^{2})$ and $g \in C^{2} (R^{2})$ . The function $u (x, t) = \partial_{t} [\frac{1}{2 π c} \int_{B_{c t} (x)} \frac{f ( y )}{c ^{2} t ^{2} - ∣ y - x ∣ ^{2}} d y] + \frac{1}{2 π c} \int_{B_{c t} (x)} \frac{g ( y )}{c ^{2} t ^{2} - ∣ y - x ∣ ^{2}} d y$ solves the two-dimensional Cauchy problem $u_{tt} - c^{2} Δ u = 0$ with $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ ^{[Poisson 1818]} ^{[Hadamard 1923]}. The integrals are area integrals over the disk of radius $c t$ centred at $x$ . The displacement at $(x, t)$ depends on the values of $f$ and $g$ inside the entire disk, not only on the boundary circle. The two-dimensional wave equation therefore does not satisfy the sharp Huygens principle: a localised initial disturbance produces a wake that persists after the wavefront has passed.

Sharp Huygens principle. The sharp Huygens principle states that the value of a solution of the wave equation at $(x, t)$ depends only on the initial data on the boundary sphere $\partial B_{c t} (x)$ , not on initial data inside or outside the ball. In Euclidean space, the sharp Huygens principle holds in every odd dimension $n \geq 3$ and fails in every even dimension $n \geq 2$ (and holds in $n = 1$ for the elementary reason that the boundary sphere is just the two-point set ${x - c t, x + c t}$ , exactly the d'Alembert evaluation points). The failure in even dimensions takes the form of a tail term: after the wavefront passes, the solution at a point continues to be nonzero, with an algebraic decay rather than the clean cutoff of the odd-dimensional case.

Counterexamples to common slips Intermediate+

Bounded continuous initial data is not enough for higher dimensions. In one space dimension the d'Alembert formula extends to $f \in C^{2}$ and $g \in C^{1}$ ; in three dimensions the Kirchhoff formula requires $f \in C^{3}$ and $g \in C^{2}$ for a classical solution. This phenomenon, the loss of one derivative when passing from one to three dimensions, was identified by Hadamard in his 1923 Lectures on Cauchy's Problem and called the regularity loss for the wave equation in odd dimensions $\geq 3$ . In two dimensions the loss is even more severe: $f \in C^{3}$ and $g \in C^{2}$ are also needed, but the descent integral with the singular factor $1/ c^{2} t^{2} - ∣ y - x ∣^{2}$ requires additional integrability conditions on the initial data at the boundary of the descent disk. The regularity loss is a defining feature of the wave equation that does not appear in the heat equation, where convolution with the smooth heat kernel restores all derivatives.
Finite propagation speed is not free. It depends on the leading symbol of the differential operator being a positive-definite quadratic form in the spatial frequencies. The standard wave operator $\partial_{t}^{2} - c^{2} Δ$ has symbol $- τ^{2} + c^{2} ∣ ξ ∣^{2}$ , which factors as $(c ∣ ξ ∣ - τ) (c ∣ ξ ∣ + τ)$ giving two real characteristic surfaces (the forward and backward light cones). For operators with a different signature (such as $\partial_{t}^{2} + c^{2} Δ$ , the elliptic Laplace equation in space-time with the wrong sign), the characteristic surfaces are complex and the Cauchy problem is ill-posed (the backward heat equation analogue from 02.13.03 counterexamples).
The energy method requires both $u$ and $u_{t}$ , not just $u$ . The conserved energy for the wave equation $u_{tt} - c^{2} Δ u = 0$ is $E (t) = \frac{1}{2} \int (u_{t}^{2} + c^{2} ∣\nabla u ∣^{2}) d x,$ the sum of a kinetic term (in $u_{t}$ ) and a potential term (in $\nabla u$ ). The kinetic term is essential: without it the conservation law would not hold (the energy of a wave equation is the sum of kinetic and potential energy, exactly as in mechanics). The energy method shows $E^{'} (t) = 0$ via integration by parts; the conservation of $E$ then gives uniqueness for the Cauchy problem and the finite-propagation-speed theorem.
The Huygens principle is not the principle that wavefronts can be reconstructed from secondary wavelets. The Huygens-Fresnel principle in physical optics says that every point on a wavefront acts as a source of secondary spherical wavelets, and the new wavefront is the envelope of these secondary wavelets. The sharp Huygens principle in mathematics says that the solution of the wave equation at a point depends only on the initial data on the boundary of the backward light cone, not on the interior. These two principles are related but not identical. The Huygens-Fresnel principle of optics is approximately correct in any dimension; the sharp mathematical Huygens principle holds only in odd dimensions $\geq 3$ . The naming collision is unfortunate but historically entrenched.
Lorentz invariance is not just a special-relativity hat tip; it is structural. The wave operator $□ = \partial_{t}^{2} - c^{2} Δ = \partial_{t}^{2} - c^{2} \sum_{i = 1}^{n} \partial_{x_{i}}^{2}$ is invariant under the Lorentz group $SO^{+} (1, n)$ that preserves the Minkowski quadratic form $- τ^{2} + c^{2} ∣ ξ ∣^{2}$ . The Galilean transformations of pre-relativistic mechanics do not preserve the wave operator; the Lorentz transformations do. The Lorentz invariance of the wave equation was first noted by Voigt (1887), then independently by Larmor and Lorentz (1900-1904), and used by Einstein in 1905 as the founding observation of special relativity. The propagation speed $c$ in the wave equation is what becomes the speed of light in the relativistic interpretation: a finite invariant speed, the same in every inertial frame, is the mathematical content of Lorentz invariance.

Key theorem with proof Intermediate+

Theorem (d'Alembert formula and uniqueness in one space dimension). Let $f \in C^{2} (R)$ and $g \in C^{1} (R)$ . The function $u (x, t) = \frac{1}{2} [f (x - c t) + f (x + c t)] + \frac{1}{2 c} \int_{x - c t}^{x + c t} g (y) d y$ is in $C^{2} (R \times R)$ , satisfies $u_{tt} - c^{2} u_{xx} = 0$ everywhere, $u (x, 0) = f (x)$ , and $u_{t} (x, 0) = g (x)$ for every $x \in R$ . The formula gives the unique classical solution of the one-dimensional Cauchy problem ^{[d'Alembert 1747]}. Moreover, the displacement at $(x_{0}, t_{0})$ depends only on the values of $f$ at the two points $x_{0} \pm c t_{0}$ and on the values of $g$ on the interval $[x_{0} - c t_{0}, x_{0} + c t_{0}]$ ; data outside this interval has no effect on $u (x_{0}, t_{0})$ (finite propagation speed in one dimension).

Proof. The proof has three steps: derivation of the formula by characteristics, verification by direct computation, and uniqueness by the energy method.

Step 1 (derivation by characteristics). The wave operator factors: $\partial_{t}^{2} - c^{2} \partial_{x}^{2} = (\partial_{t} - c \partial_{x}) (\partial_{t} + c \partial_{x}) .$ Introduce characteristic coordinates $ξ = x - c t$ and $η = x + c t$ . The two first-order transport operators become $\partial_{t} - c \partial_{x} = - 2 c \partial_{η}$ and $\partial_{t} + c \partial_{x} = 2 c \partial_{ξ}$ (chain rule: $\partial_{t} = - c (\partial_{ξ} - \partial_{η})$ and $\partial_{x} = \partial_{ξ} + \partial_{η}$ in the $(ξ, η)$ basis, then verify). The wave equation in characteristic coordinates becomes $- 4 c^{2} u_{ξ η} = 0, i.e., u_{ξ η} = 0.$ The general solution of $u_{ξ η} = 0$ is $u (ξ, η) = F (ξ) + G (η)$ for arbitrary $F, G \in C^{2}$ (one integration in $η$ shows $u_{ξ}$ depends only on $ξ$ , the second integration in $ξ$ gives the decomposition). Reverting to $(x, t)$ coordinates: $u (x, t) = F (x - c t) + G (x + c t)$ .

Now impose initial conditions. $u (x, 0) = F (x) + G (x) = f (x)$ and $u_{t} (x, 0) = - c F^{'} (x) + c G^{'} (x) = g (x)$ . Differentiate the first equation: $F^{'} (x) + G^{'} (x) = f^{'} (x)$ . Combining with the second: $G^{'} (x) = \frac{1}{2} (f^{'} (x) + g (x) / c)$ and $F^{'} (x) = \frac{1}{2} (f^{'} (x) - g (x) / c)$ .

Integrate from a reference point (say $0$ ): $F (x) = \frac{1}{2} [f (x) - f (0)] - \frac{1}{2 c} \int_{0}^{x} g (y) d y + F (0)$ , and similarly for $G$ . Substituting back $u (x, t) = F (x - c t) + G (x + c t)$ and using $F (0) + G (0) = f (0)$ gives $u (x, t) = \frac{1}{2} [f (x - c t) + f (x + c t)] + \frac{1}{2 c} \int_{x - c t}^{x + c t} g (y) d y .$ The constants of integration cancel.

Step 2 (verification). Direct computation. Compute partial derivatives: $u_{t} (x, t) = \frac{c}{2} [- f^{'} (x - c t) + f^{'} (x + c t)] + \frac{1}{2} [g (x + c t) + g (x - c t)],$ $u_{tt} (x, t) = \frac{c ^{2}}{2} [f^{''} (x - c t) + f^{''} (x + c t)] + \frac{c}{2} [g^{'} (x + c t) - g^{'} (x - c t)] .$ And $u_{x} (x, t) = \frac{1}{2} [f^{'} (x - c t) + f^{'} (x + c t)] + \frac{1}{2 c} [g (x + c t) - g (x - c t)],$ $u_{xx} (x, t) = \frac{1}{2} [f^{''} (x - c t) + f^{''} (x + c t)] + \frac{1}{2 c} [g^{'} (x + c t) - g^{'} (x - c t)] .$ Multiply $u_{xx}$ by $c^{2}$ : $c^{2} u_{xx} (x, t) = \frac{c ^{2}}{2} [f^{''} (x - c t) + f^{''} (x + c t)] + \frac{c}{2} [g^{'} (x + c t) - g^{'} (x - c t)] = u_{tt} (x, t) .$ So $u_{tt} - c^{2} u_{xx} = 0$ on $R \times R$ .

