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For a scalar equation of order $k$, Cauchy data consist of the first $k$ normal jets $u,{\partial }_{N}u,\dots ,{\partial }_N^{k-1}u$ on the initial hypersurface.

These prescribed values define the initial conditions of a Cauchy problem.

For constant wave speed $c>0$, the whole-line Cauchy problemhas the unique classical solution, for $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$,The first term evolves displacement released from rest; the second evolves initial velocity. The factor $1/(2c)$ converts an integral of velocity over distance into displacement. On a finite string the formula applies before the backward characteristic curves reach an endpoint, or globally after an extension that enforces the boundary conditions. ^[ref]

For a completely smooth example, choose a bump function with compact support:where $a>0$ sets the half-width and $A$ sets the initial peak. ThenAt $t=0$ the two copies coincide, recovering peak $A$. At $t=a/(2c)$, $u(0,t)=A{e}^{-1/3}\simeq 0.716531A$. At $t=3a/(2c)$ the pulses are separated, centred at $x=\pm 3a/2$, each of width $2a$ and peak $A/2$. Zero initial velocity forces equal left and right contributions; a purely right-moving initial profile instead requires $g=-c{f}^{\prime }$. Thus “released from rest” and “launched to the right” describe different Cauchy data.

The domain of dependence of $(x_{\ast },t_{\ast })$, for $t_{\ast }\ge 0$, is contained in $[x_{\ast }-c{t}_{\ast },x_{\ast }+c{t}_{\ast }]$ on the initial line. More precisely, displacement contributes only through the two endpoints, while initial velocity contributes throughout the interval. With $c=1$, $x_{\ast }=0.5$ and $t_{\ast }=1.5$, that interval is $[-1,2]$. Changing either initial datum outside it cannot change $u(0.5,1.5)$. If both initial data vanish outside $[-a,a]$, the solution vanishes outside $[-a-ct,a+ct]$: this is finite propagation speed. The expanding interval bounds possible influence; it need not be filled by nonzero displacement, as the separated bumps demonstrate.

Historically, d'Alembert studied the vibrating-string problem in “Recherches sur la courbe que forme une corde tendue mise en vibration”, in the Berlin Academy's volume for the year 1747, pages 214–219. The memoir introduces the differential equation and integrates it through two arbitrary one-variable functions. The original geometric problem makes the formula's enduring appeal clear: arbitrary initial shapes can be handled without first decomposing them into normal modes. ^[ref]

For a brief derivation, use characteristic coordinates $\xi =x-ct$ and $\eta =x+ct$. The chain rule gives $u_{tt}-c^2{u}_{xx}=-4c^2{u}_{\xi \eta }$, so $u=F(\xi )+G(\eta )$. The initial conditions yield $F+G=f$ and $-c{F}^{\prime }+c{G}^{\prime }=g$. Integrating these relations gives the displayed formula; the arbitrary integration constants cancel in their sum.

The positive conformal factor leaves the causal structure unchanged. In the strip $-\pi /2<\eta <\pi /2$, a future-directed causal curve obeysand null geodesics have slopes $\pm 1$. Its past and future conformal boundaries $\eta =\mp \pi /2$ are spacelike. A finite conformal time interval limits the range of light signals: the observer at $\xi =0$ can receive a signal from $(\eta ,\xi )$ before future infinity exactly when $|\xi |<\pi /2-\eta$. Equality describes its observer-dependent cosmological event horizon.

For a finite Penrose diagram of the unwrapped spatial coordinate, set $U=\arctan u$, $V=\arctan v$, $X=(U+V)/2$, and $Y=(V-U)/2$. Then the metric is conformal to $-d{Y}^2+d{X}^2$; light rays remain at $45$ degrees, and the boundaries are the images of $v-u=\pm \pi$. The figure also shows the spatially periodic de Sitter cylinder so that the global distinction is explicit.

Both the covering spacetime and the de Sitter cylinder are globally hyperbolic. To see this directly in the covering spacetime, suppose an inextendible future-directed causal curve had a future limiting conformal time strictly below $\pi /2$. The slope bound makes $\xi$ Cauchy as that time is approached, so the curve has a limiting point inside the spacetime and can be extended, a contradiction. The past endpoint similarly approaches $-\pi /2$. Thus every constant-$\eta$ line is crossed exactly once and is a Cauchy hypersurface. On the cylinder the same argument uses compactness of the spatial circle. Equivalently causal diamonds between finite events are bounded and closed within the strip, hence compact.

Initial field data on a complete Cauchy hypersurface determine a unique solution for a well-posed hyperbolic Cauchy problem, with the relevant constraints imposed for constrained systems. Spacelike conformal infinity does not require additional timelike-boundary conditions. An observer's cosmological horizon limits which data that observer can see; it does not destroy global uniqueness from data on the whole hypersurface.

