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. 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, and 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 at position \(x\) and time \(t\), and \(c \gt 0\) is a positive constant
called the wave speed. More abstractly, \(u\) is the displacement
field of a propagating wave.
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, and elliptic. The third member,
the Laplace equation, is
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, and the d'Alembert
formula, which also justifies the series solution on the interval. The energy method
then yields conservation, uniqueness, and finite speed of propagation. We close with
time reversal, where the contrast with the heat equation is sharpest.
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
\[
\begin{cases}
\partial_{tt} u = c^2\, \partial_{xx} u, & 0 \lt x \lt L,\ t \gt 0, \\\\
u(0, t) = u(L, t) = 0, & t \geq 0, \\\\
u(x, 0) = f(x), & 0 \leq x \leq L, \\\\
\partial_t u(x, 0) = g(x), & 0 \leq x \leq L,
\end{cases}
\]
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 is replaced by sustained oscillation, and a
single initial datum 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 that 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. There, an energy identity showed that every eigenvalue is real and
positive, and a three-case analysis of the ODE identified the values that occur and
produced the sine eigenfunctions. We do not repeat either argument. The result is
summarized by the previous page's
theorem on the eigenvalues and eigenfunctions of the Dirichlet Laplacian on \([0, L]\):
\[
\begin{align*}
\lambda_n &= \left(\frac{n\pi}{L}\right)^{\!2}, \\\\
\phi_n(x) &= \sin\frac{n\pi x}{L}, \quad n = 1, 2, 3, \ldots
\end{align*}
\]
What changes between the heat and wave equations is not the spatial spectrum
but how each eigenvalue acts on its mode in time. Both problems diagonalize
the same operator.
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.
We pause for 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. That family 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 \(x\). The amplitude, however, 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 \(g\). Each term now 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 \([-L, L]\), restricted to the half-interval \([0, L]\) by the evenness of
the product \(\sin \cdot \sin\), 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 \(f\) and \(g\) be continuous on \([0, L]\). Then the formal series solution
of the initial-boundary value problem
\[
\begin{cases}
\partial_{tt} u = c^2\, \partial_{xx} u, & 0 \lt x \lt L,\ t \gt 0, \\\\
u(0, t) = u(L, t) = 0, & t \geq 0, \\\\
u(x, 0) = f(x), \quad \partial_t u(x, 0) = g(x), & 0 \leq x \leq L
\end{cases}
\]
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
\[
\begin{align*}
B_n &= \frac{2}{L}\int_0^L f(x)\, \sin\frac{n\pi x}{L}\, dx, \\\\
C_n &= \frac{2}{n\pi c}\int_0^L g(x)\, \sin\frac{n\pi x}{L}\, dx.
\end{align*}
\]
Proof:
Each product solution
\[
u_n(x, t) = \sin(n\pi x/L)\bigl[B_n \cos(n\pi ct/L) + C_n \sin(n\pi ct/L)\bigr]
\]
satisfies the wave equation and the Dirichlet boundary conditions
\[
u_n(0, t) = u_n(L, t) = 0
\]
by direct differentiation, as established in the separation-of-variables
analysis above. Linearity of the PDE and the boundary conditions extends this to
the formal series by termwise differentiation.
The coefficient formulas are determined from the initial conditions. Imposing
\(u(\cdot, 0) = f\) collapses the cosine factors to one and the sine factors to
zero, yielding the Fourier sine series \(\sum_n B_n \sin(n\pi x/L) = f(x)\).
Orthogonality of the trigonometric system \(\{\sin(n\pi x/L)\}_{n \geq 1}\) on
\([0, L]\), with
\[
\int_0^L \sin(n\pi x/L)\sin(m\pi x/L)\, dx = (L/2)\delta_{nm},
\]
extracts
\[
B_n = (2/L)\int_0^L f(x)\sin(n\pi x/L)\, dx.
\]
Termwise differentiation in \(t\) and evaluation at \(t = 0\) yields
the second series
\[
\sum_n C_n (n\pi c/L)\sin(n\pi x/L) = g(x).
\]
The same orthogonality gives
\[
C_n (n\pi c/L) = (2/L)\int_0^L g(x)\sin(n\pi x/L)\, dx.
\]
Thus,
\[
C_n = (2/(n\pi c))\int_0^L g(x)\sin(n\pi x/L)\, dx.
\]
The construction is formal. It identifies the series and its
coefficients, and it leaves open whether the series converges and whether
its sum is a classical solution.
Differentiating the series twice term by term does not settle the question. In
\(x\), each differentiation multiplies the \(n\)th term by \(n\pi/L\). For
\(f \in C^2([0, L])\) with \(f(0) = f(L) = 0\), two integrations by parts give
\(|B_n| \leq M/n^2\) for a constant \(M\). This does not make \(\sum_n n^2 |B_n|\)
finite. For \(f(x) = x(L - x)\) the terms \(n^2 |B_n|\) with \(n\) odd decay only
like \(1/n\), so the twice-differentiated series is not controlled. We take a
different route. The next section solves the problem on the real line by the method
of characteristics. At its end, the
classical solution theorem on the interval
uses that solution to show that, for \(g \in C^1([0, L])\) and under the further
conditions \(f''(0) = f''(L) = 0\) and \(g(0) = g(L) = 0\), the formal series
converges to a \(C^2\) solution. These further conditions turn out to be necessary
for any \(C^2\) solution.
The contrast with the heat equation deserves emphasis. Exponential damping
\(e^{-\lambda_n k t}\) regularizes initial data of any \(L^2\) class into
\(C^\infty\) for every \(t \gt 0\), 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. Roughness in the data, including any
incompatibility with the boundary conditions, is carried along by the evolution
rather than dissipated. Initial data lying merely in \(L^2\) generate a series
convergent in \(L^2\) at each \(t\). 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 d'Alembert's formula at the end of the next section shows
the transport of roughness explicitly.