Initial conditions: $u (x, 0) = \frac{1}{2} [f (x) + f (x)] + 0 = f (x)$ ; $u_{t} (x, 0) = \frac{c}{2} [- f^{'} (x) + f^{'} (x)] + \frac{1}{2} [g (x) + g (x)] = g (x)$ . Both initial conditions are satisfied.

Step 3 (uniqueness by energy). Suppose $u_{1}, u_{2}$ are two $C^{2}$ solutions with the same initial data. Then $w = u_{1} - u_{2}$ satisfies $w_{tt} - c^{2} w_{xx} = 0$ with $w (\cdot, 0) = 0$ and $w_{t} (\cdot, 0) = 0$ . We show $w \equiv 0$ by an energy argument on a backward light cone.

Fix $(x_{0}, t_{0})$ with $t_{0} > 0$ and define the truncated cone $K_{s} = {(x, t) : 0 \leq t \leq s, ∣ x - x_{0} ∣ \leq c (t_{0} - t)}$ for $0 \leq s \leq t_{0}$ . Define the energy on the time- $s$ slice $E (s) = \frac{1}{2} \int_{∣ x - x_{0} ∣ \leq c (t_{0} - s)} (w_{t}^{2} + c^{2} w_{x}^{2}) d x .$ Compute (differentiation under the integral, using $w_{tt} = c^{2} w_{xx}$ and the chain rule on the moving boundary): $E^{'} (s) = \int_{∣ x - x_{0} ∣ \leq c (t_{0} - s)} (w_{t} w_{tt} + c^{2} w_{x} w_{x t}) d x - \frac{c}{2} [w_{t}^{2} + c^{2} w_{x}^{2}]_{x = x_{0} \pm c (t_{0} - s)} .$

The interior integral simplifies using $w_{tt} = c^{2} w_{xx}$ : $\int (w_{t} \cdot c^{2} w_{xx} + c^{2} w_{x} w_{x t}) d x = c^{2} \int \partial_{x} (w_{t} w_{x}) d x = c^{2} [w_{t} w_{x}]_{x = x_{0} \pm c (t_{0} - s)} .$ So $E^{'} (s) = c^{2} [w_{t} w_{x}]_{-} - \frac{c}{2} [w_{t}^{2} + c^{2} w_{x}^{2}]_{-},$ where $[\cdot]_{-}$ denotes the boundary values at the lateral boundary (with appropriate sign tracking; we omit the $\pm$ subscripts for compactness).

Use the algebraic identity $c^{2} ab - \frac{c}{2} (a^{2} + c^{2} b^{2}) = - \frac{c}{2} (a - c b)^{2} \leq 0$ with $a = w_{t}$ and $b = w_{x}$ . Then $E^{'} (s) \leq 0$ , so $E$ is non-increasing.

Since $w (\cdot, 0) = 0$ and $w_{t} (\cdot, 0) = 0$ , $E (0) = 0$ . So $E (s) = 0$ for every $s \in [0, t_{0}]$ , in particular $E (t_{0}) = 0$ . But $E (t_{0})$ is the integral of a non-negative quantity over the single-point set ${x_{0}}$ , which is zero; this is consistent but does not immediately give $w (x_{0}, t_{0}) = 0$ . Instead, observe that $E (s) = 0$ for every $s < t_{0}$ forces $w_{t} \equiv 0$ and $w_{x} \equiv 0$ on the open cone ${∣ x - x_{0} ∣ < c (t_{0} - s)}$ for every $s$ , hence on the open backward light cone ${(x, t) : 0 < t < t_{0}, ∣ x - x_{0} ∣ < c (t_{0} - t)}$ . Combined with $w (\cdot, 0) = 0$ , integration in $t$ gives $w \equiv 0$ on the closed backward light cone, hence $w (x_{0}, t_{0}) = 0$ by continuity.

Since $(x_{0}, t_{0})$ was arbitrary, $w \equiv 0$ on $R \times [0, \infty)$ , so $u_{1} = u_{2}$ . $□$

Bridge. The d'Alembert formula is the prototype of every closed-form wave-equation solution: a finite-dimensional integral against the initial data, supported on the domain of dependence, with the algebraic structure of the integral kernel determined by the dimension. In three dimensions the kernel is a surface measure on a sphere; in two dimensions it is a Riesz-potential-type singular weight on a disk. The Master tier below derives the higher-dimensional formulas via spherical means and the Euler-Poisson-Darboux equation, and analyses the regularity and Huygens-principle consequences. The energy method shown here for uniqueness in one dimension generalises immediately to every dimension and to the inhomogeneous problem via Duhamel's principle, and is the basis of the modern Strichartz-type a priori estimates for both linear and semilinear wave equations.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

State and verify the conservation of energy for the one-dimensional wave equation. Define $E (t) = \frac{1}{2} \int_{- \infty}^{\infty} (u_{t}^{2} + c^{2} u_{x}^{2}) d x$ and show $E^{'} (t) = 0$ for any smooth compactly supported solution.

Hint

Differentiate under the integral, substitute $u_{tt} = c^{2} u_{xx}$ , and integrate by parts. The boundary terms vanish because $u$ has compact support.

Answer

Differentiate: $E^{'} (t) = \int_{- \infty}^{\infty} (u_{t} u_{tt} + c^{2} u_{x} u_{x t}) d x .$ Substitute $u_{tt} = c^{2} u_{xx}$ : $E^{'} (t) = \int_{- \infty}^{\infty} (c^{2} u_{t} u_{xx} + c^{2} u_{x} u_{x t}) d x = c^{2} \int_{- \infty}^{\infty} \partial_{x} (u_{t} u_{x}) d x .$ For compactly supported $u$ , the boundary terms at $\pm \infty$ vanish, so the integral is zero. Therefore $E^{'} (t) = 0$ and $E (t) = E (0)$ for every $t$ .

The conserved $E$ is the total energy: the integral of the kinetic energy density $\frac{1}{2} u_{t}^{2}$ plus the potential energy density $\frac{1}{2} c^{2} u_{x}^{2}$ . Conservation reflects the time-reversibility of the wave equation. The conservation law fails for the heat equation, whose $L^{2}$ -norm is strictly decreasing in time (parabolic dissipation).

Exercise 4 (medium, symbolic).

Using the d'Alembert formula, prove the finite-propagation-speed property: if $f$ and $g$ both vanish on the interval $∣ x ∣ \leq R$ for some $R > 0$ , then $u (0, t) = 0$ for every $0 \leq t \leq R / c$ .

Hint

The d'Alembert formula evaluates $f$ at $\pm c t$ and integrates $g$ over $[- c t, c t]$ . For $c t \leq R$ , all evaluation points lie within $∣ x ∣ \leq R$ where $f$ and $g$ vanish.

Answer

The d'Alembert formula gives $u (0, t) = \frac{1}{2} [f (- c t) + f (c t)] + \frac{1}{2 c} \int_{- c t}^{c t} g (y) d y .$ If $c t \leq R$ , then $- c t \geq - R$ and $c t \leq R$ , so both $f (- c t)$ and $f (c t)$ vanish by hypothesis. And the integration interval $[- c t, c t] \subseteq [- R, R]$ lies entirely within the zero-set of $g$ , so the integral is zero. Therefore $u (0, t) = 0 + 0 = 0$ for every $0 \leq t \leq R / c$ .

The finite-propagation-speed property says that a disturbance at distance $R$ from the origin cannot affect the origin until at least time $R / c$ has passed. This is the wave-equation analogue of the special-relativistic principle that no signal can travel faster than light. The wave-equation version was understood by d'Alembert in 1747, long before its relativistic significance was recognised.

Exercise 5 (medium, symbolic).

Solve the inhomogeneous wave equation $u_{tt} - u_{xx} = h (x, t)$ on $R \times (0, \infty)$ with zero initial position and zero initial velocity, by Duhamel's principle.

Hint

Duhamel's principle for the wave equation: for each $s \in (0, t)$ , solve the homogeneous wave equation for $τ \geq 0$ with $v (\cdot, 0; s) = 0$ and $v_{τ} (\cdot, 0; s) = h (\cdot, s)$ , and integrate the solution against $s$ . The solution to the inhomogeneous problem is $u (x, t) = \int_{0}^{t} v (x, t - s; s) d s$ .

Answer

Apply the d'Alembert formula with $f = 0$ and $g = h (\cdot, s)$ at each fixed source time $s$ : $v (x, τ; s) = \frac{1}{2} \int_{x - τ}^{x + τ} h (y, s) d y .$ Then by Duhamel's principle the solution of the inhomogeneous problem is $u (x, t) = \int_{0}^{t} v (x, t - s; s) d s = \frac{1}{2} \int_{0}^{t} \int_{x - (t - s)}^{x + (t - s)} h (y, s) d y d s .$

The double integration domain is the backward light cone of $(x, t)$ : the set of space-time points $(y, s)$ with $0 \leq s \leq t$ and $∣ y - x ∣ \leq t - s$ . Geometrically, this is the triangular region in space-time bounded above by $(x, t)$ and bounded below by the line $s = 0$ , with sides given by the two characteristics through $(x, t)$ . The wave equation respects this causal structure: the displacement at $(x, t)$ is the integral of the source $h$ over the backward light cone, with no contribution from sources outside the cone.

Exercise 6 (medium, symbolic).

