## 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. [definition:Spherical Mean] Let $v: \mathbb{R}^n \to \mathbb{R}$ be a [continuous](/page/Continuity) function and let $x \in \mathbb{R}^n$. The **spherical mean** of $v$ over $\partial B(x, r)$ is \begin{align*} (M_r v)(x) := \frac{1}{\mathcal{H}^{n-1}(\partial B(x, r))}\int_{\partial B(x, r)} v(y) \, d\mathcal{H}^{n-1}(y), \end{align*} where $\mathcal{H}^{n-1}(\partial B(x, r)) = n\omega_n r^{n-1}$ is the $(n-1)$-dimensional [Hausdorff measure](/page/Hausdorff%20Measure) of the sphere, and $\omega_n := \mathcal{L}^n(B(0, 1)) = \pi^{n/2}/\Gamma(n/2 + 1)$ is the volume of the unit ball in $\mathbb{R}^n$. [/definition] The spherical mean can be rewritten as an integral over the unit sphere $\partial B(0, 1)$ via the change of variables $\varphi: \partial B(0, 1) \to \partial B(x, r)$, $\omega \mapsto x + r\omega$. Since $\varphi$ is a dilation by $r$ followed by a translation by $x$, the area element transforms as $d\mathcal{H}^{n-1}(y) = r^{n-1} \, d\mathcal{H}^{n-1}(\omega)$. Substituting into the definition: \begin{align*} (M_r v)(x) &= \frac{1}{n\omega_n r^{n-1}}\int_{\partial B(0, 1)} v(x + r\omega) \, r^{n-1} \, d\mathcal{H}^{n-1}(\omega) = \frac{1}{n\omega_n}\int_{\partial B(0, 1)} v(x + r\omega) \, d\mathcal{H}^{n-1}(\omega). \end{align*} Since $\mathcal{H}^{n-1}(\partial B(0, 1)) = n\omega_n$, this is simply the average of $v \circ \varphi$ over the unit sphere. This form is often more convenient for differentiation with respect to $r$, since the domain of integration no longer depends on $r$. The connection between spherical means and the Laplacian is the identity \begin{align*} (M_r(\Delta v))(x) = \left(\partial_r^2 + \frac{n - 1}{r}\partial_r\right)(M_r v)(x), \end{align*} valid for any $v \in C^2(\mathbb{R}^n)$. This is proved by differentiating the surface integral with respect to $r$ and applying the divergence theorem on $B(x, r)$. If $u$ solves $\partial_t^2 u = \Delta u$ and we set $U(x; r, t) := (M_r u(\cdot, t))(x)$, then: \begin{align*} \partial_t^2 U = M_r(\Delta u) = \partial_r^2 U + \frac{n - 1}{r}\partial_r U. \end{align*} This is the **Euler–Poisson–Darboux equation**: a second-order PDE in $(r, t)$ with a singular coefficient $(n-1)/r$ at $r = 0$. For general $n$, the singularity makes direct solution difficult. But for $n = 3$, a simple substitution eliminates it. ### Kirchhoff's Formula For $n = 3$, the EPD equation reads $\partial_t^2 U = \partial_r^2 U + \frac{2}{r}\partial_r U$. The substitution $\tilde{U} := rU$ transforms this into $\partial_t^2 \tilde{U} = \partial_r^2 \tilde{U}$ — exactly the one-dimensional wave equation. We can therefore apply d'Alembert's formula to $\tilde{U}$, express the result in terms of spherical means of the initial data, and recover $u(x, t) = \lim_{r \to 0} U(x; r, t)$ by a limiting argument. The outcome is Kirchhoff's formula, the fundamental representation of solutions in three dimensions. [quotetheorem:666] Kirchhoff's formula has a striking structural feature: the value $u(x, t)$ depends on the initial data only through their spherical averages over $\partial B(x, t)$. The data on the *interior* of $B(x, t)$ — and certainly the data outside $B(x, t)$ — play no role. This is much sharper than finite propagation speed alone would guarantee: not only does information travel at speed at most $1$, but in three dimensions, information from a point source travels at speed *exactly* $1$. A sharp wave front emitted at the origin at $t = 0$ passes through the point $x$ at time $t = |x|$ and is gone — there is no residual disturbance. This is **Huygens' principle** in its strong form. ### Hadamard's Method of Descent Having solved the wave equation in three dimensions, Hadamard observed that the two-dimensional case can be obtained by a simple dimensional reduction. The idea is to view a function $v(x_1, x_2)$ of two variables as a function $\tilde{v}(x_1, x_2, x_3) := v(x_1, x_2)$ of three variables that happens to be independent of $x_3$. If $\tilde{u}$ solves the three-dimensional wave equation with this extended data, then by uniqueness $\tilde{u}$ is also independent of $x_3$, and its restriction to $x_3 = 0$ solves the two-dimensional problem. Applying Kirchhoff's formula and projecting the sphere $\partial B(x, t) \subset \mathbb{R}^3$ onto the disk $B(\bar{x}, t) \subset \mathbb{R}^2$ yields Poisson's formula. [quotetheorem:667] The qualitative difference from Kirchhoff's formula is immediate and significant. In three dimensions, $u(x, t)$ depends on the data restricted to the *sphere* $\partial B(x, t)$. In two dimensions, $u(x, t)$ depends on the data throughout the *disk* $B(x, t)$ — the full interior is involved, weighted by the singular kernel $(t^2 - |y - x|^2)^{-1/2}$. This means that in two dimensions, a sharp initial disturbance leaves a residual "wake" behind the leading wave front. A stone dropped in a pond produces an expanding circular wave, but the water inside the circle does not return immediately to rest — it continues to oscillate (weakly) long after the leading front has passed. ### Huygens' Principle and the Even–Odd Dichotomy The contrast between Kirchhoff's formula (data on the sphere) and Poisson's formula (data on the disk) is the simplest instance of a deep dimensional dichotomy. Huygens' principle — the assertion that the domain of dependence of $(x, t)$ is exactly the sphere $\partial B(x, t)$, not the ball $B(x, t)$ — holds in odd dimensions $n \geq 3$ but fails in even dimensions (and in $n = 1$, where the domain of dependence is the two-point set $\{x - t, x + t\}$, which is the zero-dimensional "sphere"). The method of descent explains why: passing from dimension $n$ to $n - 1$ involves integrating over a fibre, which always fills in the interior. So the sharp sphere-dependence in odd dimensions $n$ gets smeared into disk-dependence in even dimensions $n - 1$. This means signals in three-dimensional space (sound, light) propagate cleanly, while signals on a two-dimensional membrane leave lingering traces. [example:Kirchhoff Applied To A Radial Gaussian] Consider the Cauchy problem in $\mathbb{R}^3$ with $g(x) = e^{-|x|^2}$ and $h(x) = 0$. By [Kirchhoff's Formula](/theorems/666): \begin{align*} u(x, t) = \partial_t\bigl[t \, (M_t g)(x)\bigr]. \end{align*} Since $g$ is radially symmetric, $g(y) = e^{-|y|^2}$, and the spherical mean of $g$ over $\partial B(x, t)$ can be computed by the identity \begin{align*} (M_t g)(x) = \frac{1}{4\pi t^2}\int_{\partial B(x, t)} e^{-|y|^2} \, d\mathcal{H}^2(y). \end{align*} For $x = 0$ (the origin), the integrand is constant on $\partial B(0, t)$ with value $e^{-t^2}$, so $(M_t g)(0) = e^{-t^2}$. Therefore: \begin{align*} u(0, t) = \partial_t\bigl[t \, e^{-t^2}\bigr] = e^{-t^2} - 2t^2 e^{-t^2} = (1 - 2t^2)e^{-t^2}. \end{align*} At $t = 0$, $u(0, 0) = 1 = g(0)$, consistent with the initial condition. The solution at the origin changes sign at $t = 1/\sqrt{2}$ and decays to zero as $t \to \infty$. Unlike the heat equation — where the solution at the origin would be $u(0, t) = (1 + 4t)^{-3/2}$ and remain strictly positive — the wave solution oscillates through zero. After $t = 1/\sqrt{2}$, the initial Gaussian has passed through the origin and moved outward, leaving a negative displacement behind it. By $t \approx 2$, the displacement at the origin is essentially zero: the wave has passed. This is Huygens' principle at work — in three dimensions, the signal clears completely. [/example] ## The Nonhomogeneous Problem ### Duhamel's Principle Many physical wave problems include a forcing term: $\partial_t^2 u - \Delta u = f$, where $f(x, t)$ represents an external source (e.g., a vibrating membrane driven by an oscillating pressure). The homogeneous representation formulas — d'Alembert, Kirchhoff, Poisson — solve only the unforced equation with nonzero initial data. To handle forcing, we need a principle that reduces the nonhomogeneous problem to a family of homogeneous ones. The key observation is that linearity allows us to decompose the effect of the source $f$ into infinitesimal impulses: at each time $s$, the source $f(\cdot, s)$ acts as an instantaneous kick to the velocity, which then propagates forward in time by the homogeneous equation. Integrating over $s$ assembles the full solution. [quotetheorem:668] The structure of Duhamel's principle mirrors