Insight: Standing Waves and the Vibrating String
Each term in the series solution,
\[
\sin\frac{n\pi x}{L}\, \bigl[B_n \cos\omega_n t + C_n \sin\omega_n t\bigr],
\quad \omega_n = \frac{n\pi c}{L},
\]
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. Its harmonics are the modes
\(n \geq 2\), with frequencies \(\omega_n = n \omega_1\) organized as integer
multiples of the fundamental. The \(n\)th mode has exactly \(n - 1\) interior
zeros, its nodes, at which the string does not move. 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 \(L\). For a string of tension \(T_0\) and linear mass density
\(\rho_0\), Newton's second law applied to a short segment of the string gives
\(c^2 = T_0 / \rho_0\), a relation we take from physics without derivation. The
fundamental frequency is therefore \(\omega_1 = (\pi / L)\sqrt{T_0 / \rho_0}\).
A string sounds higher when tightened (larger \(T_0\)), when shortened (smaller
\(L\)), or when made of a lighter material (smaller \(\rho_0\)).
The formula separates two ways of changing the pitch. A violinist who presses
the string at a chosen point shortens its vibrating length \(L\), which changes
the Dirichlet eigenvalues \(\lambda_n = (n\pi/L)^2\) and with them every
frequency. A guitarist who tunes a string changes its tension \(T_0\), which
leaves the eigenvalues unchanged and changes the wave speed \(c\) in
\(\omega_n = c\sqrt{\lambda_n}\).
A single standing wave admits an alternative description as the
superposition of two traveling 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 sine-cosine combination of the other standing-wave term.
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
traveling at speed \(c\). This decomposition is not an artifact of the
trigonometric identity but 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 traveling 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:
\[
\partial_{tt} u = c^2\, \partial_{xx} u, \quad
x \in \mathbb{R},\ t \gt 0,
\]
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 traveling 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
Assume, for the purposes of this sketch, that \(u(\cdot, t)\) lies in the Schwartz
class for each \(t\) and that \(t\)-derivatives may be taken inside the transform,
so that the
differentiation rule
\(\partial_x \mapsto -i\xi\) applies. We take the spatial Fourier transform in \(x\)
with the convention
\(\hat u(\xi, t) = \int_{\mathbb{R}} u(x, t)\, e^{i x \xi}\, dx\). The PDE becomes
the family of second-order ODEs in \(t\),
\[
\partial_{tt} \hat u(\xi, t) = -c^2 \xi^2\, \hat u(\xi, t),
\quad \xi \in \mathbb{R},
\]
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, namely
\(A = \hat f\) and \(B(\xi) = \hat g(\xi)/(c\xi)\) for \(\xi \neq 0\). At
\(\xi = 0\) the ODE reduces to \(\partial_{tt} \hat u = 0\), whose solutions are
affine in \(t\), and the displayed form degenerates. The product
\(B(\xi)\sin(c\xi t)\) extends continuously to \(\xi = 0\) with the value
\(\hat g(0)\, t\), so the displayed form, read with this value, gives the correct
solution \(\hat f(0) + \hat g(0)\, t\) at \(\xi = 0\). The separate amplitudes below
carry the factor \(1/\xi\), however, so when \(\int_{\mathbb{R}} g \neq 0\) their
inverse transforms need not exist as functions. The sketch is heuristic at this
point, and the construction that follows does not depend on it.
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
its inverse transform is a function of \(x - ct\) or \(x + ct\) alone. That property
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.
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 \(\widetilde G(x + ct)\). Both are 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 traveling waves. The calculation does not tell us
how to construct that superposition 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
Throughout this section, \(I\) denotes an interval of positive length that contains
\(0\). It may be \([0, \infty)\), a bounded interval \([0, T]\), or all of
\(\mathbb{R}\), and every result below holds for each such choice. Allowing
negative times costs nothing in the proofs, and negative times are needed for time
reversal at the end of this page. A function on a product \(J \times I\) of
intervals is of class \(C^k\) if its partial derivatives up to order \(k\) exist
and are continuous, with one-sided derivatives at any endpoint of \(J\) or \(I\)
that belongs to the domain.
Before factoring the second-order wave operator, we record the elementary fact that
drives the entire construction.
Theorem: Solution of the Transport Equation
Let \(\psi_0 \in C^1(\mathbb{R})\), and let \(I\) be an interval of positive
length containing \(0\). The unique \(C^1\) solution on \(\mathbb{R} \times I\)
of the first-order linear initial-value problem
\[
\begin{cases}
\partial_t \psi + c\, \partial_x \psi = 0, & x \in \mathbb{R},\ t \in I, \\\\
\psi(x, 0) = \psi_0(x), & x \in \mathbb{R},
\end{cases}
\]
is the rigid translation
\[
\psi(x, t) = \psi_0(x - ct).
\]
The analogous equation \(\partial_t \psi - c\, \partial_x \psi = 0\) with the
same initial datum is solved by the left-translation
\(\psi(x, t) = \psi_0(x + ct)\).
Proof:
The verification is direct. For any \(C^1\) function \(\psi_0\), the chain rule
applied to \(\psi(x, t) = \psi_0(x - ct)\) gives
\[
\begin{align*}
\partial_t \psi &= -c\, \psi_0'(x - ct), \\\\
\partial_x \psi &= \psi_0'(x - ct),
\end{align*}
\]
and therefore \(\partial_t \psi + c\, \partial_x \psi = 0\).
Conversely, let \(\psi\) be a \(C^1\) solution on \(\mathbb{R} \times I\)
and fix \((x, t) \in \mathbb{R} \times I\). With \(x_0 = x - ct\), the
chain rule gives
\[
\frac{d}{ds}\, \psi(x_0 + cs, s) = c\, \partial_x \psi + \partial_t \psi = 0
\]
for \(s \in I\), so \(\psi\) is constant along the characteristic line through
\((x_0, 0)\). Since \(I\) is an interval containing both \(0\) and \(t\), this
forces \(\psi(x, t) = \psi(x_0, 0) = \psi_0(x - ct)\) and establishes
uniqueness. The case of the reversed sign follows by the substitution
\(c \mapsto -c\).