Use the Kirchhoff formula to compute the solution of the three-dimensional wave equation $u_{tt} - Δ u = 0$ on $R^{3} \times (0, \infty)$ with initial position $f = 0$ and initial velocity $g (y) = 1$ for $∣ y ∣ \leq 1$ , $g (y) = 0$ for $∣ y ∣ > 1$ . Evaluate $u (0, t)$ for $t > 1$ .

Hint

With $f = 0$ and wave speed $c = 1$ , $u (x, t) = \frac{1}{4 π t} \int_{\partial B_{t} (x)} g (y) d S (y)$ . For $x = 0$ and $t > 1$ , the sphere $\partial B_{t} (0)$ has radius $t > 1$ and lies entirely outside the support of $g$ .

Answer

The Kirchhoff formula with $f = 0$ , $c = 1$ , $x = 0$ : $u (0, t) = \frac{1}{4 π t} \int_{\partial B_{t} (0)} g (y) d S (y) .$ The sphere $\partial B_{t} (0)$ has radius $t$ . For $t > 1$ , this sphere lies entirely in the region $∣ y ∣ > 1$ where $g = 0$ . The surface integral is zero, so $u (0, t) = 0$ for every $t > 1$ .

The result illustrates the sharp Huygens principle in three space dimensions: after the wavefront from a localised disturbance has passed (at time $t = 1$ in our example, when the sphere of radius $1$ first reaches the origin from outside), the medium returns exactly to rest. No tail, no afterglow, no residual ripple. This is the analytic statement of the perfect clean propagation of sound and light in three space dimensions.

For $0 \leq t \leq 1$ , the sphere $\partial B_{t} (0)$ lies entirely inside the support of $g$ , so the integral evaluates to the surface area $4 π t^{2} \cdot 1$ and $u (0, t) = \frac{4 π t ^{2}}{4 π t} = t$ , a linearly increasing displacement.

For $t$ slightly greater than $1$ , the sphere $\partial B_{t} (0)$ intersects the support of $g$ in a spherical zone; the integral picks up a partial-area factor that drops to zero at $t =$ the radius beyond which the sphere no longer intersects (which depends on the geometry; for the unit ball centred at the origin, the sphere $\partial B_{t} (0)$ ceases to intersect at $t = 1$ since the centres coincide). The clean cutoff at the moment the wavefront leaves the support of $g$ is the signature of the three-dimensional Huygens principle.

Exercise 7 (hard, symbolic).

Prove the parallelogram identity for the one-dimensional wave equation. Let $u$ solve $u_{tt} - c^{2} u_{xx} = 0$ and let $A, B, C, D$ be the four corners of a characteristic parallelogram, namely a parallelogram in the $(x, t)$ -plane whose sides lie along the two characteristic directions $x \mp c t =$ const. Show that $u (A) + u (C) = u (B) + u (D)$ , where $A$ and $C$ are opposite corners.

Hint

Use the d'Alembert form $u (x, t) = F (x - c t) + G (x + c t)$ and exploit the fact that opposite corners of the characteristic parallelogram share the same $x - c t$ value (for one diagonal) and the same $x + c t$ value (for the other diagonal).

Answer

Label the parallelogram so the side from $A$ to $B$ is a leftward-going characteristic ( $x + c t =$ const, so $A$ and $B$ have the same value of $x + c t$ , call it $η_{1}$ ). The side from $A$ to $D$ is a rightward-going characteristic ( $x - c t =$ const, so $A$ and $D$ have the same value of $x - c t$ , call it $ξ_{1}$ ). Then $C$ is the opposite corner, with $x - c t = ξ_{2}$ (the $ξ$ -value of $B$ ) and $x + c t = η_{2}$ (the $η$ -value of $D$ ).

In characteristic coordinates: $A = (ξ_{1}, η_{1})$ , $B = (ξ_{2}, η_{1})$ , $D = (ξ_{1}, η_{2})$ , $C = (ξ_{2}, η_{2})$ .

Apply the d'Alembert form $u = F (ξ) + G (η)$ : $u (A) = F (ξ_{1}) + G (η_{1}), u (C) = F (ξ_{2}) + G (η_{2}),$ $u (B) = F (ξ_{2}) + G (η_{1}), u (D) = F (ξ_{1}) + G (η_{2}) .$

Add: $u (A) + u (C) = F (ξ_{1}) + F (ξ_{2}) + G (η_{1}) + G (η_{2})$ and $u (B) + u (D) = F (ξ_{2}) + F (ξ_{1}) + G (η_{1}) + G (η_{2})$ . Both sums are equal.

The parallelogram identity is a discrete analogue of the wave equation $u_{ξ η} = 0$ in characteristic coordinates. It is the basis of explicit finite-difference schemes for the wave equation (the leapfrog scheme), and provides a geometric proof of uniqueness on a characteristic parallelogram given boundary values on two adjacent sides.

Exercise 8 (hard, symbolic).

Prove the uniqueness of the solution of the Cauchy problem for the wave equation in three space dimensions by the energy method on a backward light cone.

Hint

Imitate Step 3 of the Key Theorem proof, but with the spatial domain $B_{c (t_{0} - s)} (x_{0})$ replacing the one-dimensional interval. Use the divergence theorem in three dimensions (rather than integration by parts in one dimension) and the elementary inequality $∣ a \cdot b ∣ \leq \frac{1}{2} (∣ a ∣^{2} + ∣ b ∣^{2})$ for the lateral boundary term.

Answer

Let $w = u_{1} - u_{2}$ be the difference of two $C^{2}$ solutions with the same initial data: $w_{tt} - c^{2} Δ w = 0$ , $w (\cdot, 0) = 0$ , $w_{t} (\cdot, 0) = 0$ . Fix $(x_{0}, t_{0})$ with $t_{0} > 0$ and define the truncated cone $K_{s} = {(x, t) : 0 \leq t \leq s, ∣ x - x_{0} ∣ \leq c (t_{0} - t)}$ .

Define the energy on the time- $s$ slice: $E (s) = \frac{1}{2} \int_{∣ x - x_{0} ∣ \leq c (t_{0} - s)} (w_{t}^{2} + c^{2} ∣\nabla w ∣^{2}) d x .$

Compute $E^{'} (s)$ . Using Reynolds transport with the moving boundary and $w_{tt} = c^{2} Δ w$ : $E^{'} (s) = \int_{∣ x - x_{0} ∣ \leq c (t_{0} - s)} (w_{t} w_{tt} + c^{2} \nabla w \cdot \nabla w_{t}) d x - \frac{c}{2} \int_{∣ x - x_{0} ∣ = c (t_{0} - s)} (w_{t}^{2} + c^{2} ∣\nabla w ∣^{2}) d S .$

The interior integrand becomes, using the divergence theorem on $c^{2} \nabla w \cdot \nabla w_{t} = c^{2} div (w_{t} \nabla w) - c^{2} w_{t} Δ w$ : $\int (w_{t} w_{tt} + c^{2} \nabla w \cdot \nabla w_{t}) d x = \int [w_{t} (w_{tt} - c^{2} Δ w) + c^{2} div (w_{t} \nabla w)] d x$ $= 0 + c^{2} \int_{∣ x - x_{0} ∣ = c (t_{0} - s)} w_{t} (ν \cdot \nabla w) d S = c^{2} \int w_{t} \partial_{ν} w d S .$

So $E^{'} (s) = c^{2} \int_{∣ x - x_{0} ∣ = c (t_{0} - s)} w_{t} \partial_{ν} w d S - \frac{c}{2} \int (w_{t}^{2} + c^{2} ∣\nabla w ∣^{2}) d S .$

Bound $w_{t} \partial_{ν} w \leq ∣ w_{t} ∣ \cdot ∣\nabla w ∣ \leq \frac{1}{2} (w_{t}^{2} / c + c ∣\nabla w ∣^{2}) = \frac{1}{2 c} (w_{t}^{2} + c^{2} ∣\nabla w ∣^{2})$ by the AM-GM inequality with the weight $c$ chosen to match the energy. Then $c^{2} w_{t} \partial_{ν} w \leq \frac{c}{2} (w_{t}^{2} + c^{2} ∣\nabla w ∣^{2}),$ so $E^{'} (s) \leq 0$ . The energy is non-increasing.

$E (0) = 0$ from the zero initial data, $E (s) \geq 0$ as the integral of a non-negative quantity. So $E (s) = 0$ for every $s \in [0, t_{0}]$ . This forces $w_{t} \equiv 0$ and $\nabla w \equiv 0$ on the open backward light cone, hence $w$ is constant on the cone. The zero initial data and continuity force $w \equiv 0$ on the closed cone, in particular $w (x_{0}, t_{0}) = 0$ . Since $(x_{0}, t_{0})$ was arbitrary, $u_{1} = u_{2}$ .

The energy-cone argument is the standard uniqueness proof for the wave equation in every dimension and is the analytic basis of the finite-propagation-speed theorem (the influence cone has aperture $c$ , the wave speed).

Advanced results Master

The modern theory of the wave equation organises around six pillars: the spherical-means apparatus and the Euler-Poisson-Darboux equation, the explicit representation formulas of Kirchhoff and Poisson in three and two dimensions, the regularity loss in even dimensions and Hadamard's method of descent, the energy method for uniqueness and finite propagation speed, the Duhamel principle for the inhomogeneous problem and the Strichartz-type a priori estimates, and the connection to Lorentz invariance and the semilinear wave equation theory.

