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, the d'Alembert
formula, and the proof that mechanical energy is exactly conserved. We close 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
\[
\begin{cases}
\partial_{tt} u = c^2\, \partial_{xx} u, & 0 \lt x \lt L,\ t \gt 0, \\\\
u(0, t) = u(L, t) = 0, & t \geq 0, \\\\
u(x, 0) = f(x), & 0 \leq x \leq L, \\\\
\partial_t u(x, 0) = g(x), & 0 \leq x \leq L,
\end{cases}
\]
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. The three-case analysis (\(\lambda \lt 0\), \(\lambda = 0\),
\(\lambda \gt 0\)) 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 \([0, L]\):
\[
\begin{align*}
\lambda_n &= \left(\frac{n\pi}{L}\right)^{\!2}, \\\\
\phi_n(x) &= \sin\frac{n\pi x}{L}, \quad n = 1, 2, 3, \ldots
\end{align*}
\]
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 \([0, L]\) 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
\[
\begin{align*}
f(0) = f(L) &= 0, \\\\
g(0) = g(L) &= 0.
\end{align*}
\]
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 identifies the series and its coefficients. Convergence of the
full infinite series together with its first and second derivatives to a
classical \(C^2\) solution requires the additional regularity discussed in the
following paragraph.
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, together with higher-order boundary compatibility
conditions (the second-derivative compatibility \(f''(0) = f''(L) = 0\), required so
that the odd \(2L\)-periodic extension of \(f\) is itself \(C^2\) and the
term-by-term second differentiation of the sine series converges).
With no exponential damping to override these requirements, the convergence question
is intrinsically harder on the bounded interval than for the heat equation. We
pursue a different problem in the next section. On the entire real line, the method
of characteristics yields an explicit \(C^2\) solution formula in closed form, with
no infinite series to be checked term by term.
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. The regularity of \(u(\cdot, t)\) at any time
\(t \gt 0\) 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,
\[
\sin\frac{n\pi x}{L}\, \bigl[B_n \cos\omega_n t + C_n \sin\omega_n t\bigr],
\quad \omega_n = \frac{n\pi c}{L},
\]
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. 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, 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 \(L\). 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 \(\rho_0\)).
That relation 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. Both
are physical operations whose effect on pitch is exactly a change in 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 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
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:
\[
\partial_{tt} u = c^2\, \partial_{xx} u, \quad
x \in \mathbb{R},\ t \gt 0,
\]
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 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
\(t\),
\[
\partial_{tt} \hat u(\xi, t) = -c^2 \xi^2\, \hat u(\xi, t),
\quad \xi \in \mathbb{R},
\]
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 \(\hat f, \hat g\).
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
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 travelling 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
Before factoring the second-order wave operator, we record the elementary fact that
drives the entire construction. Consider the first-order linear PDE
\[
\partial_t \psi + c\, \partial_x \psi = 0,
\quad x \in \mathbb{R},\ t \gt 0,
\]
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
\(\psi_0\) is 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
\[
\begin{align*}
\partial_t \psi &= -c\, \psi_0'(x - ct), \\\\
\partial_x \psi &= \psi_0'(x - ct),
\end{align*}
\]
and therefore \(\partial_t \psi + c\, \partial_x \psi = 0\).
Conversely, 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
\[
\begin{align*}
\partial_{tt} - c^2 \partial_{xx}
&= (\partial_t - c\, \partial_x)(\partial_t + c\, \partial_x) \\\\
&= (\partial_t + c\, \partial_x)(\partial_t - c\, \partial_x),
\end{align*}
\]
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.
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 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
\[
\begin{align*}
F(s) &:= -\frac{1}{2c}\mathcal{P}(s), \\\\
G(s) &:= \frac{1}{2c}\mathcal{Q}(s),
\end{align*}
\]
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 \(\mathbb{R} \times [0, \infty)\) 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 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 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 \(u(x, t) = F(x - ct) + G(x + ct)\),
\[
\begin{align*}
\partial_{tt} u &= c^2 F''(x - ct) + c^2 G''(x + ct) \\\\
c^2 \partial_{xx} u &= c^2 F''(x - ct) + c^2 G''(x + ct).
\end{align*}
\]
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 \(g \in C^1(\mathbb{R})\). The unique \(C^2\)
solution of the initial-value problem
\[
\begin{cases}
\partial_{tt} u = c^2\, \partial_{xx} u, & x \in \mathbb{R},\ t \gt 0, \\\\
u(x, 0) = f(x), & x \in \mathbb{R}, \\\\
\partial_t u(x, 0) = g(x), & x \in \mathbb{R}
\end{cases}
\]
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
\(C^2(\mathbb{R} \times [0, \infty))\), and the derivation above shows it solves
the PDE and matches both initial conditions. The integral term lies in \(C^2\)
by the fundamental theorem of calculus applied to its \(C^1\) integrand.
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 holds 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, whose 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, these are the 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 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 \(\mathbb{R}\), and dually the range of influence of \(f(x_0)\)
is the full upper half-plane \(\{t \gt 0\}\). The parabolic counterpart of finite
propagation speed is therefore its absolute negation, instantaneous global coupling.
The qualitative inversions enumerated alongside in the introduction follow as
consequences.
Energy Conservation
Those qualitative inversions, articulated as structural features of the wave
equation in contrast with the parabolic case, are now sharpened 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 exactly, for all time, in both
forward and backward time directions. That functional is the mechanical
energy of the solution.
For a sufficiently regular solution \(u\) of the wave equation on a domain
\(\Omega\), define the mechanical energy at time \(t\) by
\[
E(t) := \frac{1}{2}\int_\Omega
\Bigl[\bigl(\partial_t u(x, t)\bigr)^2
+ c^2 \bigl(\partial_x u(x, t)\bigr)^2\Bigr] dx.
\]
Here \(\Omega\) is 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.
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 local strain, which is 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:
\[
\frac{dE}{dt}(t) = 0
\quad\text{for all } t \geq 0,
\]
and consequently \(E(t) = E(0)\) for every \(t \geq 0\).
Proof:
The regularity of \(u\) and, in case (ii), the decay assumption justify
differentiation of \(E(t)\) under the integral sign, which 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
\[
M(t) = \tfrac{1}{2}\int_\Omega u(x, t)^2\, dx.
\]
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 same energy-method framework that established uniqueness for the heat equation
on the previous page now delivers uniqueness for the wave equation as an immediate
corollary of conservation. There it yielded \(dM/dt \leq 0\) with strict inequality
generically. Here it yields \(dE/dt = 0\) with no inequality. 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, since the
integrand is non-negative and continuous, hence zero pointwise. Therefore \(\eta\)
is constant in spacetime, and 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.
In fact, the finite speed of propagation allows this uniqueness to be established
purely locally. Applying the energy argument over the truncated backward light cone
\[
\{(x, t) : |x - x_0| \leq c(t_0 - t),\ 0 \leq t \leq t_0\},
\]
on which the energy is non-increasing, we find that the value of \(u\) at any point
\((x_0, t_0)\) is determined entirely by the initial data on
\([x_0 - c t_0,\ x_0 + c t_0]\), without any hypothesis on the behaviour of \(f, g\)
at infinity. This locality is the hyperbolic counterpart to the heat equation's
instantaneous global smoothing. Information propagates at finite speed, and the
domain of dependence is bounded. The hyperbolic uniqueness theorem is a one-line
consequence of the conservation theorem.