The initial profile \(\psi_0\) is carried unchanged to the right at speed \(c\)
(to the left, for the reversed sign). The lines along which \(\psi\) is preserved
are the characteristics of the equation, parametrized by
\(x_0 = x - ct = \) constant in the right-moving case and by \(x + ct = \)
constant in the left-moving one. We will use both directions.
Reduction to Two Transport Equations
The wave equation contains both directions at once. Observe the operator
factorization
\[
\begin{align*}
\partial_{tt} - c^2 \partial_{xx}
&= (\partial_t - c\, \partial_x)(\partial_t + c\, \partial_x) \\\\
&= (\partial_t + c\, \partial_x)(\partial_t - c\, \partial_x),
\end{align*}
\]
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
\[
\begin{align*}
w &:= \partial_t u - c\, \partial_x u, \\\\
v &:= \partial_t u + c\, \partial_x u.
\end{align*}
\]
These are not separate unknowns but 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
\[
\begin{align*}
(\partial_t + c\, \partial_x) w
&= (\partial_t + c\, \partial_x)(\partial_t - c\, \partial_x) u \\\\
&= \partial_{tt} u - c^2 \partial_{xx} u \\\\
&= 0,
\end{align*}
\]
where the last equality is the wave equation itself. The second equality uses that
the mixed partial derivatives of a \(C^2\) function agree, a standard fact of
calculus that we take for granted. At boundary points of the domain this agreement
follows by continuity.
Therefore \(w\) satisfies the first-order transport equation
\[
\partial_t w + c\, \partial_x w = 0,
\]
whose solution, by the transport result above, is \(w(x, t) = P(x - ct)\) for some
function \(P\) determined by the initial data:
\[
\begin{align*}
P(x)
&= w(x, 0) \\\\
&= \partial_t u(x, 0) - c\, \partial_x u(x, 0) \\\\
&= g(x) - c f'(x),
\end{align*}
\]
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
\[
\begin{align*}
\partial_t u &= (v + w)/2 = [P(x - ct) + Q(x + ct)]/2, \\\\
\partial_x u &= (v - w)/(2c) = [Q(x + ct) - P(x - ct)]/(2c).
\end{align*}
\]
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 traveling wave.
Let \(\mathcal{P}\) and \(\mathcal{Q}\) be antiderivatives of \(P\) and \(Q\),
respectively, so that \(\mathcal{P}' = P\) and \(\mathcal{Q}' = Q\). Define
\[
\begin{align*}
F(s) &:= -\frac{1}{2c}\mathcal{P}(s), \\\\
G(s) &:= \frac{1}{2c}\mathcal{Q}(s),
\end{align*}
\]
and consider the function \(\tilde u(x, t) := F(x - ct) + G(x + ct)\). Direct
differentiation gives
\[
\begin{align*}
\partial_t \tilde u
&= -c\, F'(x-ct) + c\, G'(x+ct) \\\\
&= \frac{P(x-ct) + Q(x+ct)}{2} \\\\
&= \partial_t u,
\end{align*}
\]
using \(F' = -P/(2c)\) and \(G' = Q/(2c)\), and similarly
\[
\begin{align*}
\partial_x \tilde u
&= F'(x-ct) + G'(x+ct) \\\\
&= \frac{-P(x-ct) + Q(x+ct)}{2c} \\\\
&= \partial_x u.
\end{align*}
\]
Thus \(u\) and \(\tilde u\) have the same first-order partial derivatives
everywhere on the convex set \(\mathbb{R} \times I\) 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
Let \(I\) be an interval of positive length containing \(0\). Every \(C^2\)
solution \(u : \mathbb{R} \times I \to \mathbb{R}\) 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 \(\mathbb{R} \times I\). 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 result
established earlier in this section. 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 convex set
\(\mathbb{R} \times I\), hence is a global constant. Absorbing the constant into
\(F\) or \(G\) yields the stated representation.
For uniqueness, suppose that also
\(u(x, t) = \tilde F(x - ct) + \tilde G(x + ct)\) with
\(\tilde F, \tilde G \in C^2(\mathbb{R})\). Setting \(t = 0\), which lies in
\(I\), gives \(F + G = \tilde F + \tilde G\) on \(\mathbb{R}\), and
differentiating this identity gives \(F' + G' = \tilde F' + \tilde G'\).
Differentiating both representations in \(t\) and setting \(t = 0\) gives
\(-cF' + cG' = -c\tilde F' + c\tilde G'\). Since \(c \neq 0\), the two linear
relations force \(F' = \tilde F'\) and \(G' = \tilde G'\). Hence
\(\tilde F = F + \kappa_1\) and \(\tilde G = G + \kappa_2\) for constants
\(\kappa_1, \kappa_2\), and the relation at \(t = 0\) gives
\(\kappa_1 + \kappa_2 = 0\).
The converse is a direct computation. For any \(F, G \in C^2(\mathbb{R})\)
and \(u(x, t) = F(x - ct) + G(x + ct)\),
\[
\begin{align*}
\partial_{tt} u &= c^2 F''(x - ct) + c^2 G''(x + ct) \\\\
c^2 \partial_{xx} u &= c^2 F''(x - ct) + c^2 G''(x + ct).
\end{align*}
\]
Thus \(\partial_{tt} u = c^2 \partial_{xx} u\) 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
\[
\begin{align*}
F'(x) &= \frac{1}{2}f'(x) - \frac{1}{2c}g(x), \\\\
G'(x) &= \frac{1}{2}f'(x) + \frac{1}{2c}g(x).
