D’Alambert’s Formula
1 What is D’Alembert’s Formula?
D’Alembert’s formula is the explicit solution of the one-dimensional wave equation on the whole real line, \(u_{tt}=c^2u_{xx}\), in terms of the initial shape \(\varphi\) and initial velocity \(\psi\) of the string.
The problem \(u_{tt}=c^2u_{xx}\) on \(\mathbb{R}\) with \(u(0,x)=\varphi(x)\) and \(u_t(0,x)=\psi(x)\) has the solution \[ u(t,x) = \frac12\big[\varphi(x+ct)+\varphi(x-ct)\big]+ \frac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)~ds. \]
2 Derivation of the General Solution
The equation factorises as
\[ \left(\partial_t-c\partial_x\right)\left(\partial_t+c\partial_x\right)u=0, \]
so it is built from two transport equations, one moving left and one moving right. To solve it change variables to
\[ \xi=x-ct,\qquad\eta=x+ct. \] By the chain rule, \(\partial_x=\partial_\xi+\partial_\eta\) and \(\partial_t=-c\partial_\xi+c\partial_\eta\), so \[ u_{tt}-c^2u_{xx} = c^2\left(\partial_\eta-\partial_\xi\right)^2u - c^2\left(\partial_\eta+\partial_\xi\right)^2u=-4c^2u_{\xi\eta}. \] Hence the equation becomes \(u_{\xi\eta}=0\). Integrating in \(\eta\) shows \(u_\xi\) depends only on \(\xi\), and then integrating in \(\xi\) gives \[ u(t,x)=F(x-ct)+G(x+ct) \] for arbitrary (twice differentiable) functions \(F,G\): a wave travelling right at speed \(c\) plus a wave travelling left at speed \(c\).
3 The Initial Value Problem
Proof. Write \(u=F(x-ct)+G(x+ct)\). At \(t=0\),
\[ F(x)+G(x)=\varphi(x),\qquad -cF'(x)+cG'(x)=\psi(x). \]
Integrating the second, \(G(x)-F(x)=\frac1c\int_0^x\psi(s)~ds+C\). Solving with the first,
\[ G(x)=\frac12\varphi(x)+\frac{1}{2c}\int_0^x\psi+\frac C2,\qquad F(x)=\frac12\varphi(x)-\frac{1}{2c}\int_0^x\psi-\frac C2. \]
Therefore
\[ F(x-ct)+G(x+ct)=\frac12\big[\varphi(x-ct)+\varphi(x+ct)\big]+ \frac{1}{2c}\left(\int_0^{x+ct}\psi-\int_0^{x-ct}\psi\right), \]
which is the claimed formula.
A string at rest (\(\psi=0\)) is released from the shape \(\varphi\). Then \[ u(t,x)=\tfrac12\varphi(x-ct)+\tfrac12\varphi(x+ct), \] so the initial shape splits into two copies of half the height, one travelling right and one left at speed \(c\). For a plucked string with a triangular bump, two smaller triangles move apart, and they pass through each other without changing shape (linearity).
If \(\varphi=0\) and \(\psi\equiv1\) then \[ u(t,x)=\frac{1}{2c}\int_{x-ct}^{x+ct}1~ds=t, \] so the whole string rises at the constant speed \(1\) it was given. More generally a localised initial velocity produces a region that spreads outward at speed \(c\).
Let \(\psi=0\) and \(\varphi=\mathbf 1_{[-1,1]}\) (a block of height \(1\)). Then \(u=\tfrac12\mathbf 1_{[-1,1]}(x-ct)+\tfrac12\mathbf 1_{[-1,1]}(x+ct)\). At \(t=0\) the two blocks sit on top of each other (height \(\tfrac12+\tfrac12=1\)). They separate, so for \(t<1/c\) there is a region of height \(1\) in the middle and two shoulders of height \(\tfrac12\). For \(t>1/c\) the blocks have fully separated into two pulses of height \(\tfrac12\) centred at \(\pm ct\), and the string between them is back at rest, \(u=0\).