the variation-of-parameters formula for ODEs. For the ODE $u'' = f(t)$ with $u(0) = u'(0) = 0$, the solution is $u(t) = \int_0^t (t - s)f(s) \, d\mathcal{L}^1(s)$, which is the integral of the homogeneous propagator $(t - s) \cdot f(s)$ over $s$. The PDE version replaces the scalar propagator $(t - s)$ with the solution operator of the homogeneous wave equation, but the idea is identical. The regularity requirement on $f$ ensures that each constituent $w(\cdot, \cdot; s)$ is smooth enough that integration over $s$ produces a $C^2$ function. By linearity, the general solution of $\partial_t^2 u - \Delta u = f$ with initial data $(g, h)$ is the sum of: the homogeneous solution with data $(g, h)$ (given by d'Alembert, Kirchhoff, or Poisson), plus the Duhamel correction with zero initial data and source $f$. [example:One Dimensional Forced Wave Equation] Consider the problem in $n = 1$ with $g = h = 0$ and $f(x, t) = \sin(t)$ for all $x \in \mathbb{R}$, $t > 0$. By [Duhamel's Principle For The Wave Equation](/theorems/668), with $w(x, t; s)$ solving the homogeneous wave equation with initial velocity $\sin(s)$ (constant in $x$), d'Alembert's formula gives: \begin{align*} w(x, t; s) = \frac{1}{2}\int_{x - (t-s)}^{x + (t-s)} \sin(s) \, d\mathcal{L}^1(y) = (t - s)\sin(s). \end{align*} Here we used that the initial velocity is the constant function $y \mapsto \sin(s)$, so the integral over an interval of length $2(t - s)$ equals $2(t - s)\sin(s)$, and the factor $\frac{1}{2}$ from d'Alembert's formula yields $(t - s)\sin(s)$. Therefore: \begin{align*} u(x, t) = \int_0^t (t - s)\sin(s) \, d\mathcal{L}^1(s). \end{align*} [Integration by parts](/theorems/210) (with $u_{\text{ibp}} = t - s$, $dv_{\text{ibp}} = \sin(s) \, ds$): \begin{align*} u(x, t) &= \bigl[-(t - s)\cos(s)\bigr]_0^t + \int_0^t (-\cos(s)) \, d\mathcal{L}^1(s) \\ &= 0 - (-t \cdot 1) + \bigl[-\sin(s)\bigr]_0^t \\ &= t - \sin(t). \end{align*} One verifies: $\partial_t^2 u = \sin(t)$ and $u(x, 0) = 0$, $\partial_t u(x, 0) = 1 - \cos(0) = 0$. The solution is spatially homogeneous (independent of $x$), which reflects the fact that the forcing $f = \sin(t)$ is itself independent of $x$ — the source drives every point identically, so no spatial variation is created. [/example] ## Energy Methods ### The Energy Functional The representation formulas of the preceding sections give *explicit* solutions, but they require explicit knowledge of the fundamental solution (or spherical means) and impose specific regularity on the data. Energy methods take a complementary approach: they give *qualitative* information — uniqueness, stability, propagation speed — without ever writing down a formula for $u$. The central object is the total energy, which packages the kinetic and potential energies of the wave into a single conserved quantity. [definition:Energy Of The Wave Equation] Let $u \in C^2(\mathbb{R}^n \times [0, T]; \mathbb{R})$ solve $\partial_t^2 u - \Delta u = 0$. The **energy** of $u$ at time $t$ is \begin{align*} e: [0, T] &\to [0, \infty) \\ t &\mapsto \frac{1}{2}\int_{\mathbb{R}^n}\bigl((\partial_t u(x, t))^2 + |\nabla u(x, t)|^2\bigr) \, d\mathcal{L}^n(x). \end{align*} The first term $\frac{1}{2}\int (\partial_t u)^2 \, d\mathcal{L}^n$ is the **kinetic energy** and the second $\frac{1}{2}\int |\nabla u|^2 \, d\mathcal{L}^n$ is the **potential (elastic) energy**. [/definition] A direct computation shows that $e'(t) = 0$ for solutions of the homogeneous wave equation: differentiating under the integral and integrating by parts yields \begin{align*} e'(t) = \int_{\mathbb{R}^n} \partial_t u \bigl(\partial_t^2 u - \Delta u\bigr) \, d\mathcal{L}^n = 0. \end{align*} Energy conservation is the fundamental identity of wave mechanics. Unlike the heat equation — where the energy $\frac{1}{2}\int u^2 \, d\mathcal{L}^n$ is monotonically *decreasing* (reflecting dissipation) — the wave energy is exactly conserved. No energy is created or destroyed; it merely converts between kinetic and potential forms as the wave oscillates. ### Uniqueness The representation formulas establish uniqueness in dimensions $n = 1, 2, 3$ as a byproduct of providing explicit solutions. But explicit formulas