\end{align*}
\]
Integrating from \(0\) to \(s\) yields
\[
\begin{align*}
F(s) &= \frac{1}{2}f(s) - \frac{1}{2c}\int_0^s g(\sigma)\, d\sigma + \kappa, \\\\
G(s) &= \frac{1}{2}f(s) + \frac{1}{2c}\int_0^s g(\sigma)\, d\sigma - \kappa,
\end{align*}
\]
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 \(g \in C^1(\mathbb{R})\), and let \(I\) be an
interval of positive length containing \(0\). The unique \(C^2\) solution on
\(\mathbb{R} \times I\) of the initial-value problem
\[
\begin{cases}
\partial_{tt} u = c^2\, \partial_{xx} u, & x \in \mathbb{R},\ t \in I, \\\\
u(x, 0) = f(x), & x \in \mathbb{R}, \\\\
\partial_t u(x, 0) = g(x), & x \in \mathbb{R}
\end{cases}
\]
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: take \(F\) and \(G\) as in the derivation above with \(\kappa = 0\).
By the fundamental theorem of calculus, \(s \mapsto \int_0^s g\) has derivative
\(g \in C^1\), so \(F, G \in C^2(\mathbb{R})\). By the converse half of the
general solution theorem, \(u(x, t) = F(x - ct) + G(x + ct)\) is a \(C^2\)
solution on \(\mathbb{R} \times I\). Since
\(\int_0^{x + ct} g - \int_0^{x - ct} g = \int_{x - ct}^{x + ct} g\), this \(u\)
is the right-hand side of d'Alembert's formula. At \(t = 0\) we have
\(u(x, 0) = F(x) + G(x) = f(x)\) and
\(\partial_t u(x, 0) = -cF'(x) + cG'(x) = g(x)\).
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 \(u\).
On the real line, the formula settles the classical existence question directly. It
requires exactly the natural Cauchy-data class \(f \in C^2\), \(g \in C^1\), and it
involves no infinite series whose convergence would have to be checked term by term.
The same formula also settles the question that the formal series left open on the
bounded interval, as the end of this section shows.
A closed-form solution does not by itself distinguish the wave equation. The heat
equation on the real line also has one, the convolution with the Gaussian heat
kernel, and so does
the elliptic case of a
subsequent page. What distinguishes the hyperbolic case is the way the formula
samples the data, which the next two subsections examine.
Domain of Dependence and Range of Influence
For \(t \gt 0\), 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 the base of the closed triangle with apex at \((x, t)\), bounded
by the two characteristic lines through \((x, t)\) traced backward in time at speed
\(c\) and by the initial line \(t = 0\).
The same picture, traced forward in time, gives the dual notion. The closed
spacetime region of points \((x, t)\) with \(t \geq 0\) and \(|x - x_0| \leq ct\)
consists of the points reachable from \((x_0, 0)\) by a signal traveling at speed at
most \(c\) in either direction. It is called the range of influence
of the initial point \(x_0\), and 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. Suppose \(f\) and \(g\) are changed only on a small interval
\([x_0 - \delta, x_0 + \delta]\). D'Alembert's formula evaluates \(u(x, t)\) only on
the base \([x - ct, x + ct]\) of the backward triangle, which meets that interval
only if \(|x - x_0| \leq ct + \delta\). The change in the data therefore alters the
solution only in this region, which shrinks to the range of influence of \(x_0\) as
\(\delta \to 0\). Information about the initial state travels through the medium at
speed at most \(c\), and never instantaneously.
Contrast with the Heat Equation
The contrast 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 \(t \gt 0\). Consequently \(u(x, t)\) depends on the initial datum \(f\) at
every point of \(\mathbb{R}\). The domain of dependence of the heat
equation is all of \(\mathbb{R}\). Dually, if a non-negative continuous function
that is not identically zero and vanishes outside some interval around a point
\(x_0\), however short, is added to \(f\), the solution rises at every point of the
upper half-plane \(\{t \gt 0\}\). The parabolic counterpart of finite propagation
speed is therefore its absolute negation, instantaneous global coupling.
Return to the Bounded Interval
The formal series on the bounded interval can now be justified. The idea is to solve
on the whole line with data that encode the boundary conditions. For a function
\(f\) on \([0, L]\) with \(f(0) = f(L) = 0\), the odd \(2L\)-periodic
extension \(\tilde f\) is the function on \(\mathbb{R}\) with
\(\tilde f(x) = f(x)\) and \(\tilde f(-x) = -f(x)\) for \(x \in [0, L]\). It is then
extended from \([-L, L]\) to \(\mathbb{R}\) by \(\tilde f(x + 2L) = \tilde f(x)\).
The endpoint conditions make the prescriptions agree at \(0\) and at \(\pm L\).
Oddness and periodicity together give
\(\tilde f(L + s) = \tilde f(s - L) = -\tilde f(L - s)\) for every \(s\), so
\(\tilde f\) is odd about \(L\) as well as about \(0\).
The extension inherits the regularity of \(f\) once suitable endpoint conditions
hold. Suppose \(f \in C^2([0, L])\) with \(f(0) = f(L) = 0\) and
\(f''(0) = f''(L) = 0\). Between consecutive multiples of \(L\), \(\tilde f\) is
\(f\) composed with a reflection or a translation, possibly with a sign change, so
it is \(C^2\) there with one-sided derivatives at the ends. Since \(\tilde f\) is
odd about every multiple of \(L\), reflection through such a point reverses the
signs of \(\tilde f\) and \(\tilde f''\) but not of \(\tilde f'\). The one-sided
limits of \(\tilde f'\) agree, and those of \(\tilde f''\) agree because both are
\(\pm f''(0)\) or \(\pm f''(L)\), which vanish. A continuous function that is
differentiable on either side of a point, with the derivative having the same limit
from both sides, is differentiable at that point with that derivative, as the mean
value theorem shows. Applied to \(\tilde f\) and then to \(\tilde f'\), this gives
\(\tilde f \in C^2(\mathbb{R})\). The same argument, one order lower, gives
\(\tilde g \in C^1(\mathbb{R})\) for \(g \in C^1([0, L])\) with \(g(0) = g(L) = 0\).
