The previous page closed with an
observation we left as a marker. The heat equation is irreversible — fine spatial
structure is destroyed by exponential damping, and recovering it from the future state
would require an exponentially unstable amplification — and we noted that this stands
in sharp contrast to the time-reversibility of the wave equation, where the backward
problem is as well-posed as the forward one and no information is lost. We now
return to that marker. Replacing the first-order time derivative of the heat equation
by a second-order one, while retaining the same spatial Laplacian and the same Dirichlet
boundary conditions, produces an evolution that inverts the parabolic case in every
qualitative respect: oscillation in place of decay, reversibility in place of dissipation,
energy conservation in place of irreversible smoothing.
The wave equation
\[
\partial_{tt} u(x, t) = c^2\, \partial_{xx} u(x, t)
\]
is the canonical example of a hyperbolic partial differential equation,
where \(u(x, t)\) represents the transverse displacement of a vibrating string — or, more
abstractly, the displacement field of a propagating wave — at position \(x\) and time \(t\), and
\(c > 0\) is a positive constant called the wave speed. The equation
governs small-amplitude vibrations of a taut elastic string, the propagation of sound in
a one-dimensional medium, and, in higher-dimensional analogues, the propagation of
light, sound, and gravitational waves in physical space. Where the heat equation's
natural physical content is dissipation — the smoothing-out of inhomogeneities by
diffusion — the wave equation's natural physical content is propagation: a localized
disturbance travels through the medium at a definite finite speed, retaining its shape
along characteristic directions in spacetime in the one-dimensional setting treated here.
The wave equation is the second representative of the classical trichotomy of linear
second-order PDEs — parabolic, hyperbolic, elliptic — whose third member, the Laplace
equation, will be developed in a subsequent page. The present treatment is
one-dimensional and classical: separation of variables on a bounded interval, the
method of characteristics on the real line, the d'Alembert formula, and the proof
that mechanical energy is exactly conserved — closing with a contrast against the
heat equation that runs through reversibility, regularity, propagation speed, and
the conservation law itself. Higher-dimensional wave equations and the general theory
of hyperbolic Cauchy problems are not pursued here.
Wave Equation on a Bounded Interval
We begin with the prototype problem that runs in parallel with the heat conduction
setup of the previous page: transverse vibrations of a taut elastic string of length
\(L\) clamped at both ends. The mathematical formulation is the
initial-boundary value problem
\[
\begin{cases}
\partial_{tt} u = c^2\, \partial_{xx} u, & 0 < x < L,\ t > 0, \\
u(0, t) = u(L, t) = 0, & t \geq 0, \\
u(x, 0) = f(x), & 0 \leq x \leq L, \\
\partial_t u(x, 0) = g(x), & 0 \leq x \leq L,
\end{cases}
\]
where \(f\) prescribes the initial displacement and \(g\) prescribes the initial
velocity. Two initial conditions appear in place of the heat equation's single
\(u(\cdot, 0) = f\): this is the first concrete consequence of replacing a first-order
time derivative by a second-order one, since a well-posed second-order ODE in \(t\)
requires two pieces of data at \(t = 0\). The boundary conditions
\(u(0,t) = u(L,t) = 0\) are again the homogeneous Dirichlet conditions,
corresponding physically to the two ends of the string being pinned in place. Our goal is
an explicit solution formula that exposes both how the displacement evolves and the
sharp structural inversion from the parabolic case — exponential decay replaced by
sustained oscillation, single initial datum replaced by two.
Separation of Variables
We seek solutions in the product form
\[
u(x, t) = \phi(x)\, G(t),
\]
again setting aside the initial conditions and demanding only that \(u\) satisfy the PDE
and the homogeneous boundary conditions. Substituting into the wave equation gives
\(\phi(x)\, G''(t) = c^2\, \phi''(x)\, G(t)\). Dividing both sides by
\(c^2\, \phi(x)\, G(t)\) separates the variables:
\[
\frac{G''(t)}{c^2\, G(t)} = \frac{\phi''(x)}{\phi(x)} = -\lambda,
\]
where the common constant is again denoted \(-\lambda\) by a sign convention chosen with
foresight — as for the heat equation, the physically relevant eigenvalues will turn out
to be positive, but now positive \(\lambda\) drives \(G(t)\) into oscillation
rather than exponential decay. The separation produces two ordinary differential
equations,
\[
\begin{align*}
\phi''(x) + \lambda\, \phi(x) &\;=\; 0, \\\\
G''(t) + \lambda c^2\, G(t) &\;=\; 0,
\end{align*}
\]
coupled only through \(\lambda\). The Dirichlet boundary conditions translate, as before,
into \(\phi(0) = \phi(L) = 0\). The spatial equation and its boundary conditions are
identical to those of the heat case; only the temporal equation has changed — and its
change, from first order to second order, is precisely the source of every qualitative
difference between the two evolutions.