Theorem 1 (Euler-Poisson-Darboux equation; Poisson 1818, Darboux 1882). Let $u \in C^{k} (R^{n} \times (0, \infty))$ solve the wave equation $u_{tt} - c^{2} Δ u = 0$ for some integer $k \geq 2$ . For $x \in R^{n}$ and $r, t > 0$ , define the spherical mean $M (x, r, t) = \fint_{\partial B_{r} (x)} u (y, t) d S (y) = \frac{1}{ω _{n} r ^{n - 1}} \int_{\partial B_{r} (x)} u (y, t) d S (y),$ where $ω_{n} = ∣ \partial B_{1} (0) ∣$ is the surface area of the unit sphere in $R^{n}$ . Then $M$ satisfies the Euler-Poisson-Darboux equation $M_{tt} - c^{2} [M_{r r} + \frac{n - 1}{r} M_{r}] = 0$ on ${(r, t) : r > 0, t > 0}$ , with initial data $M (x, r, 0) = \fint_{\partial B_{r} (x)} f (y) d S (y)$ and $M_{t} (x, r, 0) = \fint_{\partial B_{r} (x)} g (y) d S (y)$ ^{[Poisson 1818]}. The Euler-Poisson-Darboux equation is the wave equation for the spherical mean as a function of the radius and time, with an extra Bessel-type drift term $(n - 1) / r$ that captures the spherical geometry. The spatial dimension $n$ enters only through this drift coefficient; the equation is a one-dimensional wave-like equation in $(r, t)$ that can be solved explicitly for $n = 3$ (drift coefficient 2) by the substitution $V = r M$ , which reduces the equation to the standard one-dimensional wave equation $V_{tt} = c^{2} V_{r r}$ for $V$ . Iterating the substitution gives explicit formulas for every odd $n$ ; the even- $n$ case requires the method of descent (next item).

Theorem 2 (Kirchhoff formula via spherical means; Kirchhoff 1882). Let $f \in C^{3} (R^{3})$ and $g \in C^{2} (R^{3})$ . The unique $C^{2}$ solution of the Cauchy problem $u_{tt} - c^{2} Δ u = 0$ with $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ on $R^{3}$ is $u (x, t) = \partial_{t} [t \fint_{\partial B_{c t} (x)} f (y) d S (y)] + t \fint_{\partial B_{c t} (x)} g (y) d S (y)$ ^{[Kirchhoff 1882]}. The displacement at $(x, t)$ depends only on the values of $f$ and $g$ on the boundary sphere $\partial B_{c t} (x)$ of radius $c t$ centred at $x$ , not on values inside or outside the sphere. The Kirchhoff formula is the analytic content of the sharp Huygens principle in three space dimensions: a localised initial disturbance produces a sharp expanding spherical wavefront with no support inside or outside. The formula extends to every odd $n \geq 3$ via repeated application of the Euler-Poisson-Darboux substitution, with the spherical-mean integrand multiplied by additional polynomial factors depending on $n$ .

Theorem 3 (Poisson formula and method of descent, $n = 2$ ; Hadamard 1923). Let $f \in C^{3} (R^{2})$ and $g \in C^{2} (R^{2})$ . The unique $C^{2}$ solution of the Cauchy problem $u_{tt} - c^{2} Δ u = 0$ with $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ on $R^{2}$ is $u (x, t) = \partial_{t} [\frac{1}{2 π c} \int_{B_{c t} (x)} \frac{f ( y )}{c ^{2} t ^{2} - ∣ y - x ∣ ^{2}} d y] + \frac{1}{2 π c} \int_{B_{c t} (x)} \frac{g ( y )}{c ^{2} t ^{2} - ∣ y - x ∣ ^{2}} d y$ ^{[Hadamard 1923]}. Hadamard's method of descent derives this formula from the three-dimensional Kirchhoff formula by treating a two-dimensional problem as a three-dimensional problem with initial data independent of the third coordinate. The two-dimensional displacement is the three-dimensional spherical-mean integral, projected down by integrating out the third coordinate; the projection converts the spherical surface integral into a disk area integral with the algebraic singularity $1/ c^{2} t^{2} - ∣ y - x ∣^{2}$ at the disk boundary (which arises geometrically from the Jacobian of the parametrisation of the sphere by its projection onto the $(y_{1}, y_{2})$ -plane). The integrand depends on values of $f$ and $g$ throughout the disk, not only on the boundary circle, so the two-dimensional wave equation does not satisfy the sharp Huygens principle.

Theorem 4 (sharp Huygens principle and odd-even dimension distinction; Hadamard 1923). The sharp Huygens principle holds for the wave equation on $R^{n}$ for every odd $n \geq 3$ and fails for every even $n \geq 2$ . Explicitly: the solution of the homogeneous Cauchy problem at $(x, t)$ depends only on the initial data on the boundary sphere $\partial B_{c t} (x)$ when $n \geq 3$ is odd, and depends on the initial data throughout the ball $B_{c t} (x)$ when $n \geq 2$ is even. The failure in even dimensions takes the explicit form of a tail integral over the interior of the ball with algebraic decay in $t$ . Hadamard's 1923 Lectures on Cauchy's Problem ^{[Hadamard 1923]} gave the first systematic treatment of the dimension-dependent Huygens principle, and identified the failure in even dimensions as a feature, not a bug: the descent formula remains a valid representation, but the physical interpretation differs from the odd-dimensional case. The Huygens principle on curved space-times has been a subject of active research since the mid-twentieth century, with the Schwarzschild solution and other physically relevant spacetimes satisfying the sharp principle only in special geometries.

Theorem 5 (regularity loss; Hadamard 1923, John 1955). The wave equation in $n$ space dimensions exhibits a loss of $(n - 1) /2$ derivatives in odd dimensions and a loss of $n /2$ derivatives in even dimensions. Concretely, for $f \in C^{k}$ and $g \in C^{k - 1}$ with $k$ large enough, the Kirchhoff and Poisson formulas give a classical $C^{2}$ solution only if $k \geq n /2 + 1$ in odd dimensions and $k \geq n /2 + 2$ in even dimensions. The loss of derivatives is a defining feature of the wave equation that does not appear in the heat equation, where the smoothing property of the heat semigroup restores all derivatives lost in the initial data. The regularity loss is a precise analytical version of the physical phenomenon that wavefronts can carry singularities: a discontinuity in the initial data propagates along the characteristic surfaces (the light cones) and is preserved by the wave evolution.

Theorem 6 (energy method and finite propagation speed; Friedrichs-Lewy 1928). Let $u \in C^{2} (R^{n} \times [0, T])$ solve $u_{tt} - c^{2} Δ u = 0$ with $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ . Suppose $f$ and $g$ both vanish on the ball $B_{R} (x_{0}) \subset R^{n}$ . Then $u (x_{0}, t) = 0$ for every $0 \leq t \leq R / c$ , and more generally $u \equiv 0$ on the truncated backward light cone ${(x, t) : ∣ x - x_{0} ∣ \leq R - c t, 0 \leq t \leq R / c}$ ^{[Friedrichs-Lewy 1928]}. The proof uses the conserved energy $E (x_{0}, t; r) = \frac{1}{2} \int_{B_{r - c t} (x_{0})} (u_{t}^{2} + c^{2} ∣\nabla u ∣^{2}) d x,$ shown to be non-increasing in $t$ by an integration-by-parts argument on the shrinking ball. The finite-propagation-speed theorem is the wave-equation embodiment of causality: signals from initial data outside the ball $B_{R} (x_{0})$ cannot reach $x_{0}$ in time $t < R / c$ . This is the analytic version of the special-relativistic principle that signals cannot travel faster than the wave speed $c$ (which becomes the speed of light when the wave equation is interpreted as the Maxwell field equations in vacuum).

Theorem 7 (Duhamel principle for the wave equation). Let $h \in C^{2} (R^{n} \times [0, T])$ and let $u$ solve the inhomogeneous wave equation $u_{tt} - c^{2} Δ u = h$ with zero initial data $u (\cdot, 0) = 0$ and $u_{t} (\cdot, 0) = 0$ . Then $u (x, t) = \int_{0}^{t} v (x, t - s; s) d s,$ where for each $s \in [0, t]$ , $v (\cdot, τ; s)$ solves the homogeneous wave equation $v_{τ τ} - c^{2} Δ v = 0$ for $τ \geq 0$ with $v (\cdot, 0; s) = 0$ and $v_{τ} (\cdot, 0; s) = h (\cdot, s)$ . The Duhamel formula expresses the inhomogeneous solution as a continuous superposition of homogeneous solutions, with each homogeneous solution generated by the source $h (\cdot, s)$ acting impulsively at time $s$ on the velocity. In one space dimension, substituting the d'Alembert formula gives the explicit double-integral formula of Exercise 5. In three space dimensions, substituting the Kirchhoff formula gives a triple integral over the backward light cone, weighted by the surface-measure factor of Kirchhoff's representation.

Theorem 8 (separation of variables on bounded domain). Let $U \subset R^{n}$ be a bounded smooth domain. The Cauchy-Dirichlet problem $u_{tt} - c^{2} Δ u = 0 on U \times (0, \infty), u = 0 on \partial U \times (0, \infty), u (\cdot, 0) = f, u_{t} (\cdot, 0) = g$ has the explicit solution $u (x, t) = k = 1 \sum \infty [a_{k} cos (c λ_{k} t) + \frac{b _{k}}{c λ _{k}} sin (c λ_{k} t)] φ_{k} (x),$ where ${(λ_{k}, φ_{k})}_{k = 1}^{\infty}$ is the Dirichlet eigenvalue-eigenfunction pair sequence for $- Δ$ on $U$ (with $λ_{k} > 0$ , $φ_{k} \in C^{\infty} (\overset{ˉ}{U})$ orthonormal in $L^{2} (U)$ , $λ_{k} \to \infty$ ), and $a_{k} = \int_{U} f φ_{k}$ , $b_{k} = \int_{U} g φ_{k}$ are the eigenfunction-expansion coefficients of the initial data. The solution is a superposition of standing-wave eigenmodes oscillating in time at the frequencies $c λ_{k} / (2 π)$ . The eigenfrequency spectrum is the mode structure of the bounded domain: the same data that determines the fundamental and harmonic frequencies of a guitar string ( $n = 1$ , $U = (0, L)$ , eigenvalues $(k π / L)^{2}$ ), the eigenmodes of a drumhead ( $n = 2$ , $U =$ disk, eigenvalues given by Bessel-function zeros), and the resonant cavities of musical instruments and architectural acoustics.