Theorem: Classical Solution on a Bounded Interval
Let \(f \in C^2([0, L])\) and \(g \in C^1([0, L])\) satisfy
\[
\begin{align*}
f(0) = f(L) &= 0, \\\\
f''(0) = f''(L) &= 0, \\\\
g(0) = g(L) &= 0,
\end{align*}
\]
let \(\tilde f\) and \(\tilde g\) be their odd \(2L\)-periodic extensions, and
let \(I\) be an interval of positive length containing \(0\). Then the function
\[
u(x, t) = \frac{\tilde f(x - ct) + \tilde f(x + ct)}{2}
+ \frac{1}{2c}\int_{x - ct}^{x + ct} \tilde g(\sigma)\, d\sigma,
\]
restricted to \([0, L] \times I\), is a \(C^2\) solution of the wave equation
with \(u(0, t) = u(L, t) = 0\) for \(t \in I\), \(u(\cdot, 0) = f\), and
\(\partial_t u(\cdot, 0) = g\). It satisfies
\[
|u(x, t)| \leq \sup_{[0, L]} |f| + |t| \sup_{[0, L]} |g|
\]
for all \((x, t) \in [0, L] \times I\). The formal series of the
series representation theorem
converges to \(u\) absolutely and uniformly on \([0, L] \times I\).
Proof:
Step 1: \(u\) solves the problem.
By the regularity argument before the theorem, \(\tilde f \in C^2(\mathbb{R})\)
and \(\tilde g \in C^1(\mathbb{R})\). By
d'Alembert's formula,
\(u\) is a \(C^2\) solution on \(\mathbb{R} \times I\) with
\(u(\cdot, 0) = \tilde f\) and \(\partial_t u(\cdot, 0) = \tilde g\), and
restricting to \([0, L]\) gives the equation and both initial conditions. At
\(x = 0\), oddness gives \(\tilde f(-ct) + \tilde f(ct) = 0\), and the integral
of the odd function \(\tilde g\) over \([-ct, ct]\) vanishes, so
\(u(0, t) = 0\). At \(x = L\), the oddness of \(\tilde f\) and \(\tilde g\)
about \(L\) gives the same two cancellations, so \(u(L, t) = 0\).
Step 2: the bound.
The extensions satisfy \(\sup_{\mathbb{R}} |\tilde f| = \sup_{[0, L]} |f|\) and
\(\sup_{\mathbb{R}} |\tilde g| = \sup_{[0, L]} |g|\). The first term of \(u\) is
at most \(\sup |f|\) in absolute value. The integral runs over an interval of
length \(2c|t|\), so the second term is at most \(|t| \sup |g|\).
Step 3: the series.
Let \(B_n\) be the sine coefficients of \(f\) and \(b_n\) those of \(g\), so
that \(C_n = L b_n/(n\pi c)\) in the series representation theorem. Both \(f\)
and \(g\) are continuous and piecewise \(C^1\) with vanishing endpoint values,
so by the
summability lemma
of the previous page, \(\sum_n |B_n| \lt \infty\) and
\(\sum_n |b_n| \lt \infty\), and the sine series of \(f\) and \(g\)
converge to them uniformly on \([0, L]\). The partial sums \(f_N\) and
\(g_N\) of these series are odd and \(2L\)-periodic, so
\(f_N \to \tilde f\) and \(g_N \to \tilde g\) uniformly on \(\mathbb{R}\).
The \(n\)th term of the formal series is bounded by \(|B_n| + |C_n|\) on
\([0, L] \times I\), and \(\sum_n |C_n| \leq (L/(\pi c)) \sum_n |b_n|\), so
the Weierstrass M-test, which we take for granted as on the previous page,
gives absolute and uniform convergence.
It remains to identify the sum. With \(\omega_n = n\pi c/L\), the
product-to-sum identities give
\[
\begin{align*}
\sin\frac{n\pi x}{L} \cos\omega_n t
&= \frac{1}{2}\Bigl[\sin\frac{n\pi (x - ct)}{L} + \sin\frac{n\pi (x + ct)}{L}\Bigr], \\\\
C_n \sin\frac{n\pi x}{L} \sin\omega_n t
&= \frac{1}{2c}\int_{x - ct}^{x + ct} b_n \sin\frac{n\pi \sigma}{L}\, d\sigma.
\end{align*}
\]
Summing over \(n \leq N\), the \(N\)th partial sum \(u_N\) of the formal series
is d'Alembert's formula applied to \(f_N\) and \(g_N\). Consequently
\[
|u_N(x, t) - u(x, t)| \leq \sup_{\mathbb{R}} |f_N - \tilde f|
+ |t| \sup_{\mathbb{R}} |g_N - \tilde g|,
\]
which tends to \(0\) for each fixed \((x, t)\). The series therefore
converges to \(u\).
The compatibility conditions are also necessary for a \(C^2\) solution on
\([0, L] \times I\). If \(u(0, t) = 0\) for all \(t \in I\), then
\(\partial_t u(0, t) = \partial_{tt} u(0, t) = 0\). The wave equation holds up to
\(x = 0\) by continuity of the second derivatives, so \(\partial_{xx} u(0, t) = 0\).
At \(t = 0\) these give \(f(0) = g(0) = f''(0) = 0\), and the same argument applies
at \(x = L\).
The convergence proof never differentiates the series term by term. Its sum is a
\(C^2\) solution because it coincides with the d'Alembert solution, whether or not
the twice-differentiated series converges.
The formula also makes the transport of roughness visible. It still defines \(u\)
for data outside the theorem, such as an \(f\) with a corner, a point where \(f\) is
continuous but its one-sided derivatives differ. With \(g = 0\), the function
\(u(\cdot, t)\) is the average of two translates of \(\tilde f\). A corner of
\(\tilde f\) at a point \(x_*\) therefore reappears at \(x_* - ct\) and
\(x_* + ct\), with the jump in the derivative halved, as long as no other corner of
\(\tilde f\) lies at distance \(2c|t|\) from \(x_*\). Under the heat equation the
same corner would be smoothed away at every \(t \gt 0\).