The Spatial Eigenvalue Problem
The spatial boundary value problem
\[
\begin{align*}
\phi''(x) + \lambda\, \phi(x) &\;=\; 0, \\\\
\phi(0) = \phi(L) &\;=\; 0
\end{align*}
\]
is identical to the eigenvalue problem that arose in the separation analysis of the
heat equation. The three-case analysis (\(\lambda < 0\), \(\lambda = 0\), \(\lambda > 0\))
that ruled out non-positive eigenvalues and produced the sine eigenfunctions for positive
\(\lambda\) was carried out in full there; we do not repeat it. The result is summarised
by the previous page's
theorem on the eigenvalues and eigenfunctions of the Dirichlet Laplacian on \([0, L]\):
\[
\lambda_n = \left(\frac{n\pi}{L}\right)^{\!2}, \qquad
\phi_n(x) = \sin\frac{n\pi x}{L}, \qquad n = 1, 2, 3, \ldots
\]
What changes between the heat and wave equations is not the spatial spectrum — both
problems diagonalize the same operator — but how each eigenvalue acts on its mode in
time.
The Temporal ODE
With \(\lambda = \lambda_n\) fixed, the time-dependent equation
\[
G''(t) + \lambda_n c^2\, G(t) = 0
\]
is a second-order linear ODE with constant positive coefficient \(\lambda_n c^2\). Writing
\(\omega_n := c\sqrt{\lambda_n} = n\pi c / L\) for the circular frequency of the
\(n\)th mode, the characteristic roots are \(r = \pm i \omega_n\), and the
general solution is the linear combination
\[
G_n(t) = B_n \cos\omega_n t + C_n \sin\omega_n t,
\]
with two free constants \(B_n, C_n\) reflecting the two initial conditions of the
second-order ODE. The contrast with the heat case is exact and instructive: there,
\(G'(t) + \lambda_n k\, G(t) = 0\) is a first-order equation whose general solution is
a one-parameter family \(G_n(t) = B_n e^{-\lambda_n k t}\); here, the temporal evolution
is a two-parameter family of trigonometric oscillations of frequency \(\omega_n\). The
decay rate of the parabolic case has been replaced by a temporal frequency: heat's
\(e^{-\lambda_n k t}\) is the wave equation's \(\cos(c\sqrt{\lambda_n}\, t)\) and
\(\sin(c\sqrt{\lambda_n}\, t)\), with the same eigenvalue \(\lambda_n\) playing
structurally opposite roles in the two evolutions.
A note on notation. The letter \(B_n\) is the same symbol used in the previous page
for the Fourier sine coefficient of the initial datum, and we shall presently see that
in the wave case it again denotes the Fourier sine coefficient of \(f\); the choice is
deliberate, preserving cross-page consistency. The letter \(C_n\), new to this page, is
introduced for the second family of coefficients arising from the second initial
condition; it will carry information from the initial velocity \(g\).
The \(n\)th product solution or standing wave is the
product of the spatial eigenfunction with its temporal evolution:
\[
u_n(x, t) = \sin\frac{n\pi x}{L}\, \bigl[B_n \cos\omega_n t + C_n \sin\omega_n t\bigr].
\]
Each \(u_n\) solves the wave equation and the Dirichlet boundary conditions. Where the
heat equation's modes are sine waves of fixed shape damped by a decaying amplitude, the
wave equation's modes are sine waves of fixed shape modulated by an oscillating amplitude
that does not decay. The shape \(\sin(n\pi x / L)\) is again preserved in \(x\); what is
new is that the amplitude does not shrink — it swings periodically at frequency
\(\omega_n\), and the \(n\)th mode vibrates indefinitely.
Superposition and the Series Solution
The wave equation is linear and homogeneous, and the boundary conditions are also
homogeneous; consequently any finite linear combination of product solutions is again a
solution of the PDE and the boundary conditions. The principle of
superposition extends this, formally, to infinite series. To match both
initial conditions, we must allow both the cosine and the sine temporal components in
each mode, and so the full ansatz takes the form
\[
u(x, t) = \sum_{n=1}^{\infty} \sin\frac{n\pi x}{L}\,
\Bigl[B_n \cos\frac{n\pi c t}{L} + C_n \sin\frac{n\pi c t}{L}\Bigr],
\]
in which two independent families of coefficients \(\{B_n\}\) and \(\{C_n\}\) jointly
encode the two initial conditions. The heat case carried a single Fourier family because
its first-order time evolution accommodated only one initial datum; the wave case carries
two, one for each.
To find the coefficients, we impose the initial conditions on the series. Evaluating at
\(t = 0\) collapses the cosine terms to one and the sine terms to zero, leaving
\[
u(x, 0) = \sum_{n=1}^{\infty} B_n \sin\frac{n\pi x}{L} = f(x),
\]
a Fourier sine series expansion of \(f\) on \([0, L]\). Differentiating the series
termwise in \(t\) and then evaluating at \(t = 0\) collapses the new cosine terms to
one and the new sine terms to zero, leaving
\[
\partial_t u(x, 0) = \sum_{n=1}^{\infty} C_n\, \frac{n\pi c}{L}\, \sin\frac{n\pi x}{L} = g(x),
\]
a Fourier sine series expansion of \(g\) — except that each term carries the extra
factor \(n\pi c / L\) introduced by differentiating \(\sin(n\pi ct/L)\) in \(t\). The
orthogonality of the trigonometric system
on \([0, L]\) extracts the coefficients from each series in the standard way: multiplying
by \(\sin(m\pi x / L)\), integrating over \([0, L]\), and using
\[
\int_0^L \sin\frac{n\pi x}{L} \sin\frac{m\pi x}{L}\, dx = \begin{cases} L/2, & m = n, \\ 0, & m \neq n. \end{cases}
\]
The first series gives \(B_n\) directly; the second gives \(C_n \cdot n\pi c / L\), which
we solve for \(C_n\). We collect all of this into a single statement.
