Wave Equation
1 Wave Equation
The wave equation describes how disturbances propagate through a medium at a finite speed: vibrations of a string or drum, sound in air, and light and other electromagnetic waves. It is the prototypical hyperbolic [[partial-differential-equations|PDE]], in contrast to the parabolic Heat Equation.
The wave equation for \(u(t,\boldsymbol{x})\) with wave speed \(c>0\) is \[ u_{tt}-c^2\Delta u=0, \] where \(\Delta\) is the Laplace Operator. In one dimension this is \(u_{tt}=c^2u_{xx}\). The non-homogeneous equation is \(u_{tt}-c^2\Delta u=f\), where \(f\) is an external force.
1.1 Intuition
Think of a stretched string, with \(u(t,x)\) the vertical displacement. The second derivative \(u_{xx}\) is the curvature, and a curved string pulls on itself:
- At a local peak (\(u_{xx}<0\)) the tension pulls the string back down, so \(u_{tt}<0\).
- At a local trough (\(u_{xx}>0\)) it is pulled up, so \(u_{tt}>0\).
- Where the string is straight (\(u_{xx}=0\)) the two pulls balance and there is no acceleration.
Compare with the heat equation where curvature gives the velocity \(u_t\). Here curvature gives the acceleration \(u_{tt}\), so the solution overshoots and oscillates rather than relaxing, and information travels along with the disturbance rather than spreading everywhere instantly. The speed \(c=\sqrt{T/\rho}\) is larger for a tighter (larger tension \(T\)) or lighter (smaller density \(\rho\)) string.
2 Derivation
Consider a string of constant density \(\rho\) (mass per unit length) under constant tension \(T\), with small vertical displacements \(u(t,x)\). Take a small piece between \(x\) and \(x+dx\). Let \(\theta(x)\) be the angle the string makes with the horizontal, so \[ \tan\theta=u_x. \]
Vertical forces. The tension \(T\) acts along the string at both ends, so the net vertical force on the piece is \[ T\sin\theta(x+dx)-T\sin\theta(x). \] For small displacements \(\theta\) is small, so \(\sin\theta\approx\tan\theta=u_x\) (and \(\cos\theta\approx1\), so the horizontal forces cancel). The vertical force is therefore \[ T\big(u_x(t,x+dx)-u_x(t,x)\big)\approx Tu_{xx}~dx. \]
Newton’s law. The mass of the piece is \(\rho~dx\) and its acceleration is \(u_{tt}\), so \[ \rho~dx~u_{tt}=Tu_{xx}~dx\implies u_{tt}=c^2u_{xx},\qquad c^2=\frac{T}{\rho}. \]
In higher dimensions the same argument for a membrane (drum) gives \(u_{tt}=c^2\Delta u\). Sound waves follow from conservation of mass and momentum for a gas, linearised about a state of rest, with \(c\) the speed of sound.
3 Initial and Boundary Conditions
The wave equation is second order in time, so two initial conditions are needed, the initial shape and the initial velocity: \[ u(0,x)=\varphi(x),\qquad u_t(0,x)=\psi(x). \] On a string \(0<x<L\) the usual boundary conditions are:
- Dirichlet: \(u(t,0)=0\), a fixed end (like a guitar string).
- Neumann: \(u_x(t,0)=0\), a free end (the end can slide freely on a pole).
4 d’Alembert’s Solution
4.1 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 \(c\).
4.2 Solving the Initial Value Problem
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 \[ u(t,x)=\frac12\big[\varphi(x+ct)+\varphi(x-ct)\big]+\frac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)~ds. \]
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 \(c\).
5 Properties
- Linearity and superposition: sums of solutions are solutions, so waves pass through one another unchanged.
- Finite speed of propagation: the value \(u(t_0,x_0)\) depends only on the data in the interval \([x_0-ct_0,\,x_0+ct_0]\) (the domain of dependence). Equivalently, data at \(x_0\) influences only the cone \(|x-x_0|\le ct\).