Energy Conservation
The contrast with the parabolic case now becomes quantitative. The wave equation
admits a conserved quantity, whose form is itself a structural diagnostic. For the
heat equation, the natural quadratic quantity is the squared \(L^2\) norm
\(\int u^2\,dx\), which is non-increasing under Dirichlet boundary conditions. The
wave equation instead conserves a quadratic functional exactly, at every time at
which the solution is defined, earlier and later alike. That functional is the
mechanical energy of the solution.
Let \(I\) be an interval containing \(0\), as in the previous section. For a \(C^2\)
solution \(u\) of the wave equation on \(\Omega \times I\), define the
mechanical energy at time \(t \in I\) by
\[
E(t) := \frac{1}{2}\int_\Omega
\Bigl[\bigl(\partial_t u(x, t)\bigr)^2
+ c^2 \bigl(\partial_x u(x, t)\bigr)^2\Bigr] dx.
\]
Here \(\Omega\) is either the bounded interval \([0, L]\) with homogeneous
Dirichlet boundary conditions or the entire real line. On the real line the
integral may be infinite, and the theorem below states a condition on the data
under which it is finite.
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. The second is the
potential energy density, proportional to the squared slope of the
string, the elastic energy stored by stretching the string against its tension. The
quantity \(E(t)\) is the physical energy of the string at time \(t\) divided by its
linear mass density \(\rho_0\). 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 \(I\) be an interval of positive length containing \(0\), and let \(u\) be a
\(C^2\) solution of the wave equation \(\partial_{tt} u = c^2 \partial_{xx} u\)
on \(\Omega \times I\), where either
(i) \(\Omega = [0, L]\) and \(u\) satisfies the homogeneous Dirichlet boundary
conditions \(u(0, t) = u(L, t) = 0\) for all \(t \in I\), or
(ii) \(\Omega = \mathbb{R}\) and the initial data \(f = u(\cdot, 0)\) and
\(g = \partial_t u(\cdot, 0)\) satisfy
\(\int_{\mathbb{R}} (f')^2\, dx \lt \infty\) and
\(\int_{\mathbb{R}} g^2\, dx \lt \infty\).
Then the mechanical energy is finite, and \(E(t) = E(0)\) for every \(t \in I\).
In case (ii), with \(P = g - cf'\) and \(Q = g + cf'\),
\[
E(t) = \frac{1}{4}\int_{\mathbb{R}} \bigl(P^2 + Q^2\bigr)\, dx
\quad\text{for all } t \in I.
\]
Proof:
Case (i).
Fix a compact interval \([t_1, t_2] \subseteq I\). On
\([0, L] \times [t_1, t_2]\) the integrand of \(E\) and its \(t\)-derivative
\(\partial_t u\, \partial_{tt} u + c^2\, \partial_x u\, \partial_t \partial_x u\)
are continuous and hence bounded. By the mean value theorem, the difference
quotients of the integrand in \(t\) are bounded there by the supremum of
that derivative, and
dominated convergence,
applied along every sequence of increments with limit zero, justifies
differentiation under the integral sign. The mixed partial derivatives of \(u\)
agree, as noted in the reduction to transport equations, so
\(\partial_t \partial_x u = \partial_x(\partial_t u)\), and we obtain
\[
\frac{dE}{dt}
= \int_0^L \bigl[\partial_t u \cdot \partial_{tt} u
+ c^2 \partial_x u \cdot \partial_x(\partial_t u)\bigr] dx.
\]
Substituting the wave equation \(\partial_{tt} u = c^2 \partial_{xx} u\) into
the first term and integrating the second term by parts in \(x\),
\[
\frac{dE}{dt}
= \int_0^L c^2 \partial_t u \cdot \partial_{xx} u\, dx
+ \Bigl[c^2 \partial_x u \cdot \partial_t u\Bigr]_0^L
- \int_0^L c^2 \partial_{xx} u \cdot \partial_t u\, dx.
\]
The two volume integrals are identical and cancel, leaving only
the boundary term:
\[
\frac{dE}{dt}
= \Bigl[c^2 \partial_x u(x, t) \cdot \partial_t u(x, t)\Bigr]_0^L.
\]
Since \(u(0, t) = u(L, t) = 0\) for every \(t \in I\), differentiating the
boundary conditions in \(t\) gives
\(\partial_t u(0, t) = \partial_t u(L, t) = 0\), and the boundary contribution
vanishes. Thus \(dE/dt = 0\) on \([t_1, t_2]\), and since this was an arbitrary
compact subinterval of \(I\), \(E\) is constant on \(I\).
Case (ii).
By the reduction to two transport equations, the functions
\(\partial_t u - c\, \partial_x u\) and \(\partial_t u + c\, \partial_x u\)
equal \(P(x - ct)\) and \(Q(x + ct)\) on \(\mathbb{R} \times I\), with \(P\) and
\(Q\) as in the statement. Hence
\[
\begin{align*}
\partial_t u &= \tfrac{1}{2}\bigl[P(x - ct) + Q(x + ct)\bigr], \\\\
c\, \partial_x u &= \tfrac{1}{2}\bigl[Q(x + ct) - P(x - ct)\bigr],
\end{align*}
\]
and adding the squares,
\[
(\partial_t u)^2 + c^2 (\partial_x u)^2
= \tfrac{1}{2}\bigl[P(x - ct)^2 + Q(x + ct)^2\bigr].
\]
The substitutions \(y = x - ct\) and \(y = x + ct\) in the integrals over
\(\mathbb{R}\) of the non-negative continuous functions \(P^2\) and \(Q^2\) give
the stated formula for \(E(t)\), which does not depend on \(t\). It is finite,
because \((\alpha \pm \beta)^2 \leq 2\alpha^2 + 2\beta^2\) bounds \(P^2\) and
\(Q^2\) by \(2g^2 + 2c^2 (f')^2\).