Theorem: Series Representation of the Solution on a Bounded Interval
Let \(f \in C^2([0, L])\) and \(g \in C^1([0, L])\) satisfy the boundary
compatibility conditions
\[
f(0) = f(L) = 0, \qquad g(0) = g(L) = 0.
\]
Then the formal series solution of the initial-boundary value problem
\[
\begin{cases}
\partial_{tt} u = c^2\, \partial_{xx} u, & 0 < x < L,\ t > 0, \\
u(0, t) = u(L, t) = 0, & t \geq 0, \\
u(x, 0) = f(x), \quad \partial_t u(x, 0) = g(x), & 0 \leq x \leq L
\end{cases}
\]
is given by
\[
u(x, t) = \sum_{n=1}^{\infty} \sin\frac{n\pi x}{L}\,
\Bigl[B_n \cos\frac{n\pi c t}{L} + C_n \sin\frac{n\pi c t}{L}\Bigr],
\]
where the coefficients are determined from the initial data by
\[
\begin{align*}
B_n &\;=\; \frac{2}{L}\int_0^L f(x)\, \sin\frac{n\pi x}{L}\, dx, \\\\
C_n &\;=\; \frac{2}{n\pi c}\int_0^L g(x)\, \sin\frac{n\pi x}{L}\, dx.
\end{align*}
\]
The series above is built constructively from the eigenfunction decomposition, and
each partial sum satisfies the wave equation and the Dirichlet boundary conditions
termwise, by direct differentiation; the coefficient formulas follow from the
orthogonality of the Fourier sine system applied to the initial conditions, exactly
as in the parabolic case. A rigorous proof that the full infinite series, together
with its first and second \(x\)- and \(t\)-derivatives, converges uniformly and yields
a classical \(C^2\) solution is more delicate than the parabolic analogue: it
requires initial data of strictly higher regularity than the natural class
\(f \in C^2, g \in C^1\) admitted in the hypotheses above. Existence of a classical
solution under the natural assumptions is established by a different route in the next
section, where the method of characteristics yields an explicit \(C^2\) solution formula
by a construction that sidesteps the series-convergence problem entirely.
The contrast with the heat equation, which deserves emphasis, is exactly this:
the heat equation's exponential damping \(e^{-\lambda_n k t}\) regularizes initial
data of any \(L^2\) class into \(C^\infty\) for every \(t > 0\), as established on
the previous page; any roughness or incompatibility between the initial datum and
the boundary conditions is instantaneously smoothed away. The wave equation has no
such factor. The regularity of \(u(\cdot, t)\) at any time \(t > 0\) reflects exactly
the regularity of the initial data, including any boundary incompatibility, which is
propagated forward in time rather than dissipated. Initial data lying merely in
\(L^2\) — weaker than the hypotheses of the theorem above — generate a series
convergent in \(L^2\) at each \(t\), but the resulting \(u\) need not be twice
classically differentiable, and the equation
\(\partial_{tt} u = c^2 \partial_{xx} u\) holds only in a distributional sense beyond
the present scope. This is the wave equation's first qualitative inversion of the
parabolic case, and we return to it as a structural property in the heat-vs-wave
contrast that closes the next section.
Insight: Standing Waves and the Vibrating String
Each term in the series solution,
\[
\sin\frac{n\pi x}{L}\, \bigl[B_n \cos\omega_n t + C_n \sin\omega_n t\bigr],
\qquad \omega_n = \frac{n\pi c}{L},
\]
is a standing wave: a spatial profile fixed in shape, oscillating
in time at the circular frequency \(\omega_n\). The mode \(n = 1\), with the lowest
frequency \(\omega_1 = \pi c / L\), is the fundamental; the modes
\(n \geq 2\) are its harmonics, with frequencies \(\omega_n = n \omega_1\)
organised as integer multiples of the fundamental. The \(n\)th mode has exactly
\(n - 1\) interior zeros — its nodes, points of the string that do
not move — while the points midway between adjacent nodes oscillate with maximal
amplitude.
The frequency \(\omega_n = n\pi c / L\) depends on the wave speed \(c\) and the
string length \(L\). Recalling that \(c^2 = T_0 / \rho_0\) for a string of tension
\(T_0\) and linear mass density \(\rho_0\), the fundamental frequency is
\(\omega_1 = (\pi / L)\sqrt{T_0 / \rho_0}\): a string sounds higher when tightened
(larger \(T_0\)), when shortened (smaller \(L\)), or when made of a lighter material
(smaller \(\rho_0\)). This is the mathematical content of how a guitarist tunes a
string and how a violinist fingers a note by clamping the string at a chosen point
— physical operations whose effect on pitch is exactly an
operation on the Dirichlet eigenvalue spectrum.