Fourier transform on the infinite spatial lattice giveswith polynomial roots $G_{\pm }=iA\pm \sqrt{1-A^2}$. When $|A|$ is uniformly less than one, the polynomial roots have modulus one and a uniform positive separation. Diagonalizing the companion matrix then bounds all its powers independently of $\theta$ and $n$. Parseval's identity proves stability in the discrete $L^2$ norm for arbitrary square-summable starting perturbations at both levels.

At $\mu =1/2$ the coefficients giveConsequently $\mu =1/2$ is stable.

At $\mu =3/2$,The growing polynomial root at this phase has modulusTo turn this into a Cauchy problem linear instability proof, rather than rely on a plane wave of infinite norm, choose a small interval around $\pi /3$ on which the growing polynomial root has modulus at least $1+\delta$. Choose an initial Fourier transform amplitude that is a nonzero square-integrable function supported there, and set the level-one amplitude equal to that polynomial root times the level-zero amplitude. The recurrence then multiplies by the same growing polynomial root at every step on the support. Parseval identity gives $\Vert u^n{\Vert }_2\ge (1+\delta {)}^n\Vert u^0{\Vert }_2$, whereas both initial norms are bounded independently of $h$. Real data can be obtained by adding the conjugate interval around $-\pi /3$. Since $n=T/(\mu h)$ tends to infinity at fixed $T$, $\mu =3/2$ is unstable.

Use the Fourier transform for the spatial Cauchy problem, or its periodic analogue, and regard the two starting levels as independently perturbed data. A Fourier mode with spatial factor $e^{im\theta }$ has amplification roots $G$ satisfyingPut $c=2\mu -1$ and $G=e^{i\theta /2}q$. ThenIf $|c|<1$, both roots have modulus one and their separation is bounded below uniformly in frequency:The uniform power bound from separated amplification roots now controls the two-level companion matrix for every time step. Its entries are uniformly bounded, and its eigenvector conditioning is bounded by the reciprocal root gap. The Parseval identity transfers this frequency-uniform bound to the spatial ${\ell }^2$ norm. This proves stability, rather than merely checking each root's modulus.

If $|c|>1$, the frequency $\theta =\pi$ has a root outside the unit disk, so there is exponential instability. If $|c|=1$, the two roots at $\theta =\pi$ coincide on the unit circle. The companion matrix is not a scalar matrix and has a nontrivial Jordan block; its powers grow linearly in the number of steps. Frequencies arbitrarily near that value produce the same lack of a uniform bound for localized Fourier packets, so this also invalidates Cauchy ${\ell }^2$ stability, not only periodic plane-wave stability. At $\mu =1$ the double amplification root is $-1$, and at $\mu =0$ it is $1$.

Therefore the full two-level stability range for a fixed positive Courant ratio isThe endpoint $\mu =0$ is moreover not a positive time step. Bounds deteriorate as $\mu$ approaches either endpoint; the displayed range is not a uniform claim over ratios arbitrarily close to one. A prescribed starter that removes one special parasitic component can change behavior for selected initial data, but does not establish the requested unrestricted two-level stability.

The wave equation is a second-order linear partial differential equation describing waves with finite propagation speed. In one spatial dimension its homogeneous, constant-coefficient form isHere $x$ is position, $t$ is time, $u(x,t)$ is the disturbance, and subscripts denote partial derivatives. For a wave equation on a string, $u$ is transverse displacement; in linear acoustics, it can be a pressure perturbation. In several spatial dimensions, $u_{xx}$ becomes the Laplacian $\Delta u$. The same mathematical structure connects vibrating strings and acoustic waves, although their physical assumptions and wave speeds differ. ^[ref]

For the whole-line Cauchy problem, prescribe both displacement $u(x,0)=f(x)$ and velocity $u_t(x,0)=g(x)$. A classical solution has continuous partial derivatives through order two; $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$ suffice for D'Alembert's formula to produce one. A finite string additionally needs boundary conditions. These statements concern the unforced model with constant $c$; damping, spatially varying coefficients and nonlinear terms change the solution formula.

A travelling wave $F(x-ct)$ carries its entire profile to the right at wave speed $c$; $G(x+ct)$ travels left. Their sum is again a solution by the superposition principle. A standing wave instead has a fixed mode shape whose size and sign oscillate: in $\sin (kx)\cos (ckt)$, each node of a standing wave stays stationary. The distinction is especially clear in the pulse and string diagrams below. For this ideal model, the dispersion relation is ${\omega }^2=c^2{k}^2$: each propagation branch is nondispersive, so a localized profile translates without spreading. This makes the equation a useful reference against which to understand wave dispersion and damping.

The appealing combination is local interaction with global predictability: neighbouring displacements determine acceleration through $u_{xx}$, yet D'Alembert's formula explicitly identifies exactly which initial locations can influence a later observation. Finite propagation speed turns that observation into a domain of dependence, rather than requiring knowledge of the entire initial string.