The conservation law has a structural significance that goes beyond its statement.
For the heat equation with Dirichlet conditions on \([0, L]\), the computation in
the uniqueness step of the proof of the previous page's
series solution theorem
applies to any solution in that theorem's class and gives, for \(t \gt 0\),
\[
\frac{d}{dt}\int_0^L u(x, t)^2\, dx = -2k\int_0^L (\partial_x u)^2\, dx \leq 0.
\]
The parabolic quantity can never increase. The hyperbolic quantity
is exactly preserved.
The mechanical energy is not the only conserved quadratic quantity. On the real
line, under the hypotheses of case (ii), the same expressions give
\(\partial_t u\, \partial_x u = \bigl[Q(x + ct)^2 - P(x - ct)^2\bigr]/(4c)\).
Integrating over \(\mathbb{R}\) and substituting as before yields
\[
\int_{\mathbb{R}} \partial_t u\, \partial_x u\, dx
= \frac{1}{4c}\int_{\mathbb{R}} \bigl(Q^2 - P^2\bigr)\, dx
\]
for every \(t \in I\). This momentum is conserved as well, but its
integrand can take either sign, so it cannot control the solution. What singles out
the energy is that its density is positive definite in
\((\partial_t u, \partial_x u)\). The density vanishes only where both derivatives
vanish, and that is what turns conservation into uniqueness.
The energy method that established uniqueness for the heat equation on the previous
page now delivers uniqueness for the wave equation. There the squared \(L^2\) norm
of the difference of two solutions was non-increasing. Here the energy of the
difference is constant. Let \(u_1\) and \(u_2\) be \(C^2\) solutions on
\(\Omega \times I\) with the same initial data and, when \(\Omega = [0, L]\), the
Dirichlet condition. Their difference \(\eta = u_1 - u_2\) solves the wave equation
with zero initial data, so on the real line case (ii) applies trivially, and in
either case \(E(t) = E(0) = 0\) for every \(t \in I\). The integrand is non-negative
and continuous, hence zero pointwise, so \(\partial_t \eta \equiv 0\) and
\(\partial_x \eta \equiv 0\). Therefore \(\eta\) is constant on the convex set
\(\Omega \times I\), and \(\eta(x, 0) = 0\) identifies the constant as zero.
Two differences from the parabolic argument deserve notice. First, the solutions
here are \(C^2\) up to and including \(t = 0\), whereas on the previous page the
initial datum was attained only in \(L^2\), which is why that argument had to pass
to the limit \(s \to 0^+\). Second, on the real line the argument needs no
hypothesis at infinity, because the difference has zero data. There, however, case
(ii) rests on the same reduction to transport equations, so the argument repackages
the uniqueness that d'Alembert's formula already furnished rather than proving it
independently. Its independent value is on the bounded interval, where it supplies
the uniqueness that the
classical solution theorem
did not state.
Finite Speed of Propagation
D'Alembert's formula shows that \(u(x_0, t_0)\) depends only on the data on
\([x_0 - ct_0, x_0 + ct_0]\), but it applies only to solutions defined on all of
\(\mathbb{R} \times I\). The energy method gives the same conclusion for a
solution known only on a strip \(D \times [0, t_0]\) whose spatial interval \(D\)
contains \([x_0 - ct_0, x_0 + ct_0]\), with no hypothesis at infinity. The idea
is to integrate the energy density over an interval that shrinks at speed \(c\)
from both ends.
Theorem: Finite Speed of Propagation
Let \(x_0 \in \mathbb{R}\) and \(t_0 \gt 0\), let \(D \subseteq \mathbb{R}\) be
an interval containing \([x_0 - ct_0, x_0 + ct_0]\), and let \(u\) be a \(C^2\)
solution of the wave equation on \(D \times [0, t_0]\). If \(u(\cdot, 0)\) and
\(\partial_t u(\cdot, 0)\) vanish on \([x_0 - ct_0, x_0 + ct_0]\), then \(u\)
vanishes on the backward light cone
\[
\Lambda = \{(x, t) : 0 \leq t \leq t_0,\ |x - x_0| \leq c(t_0 - t)\}.
\]
In particular, two \(C^2\) solutions on \(D \times [0, t_0]\) whose data agree
on \([x_0 - ct_0, x_0 + ct_0]\) agree at \((x_0, t_0)\).
Proof:
For \(0 \leq t \lt t_0\), let \(x_-(t) = x_0 - c(t_0 - t)\) and
\(x_+(t) = x_0 + c(t_0 - t)\), so that \(x_-(t) \lt x_+(t)\) and both lie in
\(D\). Write \(\rho = (\partial_t u)^2 + c^2 (\partial_x u)^2\) and set
\[
e(t) = \frac{1}{2}\int_{x_-(t)}^{x_+(t)} \rho(x, t)\, dx.
\]
Both \(\rho\) and \(\partial_t \rho\) are continuous, and the limits are affine
in \(t\) with \(x_-' = c\) and \(x_+' = -c\). The rule for differentiating an
integral whose integrand and limits depend on \(t\), another elementary fact
that we use without proof, therefore gives
\[
\frac{de}{dt}
= -\frac{c}{2}\bigl[\rho(x_+, t) + \rho(x_-, t)\bigr]
+ \int_{x_-}^{x_+} \bigl[\partial_t u\, \partial_{tt} u
+ c^2\, \partial_x u\, \partial_x(\partial_t u)\bigr] dx,
\]
where the mixed partial derivatives of \(u\) have been identified as before.
By the wave equation, the integrand equals
\[
c^2\bigl[\partial_t u\, \partial_{xx} u + \partial_x u\, \partial_x(\partial_t u)\bigr]
= c^2\, \partial_x(\partial_t u\, \partial_x u),
\]
so the integral is \(c^2 \bigl[\partial_t u\, \partial_x u\bigr]_{x_-}^{x_+}\).