A single standing wave admits an alternative description as the superposition of two
travelling waves moving in opposite directions. The product-to-sum identity gives
\[
\sin\frac{n\pi x}{L}\, \sin\frac{n\pi c t}{L}
\;=\; \tfrac{1}{2}\cos\frac{n\pi}{L}(x - ct) - \tfrac{1}{2}\cos\frac{n\pi}{L}(x + ct),
\]
and analogously for the cosine-cosine combination. Each standing mode is the
interference of a right-moving wave \(\cos(n\pi(x - ct)/L)\) and a left-moving wave
\(\cos(n\pi(x + ct)/L)\), each travelling at speed \(c\). This decomposition is not
an artefact of the trigonometric identity. It is the bounded-interval echo of a
structural fact: on the real line, every solution of the wave equation is a sum of
a right-moving and a left-moving travelling wave, with no eigenvalue spectrum
involved at all. We pursue that picture next.
Wave Equation on the Real Line
We turn to the wave equation on the entire real line:
\[
\partial_{tt} u = c^2\, \partial_{xx} u, \qquad
x \in \mathbb{R},\ t > 0,
\]
with initial conditions \(u(x, 0) = f(x)\) and
\(\partial_t u(x, 0) = g(x)\) and no boundary conditions imposed at any finite
point. Removing the boundary changes the picture entirely. There is no longer a
discrete spectrum to expand against; the eigenfunction expansion of the bounded
case has nothing to attach itself to. In its place a different mechanism appears,
one that is hidden by the standing-wave decomposition on the interval but central
to the hyperbolic nature of the equation: solutions are built from
travelling waves whose existence is a direct algebraic consequence of
the second-order operator's factorization. We develop this picture by first
sketching a brief Fourier-transform argument, which both yields and motivates the
right structural decomposition, and then constructing the solution rigorously by
a route that does not depend on Fourier analysis.
Motivation: A Fourier-Transform Sketch
Taking the spatial Fourier transform of the wave equation in \(x\), with the
convention \(\hat u(\xi, t) = \int_{\mathbb{R}} u(x, t)\, e^{i x \xi}\, dx\) and
the
differentiation rule
\(\partial_x \mapsto -i\xi\), the PDE becomes the family of second-order ODEs in
\(t\),
\[
\partial_{tt} \hat u(\xi, t) = -c^2 \xi^2\, \hat u(\xi, t),
\qquad \xi \in \mathbb{R},
\]
one equation per spatial frequency \(\xi\). The general solution at each
frequency is a linear combination of two trigonometric oscillations,
\[
\hat u(\xi, t) = A(\xi) \cos(c\xi t) + B(\xi) \sin(c\xi t),
\]
with \(\xi\)-dependent amplitudes determined by the initial data \(\hat f,
\hat g\). Rewriting the cosine and sine as complex exponentials,
\(\cos(c\xi t) = \tfrac{1}{2}(e^{i c\xi t} + e^{-i c\xi t})\) and similarly for
sine, regroups \(\hat u(\xi, t)\) as a sum of two terms of the form
\(\alpha(\xi)\, e^{i c\xi t}\) and \(\beta(\xi)\, e^{-i c\xi t}\). Each such
term contains a time-dependent phase factor in the frequency variable, and by
the
translation property of the Fourier transform
— which states that multiplication by the phase factor \(e^{i a \xi}\) on the
frequency side corresponds to the argument shift \(x \mapsto x - a\) on the
physical side — its inverse transform is a function
of \(x - ct\) or \(x + ct\) alone. Specifically, if \(\widetilde F\) and
\(\widetilde G\) denote the inverse transforms of \(\alpha\) and \(\beta\),
then the inverse transform of \(\alpha(\xi)\, e^{i c\xi t}\) is
\(\widetilde F(x - ct)\) and the inverse transform of
\(\beta(\xi)\, e^{-i c\xi t}\) is \(\widetilde G(x + ct)\) — waves of arbitrary
spatial shape translated rigidly at speed \(c\), to the right and to the left.
The transform calculation foretells the structure of the solution — every
component of \(u\) is a superposition of two travelling waves — without telling
us how to construct it without the transform. The construction below recovers
this structure directly from the PDE, with no reliance on Fourier machinery
and at the level of regularity natural to the problem.
The First-Order Transport Equation
Before factoring the second-order wave operator, we record the elementary fact
that drives the entire construction. Consider the first-order linear PDE
\[
\partial_t \psi + c\, \partial_x \psi = 0,
\qquad x \in \mathbb{R},\ t > 0,
\]
with arbitrary initial datum \(\psi(x, 0) = \psi_0(x)\). We claim that its
solution is the rigid translation \(\psi(x, t) = \psi_0(x - ct)\) — the initial
profile \(\psi_0\) carried unchanged to the right at speed \(c\).
The verification is direct: for any \(C^1\) function \(\psi_0\), the
chain rule applied to \(\psi(x, t) = \psi_0(x - ct)\) gives
\[
\partial_t \psi = -c\, \psi_0'(x - ct), \qquad
\partial_x \psi = \psi_0'(x - ct),
\]
and therefore \(\partial_t \psi + c\, \partial_x \psi = 0\). Conversely, the
derivative of \(\psi(x_0 + ct, t)\) along the line
\(\{(x_0 + ct, t) : t \geq 0\}\) equals
\(c\, \partial_x \psi + \partial_t \psi = 0\), so \(\psi\) is constant along
such lines, which forces \(\psi(x, t) = \psi(x - ct, 0) = \psi_0(x - ct)\). The
lines along which \(\psi\) is preserved are the characteristics
of the equation, parametrized by \(x_0 = x - ct = \) constant. Reversing the
sign of \(c\), the analogous equation
\(\partial_t \psi - c\, \partial_x \psi = 0\) is solved by left-translation:
\(\psi(x, t) = \psi_0(x + ct)\), constant along the characteristics
\(x + ct = \) constant. We will use both directions.