are not available in all settings — for instance, on bounded domains, with variable coefficients, or in general dimensions. Energy methods supply a proof of uniqueness that works in complete generality and requires no knowledge of the solution's structure. The argument is beautifully simple: if $u_1$ and $u_2$ are two solutions with the same data, their difference $w = u_1 - u_2$ solves the wave equation with zero data. The energy $e(t)$ of $w$ is therefore constant and equal to $e(0) = 0$, which forces $\partial_t w = 0$ and $\nabla w = 0$, hence $w \equiv 0$. [quotetheorem:669] The energy proof of uniqueness is robust: it extends immediately to bounded domains with Dirichlet or Neumann [boundary](/page/Boundary) conditions, to equations with lower-order terms $\partial_t^2 u - \Delta u + c(x)u = f$ (provided $c \geq 0$), and to systems of wave equations. The only ingredients are the ability to integrate by parts and the non-negativity of the energy integrand — both of which are available in far greater generality than the explicit representation formulas. ### Finite Speed of Propagation The representation formulas show, in each specific dimension, that the value $u(x_0, t_0)$ depends only on initial data within a ball of radius $t_0$ centred at $x_0$. Energy methods establish this **finite speed of propagation** in all dimensions simultaneously, without any explicit formula. The idea is to localise the energy argument to a backward cone: instead of integrating over all of $\mathbb{R}^n$, we integrate over the ball $B(x_0, t_0 - t)$ — the spatial cross-section of the cone $\{|x - x_0| \leq t_0 - t\}$ at time $t$. The shrinking domain introduces a boundary flux term, but the Cauchy–Schwarz inequality shows this flux has the right sign, so the local energy is non-increasing. [quotetheorem:670] The domain-of-dependence result says that the solution at $(x_0, t_0)$ is completely determined by the data on $\overline{B}(x_0, t_0)$. Its dual statement, the **domain of influence**, is equally important: data supported in a [set](/page/Set) $S \subset \mathbb{R}^n$ can influence the solution only within the region $\{(x, t) : \mathrm{dist}(x, S) \leq t\}$ — a thickening of $S$ that expands at unit speed. In physical terms, signals travel at speed at most $1$ (the wave speed). No faster-than-light communication is possible — a principle that is built into the mathematical structure of the wave equation through the hyperbolicity of its symbol. [example:Energy Conservation For A Compactly Supported Pulse] Let $n = 1$, $g(x) = \mathbb{1}_{[-1, 1]}(x)$, and $h(x) = 0$. Although this data is not $C^2$, the energy computation is still instructive at a formal level. The initial energy is: \begin{align*} e(0) = \frac{1}{2}\int_{\mathbb{R}} \bigl(h(x)^2 + g'(x)^2\bigr) \, d\mathcal{L}^1(x). \end{align*} Since $h = 0$, the kinetic energy is zero. The [derivative](/page/Derivative) $g'$ is zero on $(-1, 1)$ and on $(-\infty, -1) \cup (1, \infty)$, with $\delta$-function singularities at $x = \pm 1$; in the [distributional](/page/Distribution) sense, $g' = \delta_{-1} - \delta_1$, and the $L^2$ norm is not finite. This reflects the fact that $g \notin H^1(\mathbb{R})$: the energy framework requires at least one derivative of $g$ in $L^2$, which is precisely the Sobolev regularity $g \in H^1$. For a smooth approximation $g_\varepsilon$ of $\mathbb{1}_{[-1,1]}$ with $g_\varepsilon' \in L^2$, the energy $e_\varepsilon(0)$ is finite and conserved. As $\varepsilon \to 0$, $e_\varepsilon(0) \to \infty$, reflecting the infinite energy carried by the discontinuities in the indicator function. This example illustrates why the classical theory requires $g \in C^2$ and $h \in C^1$: these conditions guarantee that the energy integral converges. The weak theory in [Sobolev spaces](/page/Sobolev%20Spaces) relaxes this to $g \in H^1$ and $h \in L^2$ — the natural energy space for the wave equation. [/example] ## References Evans, L. C., *Partial Differential Equations* (2nd ed., 2010), §2.4. John, F., *Partial Differential Equations* (4th ed., 1982), Chapters 2 and 5. Courant, R. and Hilbert, D., *Methods of Mathematical Physics, Vol. II* (1962), Chapter VI.