Theorem 9 (Strichartz estimates; Strichartz 1977). Let $u$ solve the homogeneous wave equation $u_{tt} - Δ u = 0$ on $R^{n + 1}$ with initial data $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ . Then for appropriate exponent pairs $(p, q)$ satisfying a scaling-and-dimension constraint, there is a constant $C = C (n, p, q) > 0$ such that $∥ u ∥_{L_{t}^{p} L_{x}^{q} (R \times R^{n})} \leq C (∥ f ∥_{H^{s}} + ∥ g ∥_{H^{s - 1}})$ ^{[Strichartz 1977]}, where $H^{s}$ is the standard Sobolev space and the exponent $s$ is determined by the dimension and the Strichartz pair. The Strichartz estimates quantify the dispersive decay of free solutions of the wave equation: a wave packet does not stay concentrated, but spreads out over time at a rate determined by the dimension. The estimates are central to the modern theory of nonlinear dispersive equations (semilinear and quasilinear wave equations, nonlinear Schrödinger equation, Korteweg-de Vries equation), where they convert linear estimates into iterative existence and uniqueness theorems via the contraction mapping principle on a Strichartz-type space.

Theorem 10 (semilinear wave equations and blowup; John 1979, Strauss 1981). Consider the semilinear wave equation $u_{tt} - Δ u = ∣ u ∣^{p}$ on $R^{n} \times [0, T)$ with smooth compactly supported initial data $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ , with $p > 1$ a real exponent. There is a critical exponent $p_{c} (n)$ (the Strauss exponent) such that:

For $1 < p \leq p_{c} (n)$ , every nonzero small-data Cauchy problem blows up in finite time: there is a $T^{*} < \infty$ such that $sup_{x \in R^{n}} ∣ u (x, t) ∣ \to \infty$ as $t \to T^{*}$ .
For $p > p_{c} (n)$ , every sufficiently small smooth compactly supported initial data leads to a global solution that exists for all $t > 0$ .

The Strauss exponent in three space dimensions is $p_{c} (3) = 1 + 2 \approx 2.414$ . John 1979 ^{[John 1979]} proved blowup for $p = 2$ in three dimensions; Strauss 1981 conjectured the full classification. The Strauss conjecture was completed through the contributions of many authors (Kato, Sideris, Glassey, Schaeffer, Yordanov-Zhang, Zhou) over a period of three decades. The semilinear wave equation is the model nonlinear hyperbolic equation, and the Strauss exponent calibrates the boundary between focusing nonlinearity (which produces blowup) and defocusing nonlinearity (which allows global existence) for the wave equation.

Synthesis. The wave equation is the prototype hyperbolic equation, and its solution apparatus (d'Alembert formula, Kirchhoff formula, Poisson formula, spherical means, Euler-Poisson-Darboux equation, energy method, finite-propagation-speed, Duhamel principle, eigenfunction expansions on bounded domains, Strichartz estimates) is the prototype of every linear and semilinear hyperbolic-equation theory. The pattern recurs in three main escalations. First, replace the wave operator by the Klein-Gordon operator $□ + m^{2}$ (adding a mass term): the qualitative behaviour stays the same but the long-time dispersive decay rate changes, and the Strichartz estimates pick up the mass-dependent dispersion. Second, replace the wave operator by the Dirac operator (a square root of the wave operator): the Dirac equation describes spin- $1/2$ relativistic particles and is the foundation of the relativistic quantum theory of fermions. Third, replace the linear wave equation by a semilinear or quasilinear wave equation: the small-data global-existence theory (Klainerman 1985 vector-field method, the Christodoulou-Klainerman global stability of Minkowski space 1993) builds on the linear Strichartz estimates as its starting point.

The Lorentz-invariance side of the equation has been equally fertile. The wave operator is invariant under the Lorentz group, and the Cauchy problem on $R^{n}$ extends naturally to the Cauchy problem on a Lorentzian manifold (the wave equation on a curved spacetime). On a globally hyperbolic Lorentzian manifold, the d'Alembert-Kirchhoff representation generalises to the Leray-Hadamard fundamental solution, and the Huygens principle holds on a Lorentzian manifold if and only if the manifold satisfies a stringent geometric condition (essentially: the spacetime is conformally flat plus a vanishing-Bach-tensor condition). The wave equation on the Schwarzschild solution of the Einstein equations is the linearised gravitational-wave equation on a black-hole background, and its decay properties (the Price law tail, the Christodoulou-Klainerman scattering theory) are central to the modern theory of black-hole stability.

The conceptual closure is the recognition that the wave equation packages five distinct mathematical phenomena into a single equation: the spherical-means apparatus on Euclidean space (Poisson 1818, Kirchhoff 1882, Hadamard 1923), the finite-propagation-speed structure that captures causality (Friedrichs-Lewy 1928), the dispersive smoothing structure that captures the spread of wave packets (Strichartz 1977), the Lorentz-invariance structure that connects the wave equation to special relativity (Einstein 1905), and the nonlinear-equation theory that links the linear wave equation to general relativity and quantum field theory (Klainerman, Christodoulou). The arc from d'Alembert's 1747 Mémoires paper on the vibrating string to the modern theory of nonlinear hyperbolic equations on curved spacetimes is a 280-year lineage in which the same equation has been continuously refined into ever more general and ever more powerful tools.

Full proof set Master

Proposition 1 (Euler-Poisson-Darboux equation). For $u \in C^{k} (R^{n} \times (0, \infty))$ with $k \geq 2$ solving $u_{tt} - c^{2} Δ u = 0$ , the spherical mean $M (x, r, t) = \fint_{\partial B_{r} (x)} u (y, t) d S (y)$ satisfies $M_{tt} - c^{2} [M_{r r} + \frac{n - 1}{r} M_{r}] = 0$ for $r, t > 0$ .

Proof. Parametrise the sphere $\partial B_{r} (x)$ by $y = x + r ω$ for $ω \in \partial B_{1} (0)$ . Then $M (x, r, t) = \fint_{\partial B_{1} (0)} u (x + r ω, t) d S (ω) .$ Differentiate in $r$ : $M_{r} (x, r, t) = \fint_{\partial B_{1} (0)} \nabla u (x + r ω, t) \cdot ω d S (ω) = \fint_{\partial B_{r} (x)} \nabla u (y, t) \cdot \frac{y - x}{r} d S (y) .$

The integrand on the right is $\nabla u \cdot ν$ where $ν = (y - x) / r$ is the outward unit normal to $\partial B_{r} (x)$ . By the divergence theorem applied to $\nabla u$ on $B_{r} (x)$ : $\int_{\partial B_{r} (x)} \nabla u \cdot ν d S (y) = \int_{B_{r} (x)} Δ u (y, t) d y .$ Dividing by the surface area $ω_{n} r^{n - 1}$ : $M_{r} (x, r, t) = \frac{1}{ω _{n} r ^{n - 1}} \int_{B_{r} (x)} Δ u (y, t) d y .$

Multiply by $r^{n - 1}$ : $r^{n - 1} M_{r} (x, r, t) = \frac{1}{ω _{n}} \int_{B_{r} (x)} Δ u (y, t) d y = \frac{1}{ω _{n}} \int_{0}^{r} \int_{\partial B_{ρ} (x)} Δ u (y, t) d S (y) d ρ .$

Differentiate in $r$ again, using the fundamental theorem of calculus: $\partial_{r} [r^{n - 1} M_{r} (x, r, t)] = \frac{1}{ω _{n}} \int_{\partial B_{r} (x)} Δ u (y, t) d S (y) = \frac{r ^{n - 1}}{ω _{n} r ^{n - 1}} \int_{\partial B_{r} (x)} Δ u (y, t) d S (y) .$

Switch to the spherical-mean form: the right side equals $r^{n - 1} \fint_{\partial B_{r} (x)} Δ u (y, t) d S (y)$ . Using $Δ u = (1/ c^{2}) u_{tt}$ from the wave equation: $\partial_{r} [r^{n - 1} M_{r}] = r^{n - 1} \fint_{\partial B_{r} (x)} \frac{u _{tt} ( y , t )}{c ^{2}} d S (y) = \frac{r ^{n - 1}}{c ^{2}} M_{tt} (x, r, t) .$

Expand the left side: $\partial_{r} [r^{n - 1} M_{r}] = (n - 1) r^{n - 2} M_{r} + r^{n - 1} M_{r r}$ . Divide both sides by $r^{n - 1}$ : $\frac{n - 1}{r} M_{r} + M_{r r} = \frac{1}{c ^{2}} M_{tt}, i.e., M_{tt} - c^{2} [M_{r r} + \frac{n - 1}{r} M_{r}] = 0.$ This is the Euler-Poisson-Darboux equation. $□$

Proposition 2 (Kirchhoff formula, $n = 3$ ). For $f \in C^{3} (R^{3})$ and $g \in C^{2} (R^{3})$ , the function $u (x, t) = \partial_{t} [t M_{f} (x, c t)] + t M_{g} (x, c t)$ solves the three-dimensional wave equation $u_{tt} - c^{2} Δ u = 0$ with $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ , where $M_{h} (x, r) = \fint_{\partial B_{r} (x)} h (y) d S (y)$ denotes the spherical mean of $h$ at $x$ at radius $r$ .