Reduction to Two Transport Equations
The wave equation contains both directions at once. Observe the operator
factorization
\[
\partial_{tt} - c^2 \partial_{xx}
\;=\; (\partial_t - c\, \partial_x)(\partial_t + c\, \partial_x)
\;=\; (\partial_t + c\, \partial_x)(\partial_t - c\, \partial_x),
\]
where the two factorizations are equal because the constant-coefficient
first-order operators \(\partial_t \pm c\, \partial_x\) commute. The
second-order wave operator is the composition of two first-order transport
operators, one moving structure to the right at speed \(c\) and one moving it
to the left at the same speed. This algebraic factorization is the entire
content of the method of characteristics: it converts a single second-order
equation into a system of two first-order equations, each of which we know how
to solve.
Acting on this observation, we introduce the auxiliary variables
\[
w := \partial_t u - c\, \partial_x u, \qquad
v := \partial_t u + c\, \partial_x u.
\]
These are not separate unknowns. They are linear combinations of the first
derivatives of \(u\) chosen to diagonalize the second-order operator into two
independent transport problems. Applying \(\partial_t + c\, \partial_x\) to
\(w\) gives
\[
(\partial_t + c\, \partial_x) w
\;=\; (\partial_t + c\, \partial_x)(\partial_t - c\, \partial_x) u
\;=\; \partial_{tt} u - c^2 \partial_{xx} u
\;=\; 0,
\]
where the last equality is the wave equation itself. Therefore \(w\) satisfies
the first-order transport equation
\[
\partial_t w + c\, \partial_x w = 0,
\]
whose solution, by the lemma above, is \(w(x, t) = P(x - ct)\) for some
function \(P\) determined by the initial data:
\(P(x) = w(x, 0) = \partial_t u(x, 0) - c\, \partial_x u(x, 0)
= g(x) - c f'(x)\), given \(f \in C^1\) so that \(f'\) is defined. Symmetrically, applying
\(\partial_t - c\, \partial_x\) to \(v\) gives
\((\partial_t - c\, \partial_x) v = \partial_{tt} u - c^2 \partial_{xx} u = 0\),
so \(v\) satisfies \(\partial_t v - c\, \partial_x v = 0\) with general solution
\(v(x, t) = Q(x + ct)\), where \(Q(x) = v(x, 0) = g(x) + c f'(x)\).
The auxiliary variables \(w\) and \(v\) thus carry, respectively, the
right-moving and left-moving components of the dynamics, and each is determined
explicitly by the initial data. To recover \(u\) from \(w\) and \(v\) we use
their definitions as a linear system: \(v - w = 2c\, \partial_x u\) and
\(v + w = 2\, \partial_t u\), from which
\(\partial_t u = (v + w)/2 = [P(x - ct) + Q(x + ct)]/2\) and
\(\partial_x u = (v - w)/(2c) = [Q(x + ct) - P(x - ct)]/(2c)\). The wave
equation has been reduced to a pair of explicit first-order data on the
derivatives of \(u\), each living on its own characteristic family. The
integration of these data into a formula for \(u\) itself is the content of the
next step.
The General Solution
The expressions for \(\partial_t u\) and \(\partial_x u\) obtained in the previous
step have each been written as a sum of a function of \(x - ct\) and a function
of \(x + ct\). We now show that the same structural decomposition is forced on
\(u\) itself: any \(C^2\) solution of the wave equation on the real line is a
sum of a right-moving and a left-moving travelling wave.
Let \(\mathcal{P}\) and \(\mathcal{Q}\) be antiderivatives of \(P\) and \(Q\),
respectively, so that \(\mathcal{P}' = P\) and \(\mathcal{Q}' = Q\). Define
\[
F(s) := -\frac{1}{2c}\mathcal{P}(s), \qquad G(s) := \frac{1}{2c}\mathcal{Q}(s),
\]
and consider the function \(\tilde u(x, t) := F(x - ct) + G(x + ct)\). Direct
differentiation gives
\[
\partial_t \tilde u = -c\, F'(x-ct) + c\, G'(x+ct)
= \frac{P(x-ct) + Q(x+ct)}{2} = \partial_t u,
\]
using \(F' = -P/(2c)\) and \(G' = Q/(2c)\), and similarly
\[
\partial_x \tilde u = F'(x-ct) + G'(x+ct)
= \frac{-P(x-ct) + Q(x+ct)}{2c} = \partial_x u.
\]
Thus \(u\) and \(\tilde u\) have the same first-order partial derivatives
everywhere on \(\mathbb{R} \times [0, \infty)\) and therefore differ by a global
additive constant, which we absorb into the definition of either \(F\) or \(G\)
without loss of generality. The wave equation's general solution on the real
line has been determined.
Theorem: General Solution of the One-Dimensional Wave Equation
Every \(C^2\) solution \(u : \mathbb{R} \times [0, \infty) \to \mathbb{R}\)
of the wave equation
\[
\partial_{tt} u - c^2 \partial_{xx} u = 0
\]
admits a representation of the form
\[
u(x, t) = F(x - ct) + G(x + ct)
\]
for some functions \(F, G \in C^2(\mathbb{R})\), unique up to additive
constants whose sum vanishes, called the
right-moving and left-moving components
of the solution respectively. Conversely, for any \(F, G \in C^2(\mathbb{R})\),
the function \(u(x, t) = F(x - ct) + G(x + ct)\) is a \(C^2\) solution of
the wave equation on \(\mathbb{R} \times [0, \infty)\). This result is due
to d'Alembert (1747).
