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The one-dimensional wave equation has left-moving and right-moving travelling-wave solutions. Their superposition is described by the D'Alembert formula and remains valid distributionally for locally integrable profiles.

Put $u=ct-x$ and $v=ct+x$. The chain rule turns the wave equation into $4y_{uv}=0$. Integrating first in $v$ and then in $u$ gives the D'Alembert formulaAt the left endpoint, $g(u)=-f(u)$ for $u>0$. At the right endpoint, $f(ct-L)+g(ct+L)=0$, henceThus the travelling waves can be represented by $g=-f$, with $f$ and $g$ both $2L$-periodic. More precisely, the positive-time boundary conditions constrain only $f$ on $(-L,\infty )$ and $g$ on $(0,\infty )$; their unused arguments may be filled in by this periodic extension. They do not constrain arbitrary pre-existing choices outside those ranges. This is periodic reflection for a fixed-end string.

Using $g=-f$, differentiation gives $y_t=c(f^{\prime }(ct-x)-f^{\prime }(ct+x))$ and $y_x=-f^{\prime }(ct-x)-f^{\prime }(ct+x)$. The cross terms in the wave energy cancel, soThis integrates a $2L$-periodic function over exactly one period. $E(t)$ is constant. Equivalently, the wave equation gives $E^{\prime }(t)=[y_t{y}_x{]}_0^L=0$, because the fixed endpoint values imply $y_t=0$ there.

With zero initial time derivative, the integral term in the D'Alembert formula vanishes. The Heaviside step function representation givesThe rectangular pulse splitting under the wave equation consists of two width-two pulses of height $1/2$, moving left and right at speed $c$. At $t=0$ they coincide and add to the initial height-one pulse. Before $ct=1$ the overlapping interval has height one; after $ct=1$ the two pulses are separated. Values exactly at the fronts depend on the chosen $H(0)$ convention. Away from the fronts the equation holds classically, and across the fronts it holds as a weak solution of the wave equation.

Use the radial reduction of the three-dimensional wave equation $w=r\varphi$ and extend $w$ oddly in $r$. Write $w=w_h+w_F$, where $w_h$ is the homogeneous wave equation solution with the prescribed Cauchy data, and $w_F$ has zero Cauchy data. With $w_0(r)=r{\varphi }_0(r)$ and $w_1(r)=r{\varphi }_1(r)$, the D'Alembert formula givesBoth odd initial profiles have compact support, soThe Duhamel principle, applied to $w_{tt}-w_{rr}=-rF$, givesNow write $r=t+a$ with $1/2\le a\le 1$. The lower integration limit is $s+a>0$, so this domain of dependence never crosses the axis. Intersecting its integration interval with the support of the source leaves a subset of $[s+a,s+1]$, of length at most one. On this interval,The outgoing shell source estimate for a radial wave therefore yieldsFinally $r\ge t+1/2\ge (1+t)/2$, and thusThis constant depends only on the Cauchy data and the fixed unit bound for the source. The logarithm comes from integrating the shell's accumulated forcing, whereas the factor $(1+t{)}^{-1}$ comes from division by $r$.

Put $v=e^{t/2}u$. The cancellation of the damping and mass terms gives $v_{tt}=v_{xx}$, with $v(x,0)=u_0(x)$ and $v_t(x,0)=u_1(x)+u_0(x)/2$. For the wave equation, $w_{+}=v_t+v_x$ satisfies $({\partial }_t-{\partial }_x)w_{+}=0$, and $w_{-}=v_t-v_x$ satisfies $({\partial }_t+{\partial }_x)w_{-}=0$. Thus these combinations are transported on straight characteristics. Integrating them and matching the initial displacement yields the D'Alembert formula, and henceHere the initial functions are extended periodically to the real line. Direct differentiation verifies both initial conditions and the equation, and arbitrary repeated differentiation shows the solution is smooth and periodic.

This characteristic derivation also proves finite propagation speed from first principles: the value at $(x,t)$ depends only on the two endpoint displacements and initial velocity data in $[x-t,x+t]$. If two data sets agree on that interval, their solutions agree at that point. Equivalently initially supported data have no influence outside the unit-speed characteristic cone, interpreted on the periodic covering line. Multiplication by $e^{-t/2}$ does not enlarge this domain of dependence.

The method of images for the Neumann boundary conditions extends $bx$ evenly through zero and then with period $2\pi$. Define the triangular waveThe formulas agree at their endpoints. It is even, $2\pi$-periodic, and agrees with $bx$ on $[0,\pi ]$. Its Fourier cosine series has the coefficients in part (a). The cosine product identity then gives the D'Alembert formulaIt is the average of two oppositely traveling reflected waves. For $b>0$, the sketch rises from zero at even multiples of $\pi$ to $b\pi$ at odd multiples; for $b<0$ it is reflected vertically, and for $b=0$ it is zero.