Let \(\varphi=0\) and \(\psi=\mathbf 1_{[-1,1]}\) (the string is struck on \([-1,1]\)). Then \[ u(t,x)=\frac{1}{2c}\,\Big|[x-ct,x+ct]\cap[-1,1]\Big|. \] For large \(t\) and \(|x|<ct-1\) the interval \([x-ct,x+ct]\) contains all of \([-1,1]\), so \(u=\frac{2}{2c}=\frac1c\). Unlike the displacement case, the disturbance does not leave the string at rest behind it: two fronts move outward at speed \(c\) and between them the string is permanently raised to height \(1/c\).
Take \(c=1\), \(\varphi(x)=\cos x\) and \(\psi(x)=\sin x\). Then \[ u(t,x)=\tfrac12\big[\cos(x+t)+\cos(x-t)\big]+\tfrac12\int_{x-t}^{x+t}\sin s~ds =\tfrac12\big[\cos(x+t)+\cos(x-t)\big]+\tfrac12\big[\cos(x-t)-\cos(x+t)\big] =\cos(x-t). \] This is a single right-moving wave: the data was chosen so that \(G\equiv0\), which is a useful check that the formula agrees with \(u=F(x-ct)+G(x+ct)\).
For \(u_{tt}=c^2u_{xx}\) on \(x>0\) with \(u(t,0)=0\), extend \(\varphi,\psi\) to odd functions on \(\mathbb R\) and apply D’Alembert. The solution stays odd, so it vanishes at \(x=0\). For \(x>ct\) nothing changes, but for \(0<x<ct\) \[ u(t,x)=\tfrac12\big[\varphi(x+ct)-\varphi(ct-x)\big]+\frac{1}{2c}\int_{ct-x}^{x+ct}\psi(s)~ds. \] The term \(-\varphi(ct-x)\) is the left-moving pulse reflecting off the wall and coming back upside down. A free (Neumann) end uses the even extension instead, and the pulse reflects the right way up.
4 Intuition
The formula says the value at \((t,x)\) is built from the initial data at just two points and one interval:
- Shape: the average \(\tfrac12[\varphi(x-ct)+\varphi(x+ct)]\) of the initial height at the two points a distance \(ct\) away. Trace the two characteristics \(x\pm ct=\text{const}\) back to \(t=0\): half the initial shape has travelled to you from the left and half from the right.
- Velocity: \(\frac1{2c}\int_{x-ct}^{x+ct}\psi\) is \(t\) times the average of the initial velocity over \([x-ct,x+ct]\) (an interval of length \(2ct\)). Velocity data is accumulated over everything the wave could have reached you from, rather than carried along unchanged.
- Why the \(\tfrac12\): a stationary shape (\(\psi=0\)) has no preferred direction, so it splits evenly into a left and a right mover.
In the \((x,t)\) plane this is a backward cone: the solution at a point depends only on the data on the base interval \([x-ct,x+ct]\), and the shape part depends only on its two endpoints.
5 Applications
- Explicit solution in 1D: no series or transforms are needed, so it is the standard way to see how waves on an infinite string behave.
- Domain of dependence and finite speed: it makes the cone of influence exact.
- Stability: \(|u|\le\sup|\varphi|+t\sup|\psi|\), so small changes in the data give small changes in the solution (the problem is well posed).
- Propagation of singularities: a kink or jump in \(\varphi\) is carried along \(x\pm ct=\text{const}\) unchanged, which is why \(\varphi\in C^2\) is needed for a classical solution.
- Boundary problems by reflection: the odd/even extension trick above solves half-line problems, and a finite string is the same idea with a periodic extension of period \(2L\), matching the standing waves below.
- Method of characteristics: it is the model case of Method of Characteristics for a second order hyperbolic equation, and appears in PDE Characteristics.
- Limits: it is specific to one space dimension. In higher dimensions the analogues are Kirchhoff’s formula (\(\mathbb R^3\)) and Poisson’s formula (\(\mathbb R^2\)).