Since \((|\partial_t u| - c|\partial_x u|)^2 \geq 0\), we have
\(2c\, |\partial_t u\, \partial_x u| \leq \rho\), that is,
\(c^2 |\partial_t u\, \partial_x u| \leq \tfrac{c}{2}\rho\). Each endpoint term
of the integral is therefore dominated by the corresponding term
\(\tfrac{c}{2}\rho\) that is subtracted, and \(de/dt \leq 0\) on \([0, t_0)\).
At \(t = 0\), \(u(\cdot, 0)\) vanishes on \([x_-(0), x_+(0)]\), so its
\(x\)-derivative vanishes there, and \(\partial_t u(\cdot, 0) = 0\) there by
hypothesis. Hence \(e(0) = 0\). Since \(e \geq 0\) and \(e\) is non-increasing,
\(e \equiv 0\) on \([0, t_0)\). Since \(\rho\) is continuous and non-negative,
it vanishes on \([x_-(t), x_+(t)]\) for each such \(t\), and \(c \gt 0\) gives
\(\partial_t u = \partial_x u = 0\) on \(\Lambda\) minus its apex
\((x_0, t_0)\). That set is convex, so \(u\) is constant on it, and the
constant is \(0\) because \(u\) vanishes on the base
\([x_0 - ct_0, x_0 + ct_0] \times \{0\}\). By continuity, \(u\) also vanishes
at the apex. The last assertion follows by applying this to the difference of
the two solutions.
Information therefore travels at speed at most \(c\), whether or not the solution is
defined beyond the light cone. This locality is the hyperbolic counterpart of the
instantaneous global coupling of the heat equation.
Time Reversal
We can now return to the contrast drawn at the end of the previous page. The heat
equation has a preferred direction of time, since running it backward is ill-posed.
The wave equation has none, in the following exact sense.
Theorem: Time Reversal
Let \(\Omega\) be \(\mathbb{R}\) or \([0, L]\), let \(I\) be an interval of
positive length, let \(T \in I\), and let \(u\) be a \(C^2\) solution of the
wave equation on \(\Omega \times I\). Then \(v(x, s) = u(x, T - s)\) is a
\(C^2\) solution of the wave equation on \(\Omega \times (T - I)\), where
\(T - I = \{T - t : t \in I\}\) is an interval containing \(0\), and its data at
\(s = 0\) are
\[
\begin{align*}
v(\cdot, 0) &= u(\cdot, T), \\\\
\partial_s v(\cdot, 0) &= -\partial_t u(\cdot, T).
\end{align*}
\]
If \(\Omega = [0, L]\) and \(u(0, t) = u(L, t) = 0\) for \(t \in I\), then
\(v(0, s) = v(L, s) = 0\) for \(s \in T - I\).
Proof:
By the chain rule, \(\partial_s v(x, s) = -\partial_t u(x, T - s)\) and
\(\partial_{ss} v(x, s) = \partial_{tt} u(x, T - s)\), while the
\(x\)-derivatives are unchanged. Hence \(v\) is \(C^2\), and
\(\partial_{ss} v = c^2\, \partial_{xx} v\) follows from the wave equation for
\(u\) at time \(T - s\). The data and the boundary values are read off at
\(s = 0\) and at \(x = 0, L\).
The theorem turns a backward problem into a forward one. Suppose a state
\((f_T, g_T)\) with \(f_T \in C^2\) and \(g_T \in C^1\) is prescribed at a time
\(T\), and a solution taking this state at time \(T\) is sought at earlier times. By
the theorem, this is the forward problem for \(v\) with data \(f_T\) and \(-g_T\).
On the real line,
d'Alembert's formula
with \(I = \mathbb{R}\) solves that problem uniquely for all times. On the
bounded interval, the
classical solution theorem
applies whenever \((f_T, g_T)\) satisfies its compatibility conditions, and these
conditions are necessary for a \(C^2\) solution, as shown after that theorem. The
energy argument gives uniqueness. Existence and uniqueness therefore hold backward
in time as well as forward.
The third requirement of well-posedness is continuous dependence on the data. The
previous page exhibited its failure for the backward heat equation in the \(L^2\)
norm on the real line. For the wave equation on the real line, with data as in
d'Alembert's formula that in addition lie in \(L^2(\mathbb{R})\), that formula
gives the estimate
\[
\|u(\cdot, t)\|_{L^2} \leq \|f\|_{L^2} + |t|\, \|g\|_{L^2}
\]
for every \(t \in \mathbb{R}\). The first term of the formula has \(L^2\) norm at
most \(\|f\|_{L^2}\), by the triangle inequality and the substitutions
\(y = x \mp ct\). The second term \(h\) is \(\pm\frac{1}{2c}\) times the integral of
\(g\) over \([x - c|t|, x + c|t|]\), an interval of length \(2c|t|\), so the
Cauchy-Schwarz inequality gives
\(|h(x)|^2 \leq \frac{|t|}{2c}\int_{x - c|t|}^{x + c|t|} g(\sigma)^2\, d\sigma\).
Integrating in \(x\) and exchanging the order of integration by
Tonelli's theorem
gives \(\|h\|_{L^2} \leq |t|\, \|g\|_{L^2}\). Bounding the same two terms pointwise
gives \(|u(x, t)| \leq \sup |f| + |t| \sup |g|\), and on the bounded interval this
bound is part of the classical solution theorem.
Applied to the difference of two solutions, each estimate bounds the change in the
solution at any earlier or later time by the change in the data, with the same
constants in both directions. Neither estimate is a maximum principle. Both bounds
grow linearly in \(|t|\) and involve the velocity \(g\), and a nonzero \(g\) can
carry \(u\) outside the range of \(f\). The energy is constant on every interval
\(I\) containing \(0\), whether \(I\) extends forward or backward from \(0\). Apart
from the factor \(|t|\), which enters identically in both directions, nothing is
amplified when the wave equation runs backward. In this precise sense the backward
problem is as well-posed as the forward one, and no information is lost.