For the wave equation, introduce characteristic coordinates $\xi =x-ct$, $\eta =x+ct$ when $c\ne 0$. The equation becomes $u_{\xi \eta }=0$, so $u=F(x-ct)+G(x+ct)$. The initial conditions imply $F+G=\varphi$ and $-c{F}^{\prime }+c{G}^{\prime }=0$. Constants can be absorbed into $F,G$, giving the D'Alembert formulaIf $\varphi$ vanishes outside $[-\alpha ,\alpha ]$, then $u(x,t)=0$ whenever $|x|>\alpha +|c|t$. Thus Property P fails: the wave equation has finite propagation speed $|c|$. No arbitrarily short time can transmit the disturbance arbitrarily far. If $c=0$, the zero initial velocity instead gives $u(x,t)=\varphi (x)$, and the conclusion still holds.

Choose $X=x-2y$ and $T=x$, so $(\alpha ,\beta ,\gamma ,\delta )=(1,-2,1,0)$. The Jacobian determinant of the coordinate change is two, so it is an invertible linear map. By the chain rule, ${\partial }_x={\partial }_X+{\partial }_T$ and ${\partial }_y=-2{\partial }_X$. ConsequentlyThis is the wave equation with $c=1$. The D'Alembert formula is $F(X,T)=U(X+T)+V(X-T)$, with arbitrary twice differentiable functions $U,V$. Returning to the old coordinates and absorbing factors of two into the arbitrary functions givesDirect differentiation checks the cancellation. Equivalently, first integrating ${\partial }_x(f_x+f_y)=0$ produces the same two-function family.

For smooth rapidly decaying initial data, apply the Fourier transform in $x$ and differentiate under the integral. The Fourier transform of a derivative turns the wave equation intoSolving this constant-coefficient ordinary differential equation givesAt $k=0$ the quotient means its continuous limit $t$, so $\tilde{v}(0,t)=\tilde{f}(0)+t\tilde{g}(0)$.

Since $\cos kt=(e^{ikt}+e^{-ikt})/2$, the Translation property of the Fourier transform inverts the first term as $(f(x+t)+f(x-t))/2$. Also $K_t(x)=\frac{1}{2}{1}_{[-t,t]}(x)$ has Fourier transform $\sin kt/k$, including its limiting value at zero. The convolution theorem therefore inverts the second term asThis proves the D'Alembert formulaThe formula itself extends beyond the rapidly decaying class: $f\in C^2$ and $g\in C^1$ suffice for a classical solution, verified by differentiation and the two initial conditions. The decay assumption above justifies the ordinary transform derivation without distributional qualifications.

The all-time conclusion in the original PDF is false. The expanding-flow transformation of a wave equation makes both the correct forward statement and a counterexample transparent. PutFor $D={\partial }_t+\rho {\partial }_{\rho }$, the expanded operator from part (i) is $D^2+D-{\partial }_{\rho }^2$. In the new variables $D=-S{\partial }_S$ and ${\partial }_{\rho }=S{\partial }_X$, henceThus it is the ordinary wave equation on the half-plane $S>0$. By the D'Alembert formula, any $C^2$ solution has the formAt $t=0$, differentiating the zero displacement with respect to $\rho$ and using the zero velocity gives, for $-1<\rho <1$,Subtracting $(\rho -1)$ times the first equation from the second gives $2F^{\prime }(\rho +1)=0$, and then $G^{\prime }(\rho -1)=0$. Thus $F=c$ on $(0,2)$ and $G=-c$ on $(-2,0)$. If $t\ge 0$ and $-1<\rho <1$, both arguments remain in those intervals. Consequently the valid forward result isFor negative time the arguments need not stay in the initial intervals. Take $F(s)=(s-2{)}_{+}^3$ and $G=0$, where $q_{+}=\max \{q,0\}$. Thenis a global $C^2$ solution, has the prescribed zero initial displacement and velocity on $(-1,1)$, but $u(0,-\log 3)=1$. This directly disproves the assertion for $t\in \mathbb{R}$.

The connection with part (i) is directional finite propagation speed. At a future point in the strip, the two characteristics traced back to $t=0$ meet it at ${\rho }_0=e^{-t}(\rho +1)-1$ and ${\rho }_0=e^{-t}(\rho -1)+1$, both in $(-1,1)$. For negative time these intersections can lie outside that interval. The lines $\rho =\pm 1$ are characteristic barriers for this forward domain of dependence, not barriers in both time directions.

