The Wave Equation
The classical wave equation, derived by Jean le Rond d'Alembert in 1746, is: ∂²u/∂t² = v²·∂²u/∂x². Here u(x,t) is the displacement at position x and time t, and v is the wave speed. Any twice-differentiable function of the form f(x − vt) or f(x + vt) is an exact solution — representing a wave of arbitrary shape moving right or left at speed v without changing form.
In three dimensions, the ∂²/∂x² term is replaced by the Laplacian ∇²u, making the equation rotationally symmetric. This single equation governs an enormous range of phenomena: sound waves (compressions and rarefactions of air pressure), electromagnetic waves (oscillating electric and magnetic fields), seismic P-waves (compression waves through Earth's interior), and as an approximation, surface waves on water for small amplitudes.
Superposition and Interference
The wave equation is linear: if u₁(x,t) and u₂(x,t) are both solutions, then so is any linear combination αu₁ + βu₂. This linearity is the mathematical basis of the superposition principle. When two waves occupy the same region of space, they simply add their displacements at each point, independently of one another. After they pass through each other, both waves continue unchanged.
When two waves of the same frequency and wavelength meet, the result depends on their relative phase. If they are in phase (crests aligned with crests), amplitudes add: constructive interference doubles the amplitude. If they are exactly out of phase (crests aligned with troughs), amplitudes cancel: destructive interference produces zero displacement. Thomas Young exploited this in his famous 1801 double-slit experiment, demonstrating that light produces an interference pattern of alternating bright and dark fringes — conclusive proof that light is a wave.
Noise-cancelling headphones implement destructive interference electronically: a microphone picks up ambient sound, inverts it electronically (shifting phase by 180°), and the inverted signal cancels the incoming noise at the listener's ear. The effect can reduce low-frequency noise by 20 dB or more.
Standing Waves
When a wave encounters a fixed boundary (a rigid wall, the end of a clamped string), it reflects. The reflected wave travels in the opposite direction but overlaps spatially with the incoming wave. For the right frequencies, the two waves interfere to produce a standing wave: a pattern that oscillates in amplitude but does not travel.
Standing waves have nodes — points of permanent zero displacement — and antinodes — points of maximum oscillation. On a guitar string clamped at both ends, the fundamental mode has a single antinode at the center and nodes at each end, with a wavelength equal to twice the string length. Higher harmonics fit 2, 3, 4... half-wavelengths into the string length. These are the resonant frequencies; the string naturally amplifies them and suppresses others, which is why plucked strings produce pitched tones with a characteristic timbre determined by the harmonic mix.
See superposition, interference, and standing wave formation in real time with the wave simulation. Adjust frequency, amplitude, and boundary conditions to explore how standing wave patterns emerge from the interplay of incident and reflected waves.
Phase and Group Velocity
For a pure sinusoidal wave u = A·cos(kx − ωt), the phase velocity is v_p = ω/k — the speed at which the wave crests move. In a non-dispersive medium (like a taut string), all frequencies travel at the same speed and v_p is constant. In a dispersive medium, v_p depends on frequency (or equivalently, on wavenumber k), and the relationship ω(k) is called the dispersion relation.
A localized wave packet — the kind that carries energy and information — consists of a superposition of many frequencies. The envelope of the packet moves at the group velocity v_g = dω/dk. For deep water gravity waves, ω = √(gk), so v_g = (1/2)√(g/k) = v_p/2. This means wave crests travel twice as fast as the energy: they emerge from the back of the packet, cross it, and disappear at the front. You can see this in the wake of a ship or a stone dropped in still water.
For quantum mechanical particles, the de Broglie relations E = ħω and p = ħk make the group velocity exactly equal to the classical particle velocity v = p/m — a beautiful consistency check between wave mechanics and Newtonian mechanics in the appropriate limit.
Fourier Modes and Wave Packets
Any function can be decomposed into sinusoidal components via the Fourier transform. A localized wave packet — compact in space — requires a broad distribution of frequencies (wavenumbers) to build up by superposition. Conversely, a perfectly monochromatic wave (single frequency) extends infinitely in space. This trade-off is quantified by the bandwidth theorem: Δx · Δk ≥ 1/2, where Δx is the spatial width of the packet and Δk is the width of its wavenumber distribution.
In signal processing, this becomes the time-bandwidth product: Δt · Δf ≥ 1/2. Short pulses (small Δt) require broad bandwidth (large Δf). This is why high-speed fiber optic communications require broad spectral channels, and why radar designers face a trade-off between range resolution and Doppler resolution. In quantum mechanics, multiplying by ħ converts this into Heisenberg's uncertainty principle: Δx · Δp ≥ ħ/2.
de Broglie and Matter Waves
In 1924, Louis de Broglie proposed in his doctoral thesis that particles of matter have an associated wavelength: λ = h/p, where h = 6.626 × 10⁻³⁴ J·s is Planck's constant and p is the particle's momentum. This was initially considered audacious speculation — but de Broglie argued from relativistic consistency that Einstein's photon relation E = hf, combined with E = pc for photons, implied a wavelength for any particle with momentum.
The experimental confirmation came in 1927. Clinton Davisson and Lester Germer at Bell Labs were studying electron scattering from a nickel crystal when they accidentally annealed the crystal into a large single crystal with a regular atomic lattice. The scattered electrons produced diffraction peaks at exactly the angles predicted by de Broglie's wavelength — proof that electrons are waves. George Paget Thomson simultaneously observed electron diffraction through thin metal foils and shared the 1937 Nobel Prize with Davisson.
The Schrödinger equation (1926) is the wave equation for matter waves, with the de Broglie relation built in. Where the classical wave equation has v²∂²u/∂x², the Schrödinger equation has (ħ²/2m)∂²ψ/∂x², which is equivalent after substituting the de Broglie momentum. Wave mechanics thus emerges naturally from extending the physics of classical waves to the quantum domain.