Proof. In dimension $n = 3$ , the Euler-Poisson-Darboux equation for the spherical mean $M (x, r, t)$ of a solution $u$ is $M_{tt} = c^{2} [M_{r r} + \frac{2}{r} M_{r}] .$ Substitute $V (x, r, t) = r M (x, r, t)$ . Then $V_{r} = M + r M_{r}$ and $V_{r r} = 2 M_{r} + r M_{r r}$ , so $V_{r r} = r [M_{r r} + \frac{2}{r} M_{r}] = \frac{r}{c ^{2}} M_{tt} = \frac{1}{c ^{2}} V_{tt},$ i.e., $V_{tt} = c^{2} V_{r r}$ . The function $V$ satisfies the one-dimensional wave equation in the variables $(r, t)$ on the half-line $r > 0$ , with the boundary condition $V (x, 0, t) = 0 \cdot M (x, 0, t) = 0$ (since $V = r M$ vanishes at $r = 0$ ).

Apply the d'Alembert formula on the half-line with Dirichlet boundary condition at $r = 0$ . The Dirichlet condition is enforced by odd reflection: extend $V$ to negative $r$ by $V (x, - r, t) = - V (x, r, t)$ , then $V$ satisfies the wave equation on the full line. The d'Alembert formula gives $V (x, r, t) = \frac{1}{2} [V_{0} (r + c t) + V_{0} (r - c t)] + \frac{1}{2 c} \int_{r - c t}^{r + c t} V_{1} (s) d s,$ where $V_{0} (r) = r M_{f} (x, r)$ (odd-extended) and $V_{1} (r) = r M_{g} (x, r)$ (odd-extended).

For $0 < r < c t$ , the argument $r - c t$ is negative, so $V_{0} (r - c t) = - (c t - r) M_{f} (x, c t - r)$ by odd reflection, and similarly for $V_{1}$ . Substitute and simplify: $V (x, r, t) = \frac{1}{2} [(r + c t) M_{f} (x, r + c t) - (c t - r) M_{f} (x, c t - r)] + \frac{1}{2 c} \int_{c t - r}^{r + c t} s M_{g} (x, s) d s .$

(The integration limits use the odd reflection: the integral from $r - c t$ to $0$ flips sign and changes to an integral from $0$ to $c t - r$ , then combines with the integral from $0$ to $r + c t$ to give the integral from $c t - r$ to $r + c t$ after sign tracking.)

Divide by $r$ to recover $M (x, r, t) = V / r$ . Take the limit $r \to 0^{+}$ (which corresponds to evaluating $M$ at radius zero, namely $u (x, t) = M (x, 0, t)$ ): $u (x, t) = r \to 0^{+} lim \frac{V ( x , r , t )}{r} .$

L'Hôpital or direct Taylor expansion: $V (x, r, t) / r = \frac{1}{2 r} [(r + c t) M_{f} (x, r + c t) - (c t - r) M_{f} (x, c t - r)] + \frac{1}{2 r c} \int_{c t - r}^{r + c t} s M_{g} (x, s) d s$ .

For the second term, the integral over an interval of length $2 r$ centred near $c t$ divided by $2 r c$ tends to $\frac{1}{c} \cdot c t \cdot M_{g} (x, c t) = t M_{g} (x, c t)$ as $r \to 0$ (fundamental theorem of calculus).

For the first term, expand both spherical means in $r$ around $r = 0$ : $(r + c t) M_{f} (x, r + c t) = c t \cdot M_{f} (x, c t) + r [M_{f} (x, c t) + c t \cdot \partial_{r} M_{f} (x, c t)] + O (r^{2})$ , and similarly for $(c t - r) M_{f} (x, c t - r) = c t \cdot M_{f} (x, c t) - r [M_{f} (x, c t) + c t \cdot \partial_{r} M_{f} (x, c t)] + O (r^{2})$ . The difference is $2 r [M_{f} (x, c t) + c t \cdot \partial_{r} M_{f} (x, c t)] + O (r^{2})$ . Dividing by $2 r$ : $r \to 0^{+} lim \frac{1}{2 r} [(r + c t) M_{f} (x, r + c t) - (c t - r) M_{f} (x, c t - r)] = M_{f} (x, c t) + c t \cdot \partial_{r} M_{f} (x, c t) .$

The right side equals $\partial_{t} [t M_{f} (x, c t)]$ when interpreted as a function of $t$ (with $r = c t$ ): $\partial_{t} [t M_{f} (x, c t)] = M_{f} (x, c t) + t \cdot c \cdot \partial_{r} M_{f} (x, c t)$ , matching the limit.

Combining: $u (x, t) = \partial_{t} [t M_{f} (x, c t)] + t M_{g} (x, c t) .$

Direct verification that $u$ satisfies the initial conditions: $u (x, 0) = \partial_{t} [t M_{f} (x, c t)] ∣_{t = 0} + 0 = M_{f} (x, 0) = f (x)$ (the spherical mean over a zero-radius sphere is the value at the centre). And $u_{t} (x, 0)$ : differentiate $u = \partial_{t} [t M_{f} (x, c t)] + t M_{g} (x, c t)$ in $t$ and evaluate at $t = 0$ . After cancellation of the $f$ -derivative terms (which vanish at $t = 0$ by symmetry), one gets $u_{t} (x, 0) = M_{g} (x, 0) = g (x)$ . The wave equation $u_{tt} - c^{2} Δ u = 0$ holds by construction since $u$ is built from spherical means of solutions of the Euler-Poisson-Darboux equation.

The Kirchhoff formula is the explicit closed-form solution of the three-dimensional Cauchy problem for the wave equation. $□$

Proposition 3 (Hadamard's method of descent, $n = 2$ ). The two-dimensional Cauchy problem is solved by treating a two-dimensional initial datum as a three-dimensional initial datum independent of the third coordinate and applying the three-dimensional Kirchhoff formula.

Proof sketch. Let $f, g : R^{2} \to R$ be the two-dimensional initial data. Define three-dimensional initial data $\tilde{f}, \tilde{g} : R^{3} \to R$ by $\tilde{f} (y_{1}, y_{2}, y_{3}) = f (y_{1}, y_{2})$ and $\tilde{g} (y_{1}, y_{2}, y_{3}) = g (y_{1}, y_{2})$ (independent of $y_{3}$ ). The three-dimensional Cauchy problem with these initial data has a solution $\tilde{u} (x_{1}, x_{2}, x_{3}, t)$ via the Kirchhoff formula; by symmetry, $\tilde{u}$ is independent of $x_{3}$ , so we can write $\tilde{u} (x_{1}, x_{2}, x_{3}, t) = u (x_{1}, x_{2}, t)$ for some $u : R^{2} \times [0, \infty) \to R$ . The function $u$ then solves the two-dimensional wave equation (since $Δ_{R^{3}} \tilde{u} = Δ_{R^{2}} u + \partial_{x_{3}}^{2} \tilde{u} = Δ_{R^{2}} u$ given the $x_{3}$ -independence) with initial data $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ .

The Kirchhoff representation reduces to a surface integral over the sphere $\partial B_{c t} (x_{1}, x_{2}, 0)$ in $R^{3}$ . Parametrise this sphere by its projection onto the $(y_{1}, y_{2})$ -plane: $y_{3} = \pm c^{2} t^{2} - (y_{1} - x_{1})^{2} - (y_{2} - x_{2})^{2}$ , with the $\pm$ giving the upper and lower hemispheres. The surface area element on the sphere is $d S = \frac{c t}{∣ y _{3} ∣} d y_{1} d y_{2} = \frac{c t}{c ^{2} t ^{2} - ∣ y - x ∣ ^{2}} d y_{1} d y_{2}$ (writing $y = (y_{1}, y_{2})$ and $x = (x_{1}, x_{2})$ ).

Doubling for the two hemispheres (since $\tilde{f}$ and $\tilde{g}$ are independent of $y_{3}$ , the upper and lower hemispheres contribute equally) and substituting into the Kirchhoff formula gives the two-dimensional Poisson formula $u (x, t) = \partial_{t} [\frac{1}{2 π c} \int_{B_{c t} (x)} \frac{f ( y )}{c ^{2} t ^{2} - ∣ y - x ∣ ^{2}} d y] + \frac{1}{2 π c} \int_{B_{c t} (x)} \frac{g ( y )}{c ^{2} t ^{2} - ∣ y - x ∣ ^{2}} d y .$

The integrand involves values of $f$ and $g$ on the entire disk $B_{c t} (x)$ , not only the boundary circle, reflecting the failure of the sharp Huygens principle in two space dimensions. $□$

Proposition 4 (energy conservation and finite propagation speed). Let $u \in C^{2} (R^{n} \times [0, T])$ solve $u_{tt} - c^{2} Δ u = 0$ with $u (\cdot, 0) = f$ and $u_{t} (\cdot, 0) = g$ both supported in $B_{R} (0)$ . Then $u (\cdot, t)$ is supported in $B_{R + c t} (0)$ for every $t \in [0, T]$ .

Proof. Fix $x_{0} \in / B_{R + c t_{0}} (0)$ for some $t_{0} \in [0, T]$ . We show $u (x_{0}, t_{0}) = 0$ . The backward light cone from $(x_{0}, t_{0})$ down to time $0$ is the set ${(x, t) : ∣ x - x_{0} ∣ \leq c (t_{0} - t), 0 \leq t \leq t_{0}}$ . Its base (at $t = 0$ ) is the ball $B_{c t_{0}} (x_{0})$ .

Since $∣ x_{0} ∣ > R + c t_{0}$ by assumption, any $x \in B_{c t_{0}} (x_{0})$ satisfies $∣ x ∣ \geq ∣ x_{0} ∣ - c t_{0} > R$ , so $x \in / B_{R} (0)$ . Therefore $f (x) = 0$ and $g (x) = 0$ for every $x$ in the base of the backward light cone.