With $\xi =x+ct$ and $\eta =x-ct$,so the wave equation becomesThus $u=F(\xi )+G(\eta )$. At $t=0$ the initial data giveSolving for $F^{\prime }$ and $G^{\prime }$ and integrating gives d'Alembert's formula>

Take the spatial Fourier transform of the wave equation. The transformed initial-value problem isand hence

Part (c) and the convolution theorem turn the first term intoFor the second term, choose $p$ with $p^{\prime }=g$. Part (b) says $\tilde{g}=ik\tilde{p}$, soPart (c) now makes its inverse transformCombining the two terms gives the D'Alembert formula

Let $A=2V$ and $B=W-V$. Part (c) makes $A$ and $B$ independent identically distributed exponential variables of rate $\lambda$. The change of variables $(A,B)=(rs,(1-r)s)$ has Jacobian $s$, and integrating ${\lambda }^{2}s{e}^{-\lambda s}$ over $s>0$ gives density one for $0<r<1$. Therefore the uniform ratio of independent exponential variables gives

Take the Fourier transform in $x$. The wave equation becomes the ordinary differential equationwith initial data $\tilde{u}(k,0)=\tilde{f}(k)$ and ${\partial }_t\tilde{u}(k,0)=\tilde{g}(k)$. Therefore

Part (c) and the Translation property of the Fourier transform invert the first term toFor the second term choose an antiderivative $p$ of $g$. By part (b), $\tilde{g}=ik\tilde{p}$, and henceIts inverse transform isCombining the terms gives the D'Alembert formula

For radial data, seton $\mathbb{R}$. The conditions at the origin say exactly that these are smooth odd compactly supported functions. The radial reduction $w(t,r)=ru(t,r)$ satisfies the one-dimensional wave equation, and the D'Alembert formula givesIn null coordinates this isThe limits are thereforeThey are smooth and compactly supported because $f_0,f_1$ are odd.

WriteThen $A$ is an arbitrary odd test function and $B$ is an arbitrary even test function. Conversely, every odd $A\in C_c^{\infty }(\mathbb{R})$ gives $f_0=2A$, and every even $B\in C_c^{\infty }(\mathbb{R})$ gives $f_1=-2B^{\prime }$. Thus both maps are injective andSincethe radial scattering map is

The space of test functions isA sequence ${\phi }_j$ converges to $\phi$ in $\mathcal{D}$ when all supports lie eventually in one compact set $K$ andfor every multi-index $\alpha$. A distribution is a linear functional $u:\mathcal{D}\to \mathbb{C}$ such that, for every compact $K$, there are $C_K>0$ and $m_K\in {\mathbb{N}}_0$ withwhenever $\mathrm{supp}\phi \subseteq K$. Convergence in ${\mathcal{D}}^{\prime }$ is pointwise convergence on test functions.

Continuity plainly implies sequential continuity. Conversely, suppose the linear form is sequentially continuous but the displayed estimate fails for some compact $K$. For every $j$, choose ${\phi }_j$ supported in $K$ such thatThen ${\phi }_j\to 0$ in $\mathcal{D}$, while $u({\phi }_j)\nrightarrow 0$, a contradiction. Hence the seminorm estimate holds on every compact set and $u\in {\mathcal{D}}^{\prime }$.

For a translation vector $h$ and a multi-index $\alpha$, defineThese definitions extend ordinary translation and differentiation to distributions.

If ${\tau }_{t{e}_i}u=u$ for every $t$, differentiating its pairing at $t=0$ gives ${\partial }_{i}u=0$. Conversely, if ${\partial }_{i}u=0$, then for every test functionThe pairing is constant in $t$, hence ${\tau }_{t{e}_i}u=u$. This proves both directions of translation invariance and vanishing distributional derivative.

Finally, the distributional differentiation obeys the linear chain rule under the linear coordinates $s=x-y$ and $t=x+y$. Thusand likewiseas distributions. Adding the two identities givesthe one-dimensional wave equation; this is the low-regularity form of the travelling waves in the D'Alembert formula.

Spatial Fourier transformation givesHenceInverting and splitting the cosine into exponentials, or applying D'Alembert formula, yieldsThus the initial Gaussian separates into two half-amplitude Gaussian pulses travelling without distortion at speeds $c$ and $-c$.

Zero values at both endpoints of a length-$L$ string allow the D'Alembert formula to be written with opposite travelling waves and a $2L$-periodic profile. This represents reflection with sign reversal at each fixed endpoint. For positive-time boundary conditions, periodicity concerns the arguments visited by the string solution; unused profile values can be defined by periodic extension.

A rectangular initial displacement with zero initial velocity splits into two equal half-height copies moving in opposite directions. The D'Alembert formula reads $u(x,t)=[u_0(x+ct)+u_0(x-ct)]/2$. Overlap adds the heights; separated copies each retain the original width. For discontinuous fronts the formula defines a weak solution of the wave equation.