- Energy conservation: for solutions on \([0,L]\) with Dirichlet (or Neumann) ends, the energy \[ E(t)=\frac12\int_0^L\left(u_t^2+c^2u_x^2\right)dx \] is constant in time. Hence the solution is unique.
- Time reversibility: if \(u(t,x)\) is a solution, so is \(u(-t,x)\).
- No smoothing: singularities in the data travel along the characteristics \(x\pm ct=\text{const}\) and are not removed.
Proof. (2) is immediate from d’Alembert’s formula, which uses \(\varphi\) only at \(x\pm ct\) and \(\psi\) only on \([x-ct,x+ct]\).
(3) Differentiating and integrating by parts, \[ E'(t)=\int_0^L\left(u_tu_{tt}+c^2u_xu_{xt}\right)dx =\int_0^Lu_t\big(u_{tt}-c^2u_{xx}\big)dx+c^2\big[u_xu_t\big]_0^L=0, \] where the boundary term vanishes since \(u_t=0\) at a fixed end and \(u_x=0\) at a free end. If \(u_1,u_2\) solve the same problem, then \(w=u_1-u_2\) has \(E(0)=0\), so \(E\equiv0\), so \(w_t=w_x=0\) and \(w\equiv0\).
Note: Compare with the Heat Equation, where the energy \(\int u^2\) decays (smoothing, irreversible, infinite speed) whereas here \(E\) is conserved (waves persist and are reversible, finite speed).
| Heat equation | Wave equation | |
|---|---|---|
| Equation | \(u_t=ku_{xx}\) | \(u_{tt}=c^2u_{xx}\) |
| Type | parabolic | hyperbolic |
| Initial data | \(u\) | \(u\) and \(u_t\) |
| Speed of influence | infinite | \(c\) |
| Smoothing | yes | no |
| Time reversible | no | yes |
6 Standing Waves on a Finite String
For \(u_{tt}=c^2u_{xx}\) on \((0,L)\) with \(u(t,0)=u(t,L)=0\), separate variables \(u=T(t)X(x)\). Then \[ \frac{T''}{c^2T}=\frac{X''}{X}=-\lambda, \] and as for the heat equation the boundary conditions force \(X_n(x)=\sin(n\pi x/L)\) with \(\lambda_n=(n\pi/L)^2\). Now \(T''+c^2\lambda_nT=0\) is an oscillator rather than a decay, so \[ u(t,x)=\sum_{n=1}^\infty\left(A_n\cos\frac{n\pi ct}{L}+B_n\sin\frac{n\pi ct}{L}\right)\sin\frac{n\pi x}{L}, \] and matching \(u(0,x)=\varphi\), \(u_t(0,x)=\psi\) gives \[ A_n=\frac2L\int_0^L\varphi(x)\sin\frac{n\pi x}{L}~dx,\qquad B_n=\frac{2}{n\pi c}\int_0^L\psi(x)\sin\frac{n\pi x}{L}~dx. \]
For \(\varphi=\sin(\pi x/L)\) and \(\psi=0\) only \(A_1=1\) survives and \[ u(t,x)=\cos\frac{\pi ct}{L}\sin\frac{\pi x}{L}. \] This is a standing wave: the shape never moves, it just oscillates up and down with frequency \(c/(2L)\). The \(n\)th mode oscillates at \(n\) times this frequency, which are the harmonics that give a string its musical pitch. By \(\sin a\cos b=\frac12[\sin(a+b)+\sin(a-b)]\) this equals \(\frac12\sin\frac{\pi(x+ct)}{L}+\frac12\sin\frac{\pi(x-ct)}{L}\), consistent with d’Alembert: two travelling waves that reflect off the ends.
7 Higher Dimensions
In \(\mathbb{R}^3\) the solution of \(u_{tt}=c^2\Delta u\) is given by Kirchhoff’s formula, which depends only on the data on the sphere of radius \(ct\) about the point, so a sharp pulse stays sharp and sound is heard clearly (Huygens’ principle). In \(\mathbb{R}^2\) the solution depends on the whole disc, so a pulse leaves a lingering tail, as ripples on a pond do.