By the energy-method uniqueness argument of Exercise 8 (or Step 3 of the Key Theorem proof in one dimension, generalised to $n$ dimensions), the value of $u$ at $(x_{0}, t_{0})$ is determined by the initial data on the base of the backward light cone. Since both $f$ and $g$ vanish there, $u (x_{0}, t_{0}) = 0$ .

Since $x_{0} \in / B_{R + c t_{0}} (0)$ was arbitrary, $u (\cdot, t_{0})$ is supported in $B_{R + c t_{0}} (0)$ . $□$

Connections Master

Laplace equation 02.13.01. The elliptic prototype, and the steady-state limit of the wave equation in the absence of time-dependent forcing. A standing-wave eigenmode of the wave equation $u (x, t) = φ (x) e^{iω t}$ leads to the Helmholtz equation $- Δ φ = (ω / c)^{2} φ$ for the spatial part, which is the Laplace equation modified by an eigenvalue. Setting $ω = 0$ recovers the Laplace equation $Δ φ = 0$ , which describes the time-independent equilibrium state.
Poisson equation 02.13.02. The elliptic equation with a source term, related to the wave equation by the same eigenvalue-substitution as the Laplace equation. The Poisson equation $- Δ u = f$ is the time-independent reduction of the inhomogeneous wave equation with a time-independent source. The fundamental solution of the Laplace operator (Newtonian potential) and the fundamental solution of the wave operator (retarded Green function) are connected by the resolvent identity: the wave-equation Green function integrated over all time is the Laplace-equation Green function.
Heat equation 02.13.03. The parabolic cousin of the wave equation. The heat equation $u_{t} - Δ u = 0$ has a first time derivative and describes irreversible smoothing toward equilibrium; the wave equation has a second time derivative and describes reversible oscillation. The heat equation has infinite propagation speed and dissipates energy; the wave equation has finite propagation speed and conserves energy. Wick rotation $t \to i t$ formally converts the wave equation into a Helmholtz-type equation, the Euclidean version of the wave equation. The relativistic Klein-Gordon equation $\partial_{t}^{2} ψ - Δ ψ + m^{2} ψ = 0$ interpolates between the wave equation (mass zero) and gives a hyperbolic version of the Schrödinger equation under Wick rotation.
Classification of second-order linear PDEs 02.13.05. The wave equation is the prototype hyperbolic equation, parallel to the Laplace equation (elliptic prototype) and the heat equation (parabolic prototype). The classification of general second-order linear PDEs $\sum a_{ij} (x) \partial_{i} \partial_{j} u + lower order = 0$ in terms of the signature of the symbol matrix $(a_{ij} (x))$ at each point gives the elliptic-parabolic-hyperbolic trichotomy, with the wave equation as the maximally simple hyperbolic representative.
Energy methods for PDEs 02.13.06. The energy method for the wave equation, shown above for one-dimensional uniqueness and generalised in Exercise 8 to higher dimensions, extends to the entire family of hyperbolic equations. The conserved quantities are quadratic functionals of the solution and its derivatives, with conservation following from integration by parts on the underlying equation. The energy method is the foundation of the modern $H^{s}$ Sobolev-space theory of well-posedness for hyperbolic equations and the standard tool for proving uniqueness and continuous dependence in the absence of explicit representation formulas.
Fourier analysis 02.10.04. The diagonalisation framework for the wave equation on $R^{n}$ . The Fourier transform diagonalises the spatial Laplacian: $Δ u (ξ) = - ∣ ξ ∣^{2} \overset{u}{^} (ξ)$ . The wave equation $u_{tt} - c^{2} Δ u = 0$ becomes the second-order ODE $\overset{u}{^}_{tt} = - c^{2} ∣ ξ ∣^{2} \overset{u}{^}$ for each fixed frequency $ξ$ , with general solution $\overset{u}{^} (ξ, t) = A (ξ) cos (c ∣ ξ ∣ t) + B (ξ) sin (c ∣ ξ ∣ t) / (c ∣ ξ ∣)$ . The initial data fix $A = \hat{f}$ and $B = \overset{g}{^}$ , giving the explicit Fourier representation $u (x, t) = (2 π)^{- n} \int_{R^{n}} e^{i x \cdot ξ} [\hat{f} (ξ) cos (c ∣ ξ ∣ t) + \overset{g}{^} (ξ) \frac{sin ( c ∣ ξ ∣ t )}{c ∣ ξ ∣}] d ξ .$ The Fourier representation is the bridge between the explicit Kirchhoff and Poisson formulas (via inverse Fourier transform of the dispersive symbol $cos (c ∣ ξ ∣ t)$ ) and the modern Strichartz estimates (via stationary-phase analysis of the same dispersive symbol).
Schrödinger equation [12.02]. The quantum-mechanical analogue of the wave equation, related by the substitution of a first time derivative for a second time derivative (and a factor of $i$ ). The free Schrödinger equation $i u_{t} = - Δ u$ is the quantum-mechanical evolution equation, with the same spatial structure as the wave equation but a different temporal structure. The dispersive estimates for the wave equation (Strichartz 1977) and the Schrödinger equation are proven by analogous stationary-phase methods. The Schrödinger group is unitary on $L^{2}$ (preserves $L^{2}$ norm) like the wave equation's energy is conserved; both equations are time-reversible.
Maxwell equations and electromagnetic waves [10.04]. Maxwell's equations in vacuum give rise to a wave equation for each component of the electric and magnetic fields, with wave speed equal to the speed of light $c = 1/ ε_{0} μ_{0}$ . The d'Alembert and Kirchhoff representations apply directly to the electromagnetic field components. The Lorentz invariance of the wave equation is the Lorentz invariance of Maxwell's equations, and the wave-equation theory is the analytic foundation of classical electrodynamics.
Special relativity [10.05]. The wave equation $\partial_{t}^{2} u - c^{2} Δ u = 0$ is invariant under the Lorentz group $SO^{+} (1, n)$ that preserves the Minkowski metric $- c^{2} d t^{2} + d x_{1}^{2} + \dots + d x_{n}^{2}$ . The finite propagation speed $c$ is the invariant speed of the Lorentz transformations and is identified with the speed of light. Einstein's 1905 derivation of special relativity started from the observation that the Maxwell wave equation (above) has $c$ as an invariant speed in every inertial frame, leading to the postulates of special relativity. The wave equation is therefore not only an equation of mathematical physics but the analytic embodiment of one of the two pillars of modern physics.

Historical & philosophical context Master

D'Alembert's 1747 paper Recherches sur la courbe que forme une corde tendue mise en vibration in the Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin ^{[d'Alembert 1747]} is the founding document of the modern theory of the wave equation and one of the earliest closed-form solutions of any partial differential equation. D'Alembert started from Newton's second law applied to a small element of a taut string, derived the wave equation $u_{tt} = c^{2} u_{xx}$ for the transverse displacement, and discovered the general solution $u (x, t) = F (x - c t) + G (x + c t)$ as a sum of two travelling waves. The d'Alembert formula was the prototype of every closed-form PDE solution and remains the standard introduction to the wave equation in every textbook.

Euler 1748 Sur la vibration des cordes ^{[Euler 1748]} and Daniel Bernoulli 1755 Réflexions et éclaircissemens sur les nouvelles vibrations des cordes ^{[Bernoulli 1755]} extended d'Alembert's work in two directions. Euler discussed the admissibility of discontinuous and non-smooth initial data, anticipating the modern weak-solution theory by more than a century. Bernoulli proposed the representation of the solution as an infinite trigonometric series, the seed of Fourier analysis (which Fourier carried out fully in the heat-equation context in 1822). The d'Alembert-Euler-Bernoulli controversy over the admissibility of arbitrary initial data was a foundational debate of eighteenth-century analysis, and was resolved only with the development of the modern theory of distributions and weak solutions in the twentieth century (Schwartz 1950, Théorie des distributions).

Lagrange 1759 Recherches sur la nature et la propagation du son ^{[Lagrange 1759]} extended the wave equation to acoustic propagation in three space dimensions, recognising that sound waves in air obey a three-dimensional version of the same equation governing the vibrating string. Lagrange's Mécanique analytique (1788) systematised the variational and energy-method techniques that would become the standard tools for proving uniqueness and existence for the wave equation.

Poisson 1818 Mémoire sur l'intégration de quelques équations linéaires aux différences partielles ^{[Poisson 1818]} gave the first explicit representation formula for the wave equation in three space dimensions using spherical means. Poisson's 1818 paper introduced the spherical-mean apparatus that would become standard in the theory of the wave equation, and gave a precursor of the modern Kirchhoff formula. Poisson also analysed the two-dimensional case and discovered the absence of the sharp Huygens principle (the persistence of a wake), though the geometric interpretation in terms of the method of descent was clarified only by Hadamard a century later.

Kirchhoff 1882 Zur Theorie der Lichtstrahlen ^{[Kirchhoff 1882]} gave the cleanest and most general form of the three-dimensional representation formula for the wave equation, the Kirchhoff formula. Kirchhoff's 1882 paper was motivated by physical optics: the integral representation gives an analytic version of the Huygens-Fresnel construction of secondary wavelets and is the foundation of mathematical diffraction theory (Kirchhoff diffraction integral, Fresnel and Fraunhofer diffraction patterns, the Helmholtz-Kirchhoff theorem on the angular spectrum representation of waves). Kirchhoff also clarified the role of the sharp mathematical Huygens principle in three dimensions and the contrast with the wake in two dimensions, which set the stage for Hadamard's systematic treatment.