Proof:
The forward direction was established above: given a \(C^2\) solution \(u\),
the auxiliary variables \(w = u_t - c u_x\) and \(v = u_t + c u_x\) satisfy
\(w(x, t) = P(x - ct)\) and \(v(x, t) = Q(x + ct)\) by the transport lemma,
and the antiderivatives \(\mathcal{P}, \mathcal{Q}\) furnish functions
\(F = -\mathcal{P}/(2c)\) and \(G = \mathcal{Q}/(2c)\) for which
\(u(x, t) - [F(x - ct) + G(x + ct)]\) has vanishing first derivatives in
both \(x\) and \(t\) on the connected domain \(\mathbb{R} \times [0, \infty)\),
hence is a global constant. Absorbing
the constant into \(F\) or \(G\) yields the stated representation.
The converse is a direct computation. For any \(F, G \in C^2(\mathbb{R})\)
and \(u(x, t) = F(x - ct) + G(x + ct)\),
\[
\partial_{tt} u = c^2 F''(x - ct) + c^2 G''(x + ct), \qquad
c^2 \partial_{xx} u = c^2 F''(x - ct) + c^2 G''(x + ct),
\]
so \(\partial_{tt} u = c^2 \partial_{xx} u\) identically.
Solving the Initial-Value Problem
The general solution \(u(x, t) = F(x - ct) + G(x + ct)\) contains two arbitrary
\(C^2\) functions, exactly matching the two initial conditions of the wave
equation. To determine \(F\) and \(G\) from \(f\) and \(g\), we impose the
initial conditions on the general representation. At \(t = 0\),
\[
u(x, 0) = F(x) + G(x) = f(x),
\]
while
\[
\partial_t u(x, 0) = -c F'(x) + c G'(x) = g(x).
\]
Differentiating the first equation gives \(F'(x) + G'(x) = f'(x)\). Together
with the second equation, this is a linear system in \(F', G'\),
\[
\begin{cases}
F'(x) + G'(x) = f'(x), \\
-c F'(x) + c G'(x) = g(x),
\end{cases}
\]
whose solution is
\[
F'(x) = \frac{1}{2}f'(x) - \frac{1}{2c}g(x), \qquad
G'(x) = \frac{1}{2}f'(x) + \frac{1}{2c}g(x).
\]
Integrating from \(0\) to \(s\) yields
\[
F(s) = \frac{1}{2}f(s) - \frac{1}{2c}\int_0^s g(\sigma)\, d\sigma + \kappa, \qquad
G(s) = \frac{1}{2}f(s) + \frac{1}{2c}\int_0^s g(\sigma)\, d\sigma - \kappa,
\]
where the integration constants are constrained by
\(F(0) + G(0) = f(0)\), leaving one free parameter \(\kappa\) that cancels when
\(F\) and \(G\) are added. Evaluating \(F(x - ct) + G(x + ct)\) yields the
explicit formula.
Theorem: D'Alembert's Formula
Let \(f \in C^2(\mathbb{R})\) and \(g \in C^1(\mathbb{R})\). The unique
\(C^2\) solution of the initial-value problem
\[
\begin{cases}
\partial_{tt} u = c^2\, \partial_{xx} u, & x \in \mathbb{R},\ t > 0, \\
u(x, 0) = f(x), & x \in \mathbb{R}, \\
\partial_t u(x, 0) = g(x), & x \in \mathbb{R}
\end{cases}
\]
is given by d'Alembert's formula:
\[
u(x, t) = \frac{f(x - ct) + f(x + ct)}{2}
+ \frac{1}{2c}\int_{x - ct}^{x + ct} g(\sigma)\, d\sigma.
\]
Proof:
Existence: direct substitution. For \(f \in C^2\) and \(g \in C^1\), the
function on the right-hand side belongs to
\(C^2(\mathbb{R} \times [0, \infty))\) — the integral term lies in \(C^2\)
by the fundamental theorem of calculus applied to its \(C^1\) integrand —
and the derivation above shows it solves the PDE and matches both initial
conditions.
Uniqueness: any \(C^2\) solution \(\tilde u\) of the same initial-value
problem admits the form
\(\tilde u = \tilde F(x - ct) + \tilde G(x + ct)\) by the general
representation theorem above. Imposing \(\tilde u(\cdot, 0) = f\) and
\(\partial_t \tilde u(\cdot, 0) = g\) determines \(\tilde F'\) and
\(\tilde G'\) uniquely as solutions of the same linear system as \(F', G'\),
whose integration with constraint
\(\tilde F(0) + \tilde G(0) = f(0)\) gives \(\tilde F\) and \(\tilde G\)
differing from \(F, G\) only by additive constants whose sum is zero. The
sum \(\tilde u = \tilde F(x - ct) + \tilde G(x + ct)\) is therefore
identical to \(u\).
The formula resolves the classical existence question on the real line that
the bounded-interval setting left open. The series representation
given there was an existence statement of formal character; d'Alembert's formula is
the rigorous classical existence result on the real line, and it does so under
exactly the natural Cauchy-data class \(f \in C^2, g \in C^1\), with no extra
regularity required and no infinite series whose convergence to a \(C^2\)
function would have to be checked term by term. The hyperbolic equation admits
a closed-form classical solution by a finite chain of algebraic and integration
steps. This is a structural feature distinguishing the wave equation from both
the heat equation — where the closed form is the convolution with a Gaussian
heat kernel — and from the elliptic case, which we encounter in a subsequent
page.
Domain of Dependence and Range of Influence
D'Alembert's formula determines \(u(x, t)\) from values of \(f\) at the two
points \(x \pm ct\) and values of \(g\) on the interval \([x - ct, x + ct]\).
The solution at the spacetime point \((x, t)\) therefore depends only on
initial data lying within this bounded interval — no information from outside
\([x - ct, x + ct]\) reaches \((x, t)\). The interval is called the
domain of dependence of \(u\) at \((x, t)\); it is bounded by
the two characteristic lines through \((x, t)\) traced backward in time at
speed \(c\), and forms a closed triangle with apex at \((x, t)\) and base on
the initial line \(t = 0\).
The same picture, traced forward in time, gives the dual notion. The initial
data \(f(x_0), g(x_0)\) at a single initial point \(x_0\) affect the solution
\(u(x, t)\) only at spacetime points \((x, t)\) with \(t \geq 0\) satisfying
\(|x - x_0| \leq ct\) — equivalently, points reachable from \((x_0, 0)\) by a
signal travelling at speed at most \(c\) in either direction. This closed
spacetime region is the range of influence of the initial
data at \(x_0\); it is the upward cone bounded by the two characteristic lines
issuing from \((x_0, 0)\) at slopes \(\pm 1/c\).
The slope \(\pm 1/c\) of the characteristic lines in the spacetime diagram
encodes the central physical content of the hyperbolic equation:
finite propagation speed. A localized disturbance in the
initial data — say \(f\) or \(g\) supported in a small interval near \(x_0\) —
cannot influence the solution at any spacetime point outside the upward cone
issuing from \((x_0, 0)\), because the d'Alembert formula evaluates \(u(x, t)\)
only on the backward triangle, which intersects the support of the data only
if \((x, t)\) lies in the cone. Information about the initial state travels
through the medium at speed exactly \(c\), neither faster nor instantaneously.
Contrast with the Heat Equation
The dichotomy with the parabolic case is sharp and structural. The
heat kernel solution
on the real line is the convolution
\[
u(x, t) \;=\; \int_{-\infty}^{\infty} K_t(x - y)\, f(y)\, dy,
\]
in which the kernel \(K_t\) is strictly positive everywhere on \(\mathbb{R}\)
for every \(t > 0\). Consequently \(u(x, t)\) depends on the initial datum
\(f\) at every point of \(\mathbb{R}\), not on a bounded interval; the
domain of dependence of the heat equation is all of \(\mathbb{R}\).
Reciprocally, the data \(f(x_0)\) at a single point affect \(u(x, t)\) at
every \((x, t)\) with \(t > 0\) — the range of influence is the full upper
half-plane. Information propagates instantaneously: an arbitrarily small
perturbation of \(f\) on an arbitrarily small interval changes the parabolic
solution everywhere at every positive time. This is the parabolic counterpart
of the hyperbolic equation's finite-speed-of-propagation property, and it lies
at the root of every qualitative difference between the two evolutions —
irreversibility, smoothing, backward ill-posedness on the heat side;
reversibility, regularity preservation, backward well-posedness on the wave
side.
Energy Conservation
The qualitative inversions catalogued at the close of the previous section —
reversibility, regularity preservation, backward well-posedness — were
articulated as structural features of the wave equation, in direct contrast
with the parabolic case. We now sharpen the contrast with one quantitative
statement: the wave equation admits a conserved quantity, and the form this
quantity takes is itself a structural diagnostic. Where the heat equation's
natural quantity is the \(L^2\) mass \(\int u^2\,dx\), which is monotonically
non-increasing and tends to zero under Dirichlet boundary conditions, the
wave equation conserves a different quadratic functional — the
mechanical energy of the solution — exactly, for all time,
in both forward and backward time directions.
For a sufficiently regular solution \(u\) of the wave equation on a domain
\(\Omega\) — either the bounded interval \([0, L]\) with homogeneous Dirichlet
boundary conditions, or the entire real line with \(u(\cdot, t)\) and its
derivatives decaying sufficiently rapidly at infinity — define the
mechanical energy at time \(t\) by
\[
E(t) \;:=\; \frac{1}{2}\int_\Omega
\Bigl[\bigl(\partial_t u(x, t)\bigr)^2
+ c^2 \bigl(\partial_x u(x, t)\bigr)^2\Bigr] dx.
\]
The two terms have a direct physical interpretation in the vibrating string
model introduced earlier in this page: the first is the kinetic energy density,
proportional to the squared instantaneous velocity of each point of the string,
and the second is the potential energy density, proportional to
the squared local strain — the elastic energy stored in the displacement. Their
sum, integrated over the spatial domain, is the total mechanical energy of the
string at time \(t\). The factor \(c^2 = T_0/\rho_0\) in the potential term
couples the spatial slope to the physical parameters of the string in the manner
dictated by the wave equation itself.