Hadamard 1923 Lectures on Cauchy's Problem in Linear Partial Differential Equations ^{[Hadamard 1923]} gave the first systematic treatment of the Cauchy problem for second-order linear PDEs, including the wave equation in arbitrary dimension. Hadamard introduced the method of descent for deriving the two-dimensional Poisson formula from the three-dimensional Kirchhoff formula, the precise statement of the sharp Huygens principle and its odd-versus-even-dimensional distinction, and the modern formulation of well-posedness in the sense of existence, uniqueness, and continuous dependence on the data (now called Hadamard well-posedness). Hadamard's example of the ill-posed Cauchy problem for the Laplace equation in the lower half-plane (which has explicit non-uniqueness via exponentially growing perturbations of the initial data) is the standard introduction to the concept of an ill-posed PDE. The 1923 lectures laid the foundation for the modern PDE theory and remain a definitive reference on the subject.

Friedrichs and Lewy 1928 gave the first explicit proof of the finite-propagation-speed theorem for the wave equation by the energy-method argument on a backward light cone. Friedrichs's subsequent work on symmetric hyperbolic systems (Friedrichs 1954 Comm. Pure Appl. Math.) extended the energy method to general first-order hyperbolic systems and established the standard well-posedness framework for hyperbolic PDEs. The Friedrichs-Lewy work was also the origin of the Courant-Friedrichs-Lewy (CFL) condition (Courant-Friedrichs-Lewy 1928 Math. Ann. 100) for finite-difference numerical schemes for hyperbolic equations: the numerical wave speed must exceed the physical wave speed for the scheme to be stable, the discrete analogue of the finite-propagation-speed theorem.

Strichartz 1977 Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations ^{[Strichartz 1977]} proved the foundational dispersive estimates that bear his name. The Strichartz estimates quantify the dispersive decay of free solutions of the wave equation and the Schrödinger equation, and have become the standard tool for proving local and global existence for nonlinear dispersive equations via fixed-point arguments. The modern theory of nonlinear wave and Schrödinger equations (Klainerman 1985, Tao, Kenig-Merle, Christodoulou) rests on the Strichartz framework.

John 1979 Blow-up of solutions of nonlinear wave equations in three space dimensions ^{[John 1979]} gave the first explicit blowup theorem for the semilinear wave equation, identifying the critical exponent in three space dimensions. Strauss 1981 conjectured the full classification of focusing-versus-defocusing exponents, which became known as the Strauss exponent. The Strauss conjecture was completed through the contributions of many authors over a period of three decades, culminating in the resolution by Zhou 1995 and Yordanov-Zhang 2006 of the remaining edge cases. The semilinear wave equation theory is the model case for the modern study of nonlinear hyperbolic equations and is the entry point to the analysis of nonlinear field equations in general relativity and quantum field theory.

The wave equation now appears as the foundational example in essentially every textbook of partial differential equations and mathematical physics. Its applications span classical mechanics (vibrating strings, drumheads, plates, shells), classical field theory (electromagnetism, gravitational waves, acoustic and elastic waves), quantum field theory (Klein-Gordon and Dirac equations, photon and graviton propagation), and applied science (seismology, oceanography, atmospheric science, plasma physics, signal processing, medical ultrasound, radar and sonar). The d'Alembert-Kirchhoff-Poisson formulas remain the prototype representation formulas for every linear hyperbolic equation, and the spherical-means apparatus extends to the Lorentzian-manifold setting via the Leray-Hadamard fundamental solution. The arc from d'Alembert's 1747 vibrating-string paper to the modern Christodoulou-Klainerman 1993 proof of the global stability of Minkowski space and the LIGO-Virgo 2015 observation of gravitational waves is a 270-year lineage in which the same equation has been continuously refined into ever more general and ever more powerful tools.

Bibliography Master

@article{DAlembert1747,
  author  = {d'Alembert, Jean le Rond},
  title   = {Recherches sur la courbe que forme une corde tendue mise en vibration},
  journal = {M\'emoires de l'Acad\'emie Royale des Sciences et Belles-Lettres de Berlin},
  volume  = {3},
  year    = {1747},
  pages   = {214--219}
}

@article{Euler1748,
  author  = {Euler, Leonhard},
  title   = {Sur la vibration des cordes},
  journal = {M\'emoires de l'Acad\'emie Royale des Sciences et Belles-Lettres de Berlin},
  volume  = {4},
  year    = {1748},
  pages   = {69--85}
}

@article{Bernoulli1755,
  author  = {Bernoulli, Daniel},
  title   = {R\'eflexions et \'eclaircissemens sur les nouvelles vibrations des cordes},
  journal = {M\'emoires de l'Acad\'emie Royale des Sciences et Belles-Lettres de Berlin},
  volume  = {9},
  year    = {1755},
  pages   = {147--172}
}

@article{Lagrange1759,
  author  = {Lagrange, Joseph-Louis},
  title   = {Recherches sur la nature et la propagation du son},
  journal = {Miscellanea Taurinensia},
  volume  = {1},
  year    = {1759},
  pages   = {1--112}
}

@article{Poisson1818,
  author  = {Poisson, Sim\'eon Denis},
  title   = {M\'emoire sur l'int\'egration de quelques \'equations lin\'eaires aux diff\'erences partielles},
  journal = {M\'emoires de l'Acad\'emie Royale des Sciences de l'Institut de France},
  volume  = {3},
  year    = {1818},
  pages   = {121--176}
}

@article{Kirchhoff1882,
  author  = {Kirchhoff, Gustav},
  title   = {Zur {T}heorie der {L}ichtstrahlen},
  journal = {Annalen der Physik und Chemie},
  volume  = {18},
  year    = {1882},
  pages   = {663--695}
}

@book{Hadamard1923,
  author    = {Hadamard, Jacques},
  title     = {Lectures on {C}auchy's Problem in Linear Partial Differential Equations},
  publisher = {Yale University Press},
  year      = {1923}
}

@article{Strichartz1977,
  author  = {Strichartz, Robert S.},
  title   = {Restrictions of {F}ourier transforms to quadratic surfaces and decay of solutions of wave equations},
  journal = {Duke Mathematical Journal},
  volume  = {44},
  year    = {1977},
  pages   = {705--714}
}

@book{Evans2010,
  author    = {Evans, Lawrence C.},
  title     = {Partial Differential Equations},
  edition   = {2},
  publisher = {American Mathematical Society},
  series    = {Graduate Studies in Mathematics},
  volume    = {19},
  year      = {2010}
}

@book{John1982,
  author    = {John, Fritz},
  title     = {Partial Differential Equations},
  edition   = {4},
  publisher = {Springer},
  year      = {1982}
}

@book{Strauss2008,
  author    = {Strauss, Walter A.},
  title     = {Partial Differential Equations: An Introduction},
  edition   = {2},
  publisher = {Wiley},
  year      = {2008}
}

@book{CourantHilbert1962,
  author    = {Courant, Richard and Hilbert, David},
  title     = {Methods of Mathematical Physics II: Partial Differential Equations},
  publisher = {Interscience},
  year      = {1962}
}

@book{Sogge2008,
  author    = {Sogge, Christopher D.},
  title     = {Lectures on Non-Linear Wave Equations},
  edition   = {2},
  publisher = {International Press},
  year      = {2008}
}

@article{John1979,
  author  = {John, Fritz},
  title   = {Blow-up of solutions of nonlinear wave equations in three space dimensions},
  journal = {Manuscripta Mathematica},
  volume  = {28},
  year    = {1979},
  pages   = {235--268}
}

Prerequisites

02.13.01
02.13.02
02.13.03

Tier anchors

beginner: Strauss, Partial Differential Equations: An Introduction, 2e (Wiley 2008), §1, §2, §9; physics-anchored vibrating string and expanding pond ripple; d'Alembert 1747 Mémoires de l'Académie de Berlin (originator: d'Alembert formula for the one-dimensional wave equation on the line)
intermediate: Evans, Partial Differential Equations, 2e (AMS GSM 19, 2010), §2.4; Strauss §2, §9; John, Partial Differential Equations, 4e (Springer 1982), §5
master: Evans §2.4; John §5; Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale UP 1923); Courant-Hilbert, Methods of Mathematical Physics II (Interscience 1962), §VI; Strichartz, A Guide to Distribution Theory and Fourier Transforms (CRC 1994); Sogge, Lectures on Non-Linear Wave Equations, 2e (International Press 2008)

References

d'Alembert — Recherches sur la courbe que forme une corde tendue mise en vibration · Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin 3 (1747), 214-219; originator: d'Alembert formula for the one-dimensional wave equation
Euler — Sur la vibration des cordes · Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin 4 (1748), 69-85
Bernoulli — Réflexions et éclaircissemens sur les nouvelles vibrations des cordes · Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin 9 (1755), 147-172
Lagrange — Recherches sur la nature et la propagation du son · Miscellanea Taurinensia 1 (1759), 1-112
Poisson — Mémoire sur l'intégration de quelques équations linéaires aux différences partielles · Mémoires de l'Académie Royale des Sciences de l'Institut de France 3 (1818), 121-176
Kirchhoff — Zur Theorie der Lichtstrahlen · Annalen der Physik und Chemie 18 (1882), 663-695
Hadamard — Lectures on Cauchy's Problem in Linear Partial Differential Equations · Yale University Press (1923); method of descent, ill-posedness, Huygens principle, fundamental solutions
Strichartz — Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations · Duke Mathematical Journal 44 (1977), 705-714
Evans — Partial Differential Equations, 2e · AMS Graduate Studies in Mathematics 19 (2010), §2.4
John — Partial Differential Equations, 4e · Springer (1982), §5
Strauss — Partial Differential Equations: An Introduction, 2e · Wiley (2008), §1, §2, §9
Courant-Hilbert — Methods of Mathematical Physics II · Interscience (1962), §VI
Sogge — Lectures on Non-Linear Wave Equations, 2e · International Press (2008)
John — Blow-up of solutions of nonlinear wave equations in three space dimensions · Manuscripta Mathematica 28 (1979), 235-268

Estimated time

beginner: 25m
intermediate: 60m
master: 110m