Theorem: Conservation of Mechanical Energy
Let \(u\) be a \(C^2\) solution of the wave equation
\(\partial_{tt} u = c^2 \partial_{xx} u\) on \(\Omega \times [0, \infty)\),
where either
(i) \(\Omega = [0, L]\) and \(u\) satisfies the homogeneous Dirichlet
boundary conditions \(u(0, t) = u(L, t) = 0\) for all \(t \geq 0\); or
(ii) \(\Omega = \mathbb{R}\) and, for each \(t \geq 0\), both
\(\partial_t u(\cdot, t)\) and \(\partial_x u(\cdot, t)\) decay
sufficiently rapidly at infinity that their product vanishes as
\(|x| \to \infty\).
Then the mechanical energy is conserved:
\[
\frac{dE}{dt}(t) \;=\; 0
\qquad\text{for all } t \geq 0,
\]
and consequently \(E(t) = E(0)\) for every \(t \geq 0\).
Proof:
Differentiating \(E(t)\) under the integral sign — justified by the
regularity of \(u\) and, in case (ii), the decay assumption — gives
\[
\frac{dE}{dt}
\;=\; \int_\Omega \bigl[\partial_t u \cdot \partial_{tt} u
+ c^2 \partial_x u \cdot \partial_x(\partial_t u)\bigr] dx.
\]
Substituting the wave equation \(\partial_{tt} u = c^2 \partial_{xx} u\)
into the first term and integrating the second term by parts in \(x\),
\[
\frac{dE}{dt}
\;=\; \int_\Omega c^2 \partial_t u \cdot \partial_{xx} u\, dx
+ \Bigl[c^2 \partial_x u \cdot \partial_t u\Bigr]_{\partial\Omega}
- \int_\Omega c^2 \partial_{xx} u \cdot \partial_t u\, dx.
\]
The two volume integrals are identical and cancel, leaving only the
boundary term:
\[
\frac{dE}{dt}
\;=\; \Bigl[c^2 \partial_x u(x, t) \cdot \partial_t u(x, t)\Bigr]_{\partial\Omega}.
\]
In case (i), \(\partial_t u\) vanishes at both endpoints because
\(u(0, t) = u(L, t) = 0\) for every \(t\) implies \(\partial_t u(0, t) =
\partial_t u(L, t) = 0\) by differentiating the boundary conditions in
\(t\); the boundary contribution vanishes. In case (ii), the assumed
decay of \(\partial_x u\) and \(\partial_t u\) at infinity makes their
product vanish as \(|x| \to \infty\), and again the boundary contribution
vanishes. In either case, \(\frac{dE}{dt} = 0\). Integrating from \(0\)
to \(t\) yields \(E(t) = E(0)\).
The conservation law has a structural significance that goes beyond its
statement. The heat equation also admits a natural quadratic quantity, the
\(L^2\) mass
\(M(t) = \tfrac{1}{2}\int_\Omega u(x, t)^2\, dx\), and the
corresponding identity — obtained by multiplying the heat equation by \(u\)
and integrating by parts — yields \(\frac{dM}{dt} = -k\int_\Omega
(\partial_x u)^2\, dx \leq 0\), strictly negative whenever \(u\) is not
spatially constant. The parabolic quantity is monotonically non-increasing
and, under Dirichlet conditions, dissipates to zero; the hyperbolic quantity
is exactly preserved. The wave equation's natural invariant is not the
\(L^2\) mass of the solution but the more elaborate mechanical energy
involving both temporal and spatial derivatives — kinetic and
potential, balanced against each other. This is the precise sense in which
the wave equation, unlike the heat equation, has no preferred direction in
time: every quantity quadratic in \((\partial_t u, \partial_x u)\)
that is conserved must treat the two derivatives symmetrically, and the
mechanical energy is the canonical such invariant.
The conservation law closes a marker placed in the previous page. There the
energy estimate
\(\frac{d}{dt}\!\int_0^L u(x, t)^2\, dx \leq 0\)
was named as the standard machinery underlying uniqueness for the heat equation,
with its proof deferred. The wave
equation analogue \(\frac{dE}{dt} = 0\) — the equality form rather than an
inequality — has now been proved in full, and the structural shift from
\(\int u^2\) to
\(\int [(\partial_t u)^2 + c^2 (\partial_x u)^2]\)
is the quantitative content of the heat-vs-wave inversion. Uniqueness for the
wave equation in case (i) and case (ii) follows from this conservation
immediately: if two solutions \(u_1, u_2\) share the same initial data, their
difference \(\eta = u_1 - u_2\) solves the wave equation with zero initial data,
has zero energy at \(t = 0\), and therefore zero energy for all \(t\); but
\(E(t) = 0\) forces \(\partial_t \eta \equiv 0\) and \(\partial_x \eta \equiv 0\)
everywhere (the integrand is non-negative and continuous, hence zero pointwise),
so \(\eta\) is constant in spacetime; the initial condition \(\eta(x, 0) = 0\)
identifies the constant as zero, so \(\eta \equiv 0\). On the real line, this
energy-based argument applies under the decay hypothesis of the theorem and
recovers, by a qualitative route, the uniqueness already furnished
constructively by d'Alembert's formula. The hyperbolic uniqueness
theorem is a one-line consequence of the conservation theorem.