The Wave Equation

Guitar sound, earthquakes in Japan, WiFi signals, and starlight-all described by the same equation. The only difference is the speed c and spatial dimension. The wave equation is the most universal in physics.

• **Sound synthesis:** instrument timbre = set of harmonics with coefficients Aₙ, Bₙ. Physical modeling synthesis in Logic Pro/Ableton solves the wave equation for the string numerically.
• **Seismic simulation:** FDTD simulation of seismic wave propagation for oil exploration and earthquake prediction.
• **FDTD in electronics:** simulating signals in PCB traces, antennas, and optical waveguides-the main tool in RF component design.

Предварительные знания

• The Heat Equation

Deriving the Wave Equation

Consider a string with linear density ρ under tension T. The transverse force on element [x, x+dx] is T·∂²u/∂x²·dx. Newton's second law: ρ·dx·∂²u/∂t² = T·∂²u/∂x²·dx. Canceling dx: **∂²u/∂t² = c²∂²u/∂x²**, where **c = √(T/ρ)** is the wave speed.

The wave equation is a **hyperbolic** PDE. Unlike the heat equation (dissipation), it describes conservative oscillations. The speed c is the single parameter: √(T/ρ) for strings, √(E/ρ) for elastic waves, 1/√(με) for EM waves.

D'Alembert's Solution

The general solution of ∂²u/∂t² = c²∂²u/∂x² on the whole line: **u(x,t) = f(x-ct) + g(x+ct)**. The function f(x-ct) is a right-traveling wave with speed c; g(x+ct) is a left-traveling wave. This is **d'Alembert's formula** (1747).

With initial conditions u(x,0) = φ(x) and ∂u/∂t(x,0) = ψ(x): **u(x,t) = [φ(x+ct) + φ(x-ct)]/2 + (1/2c)∫_{x-ct}^{x+ct} ψ(s)ds**.

**Huygens' principle** (in 3D): a point source produces a spherical wave. In 3D information arrives exactly at time r/c-a 'sharp front'. In 2D (or 1D) there is 'afterglow'-the solution is nonzero behind the wave front. This is a fundamental 3D vs 2D difference.

Standing Waves

On [0, L] with u(0,t) = u(L,t) = 0, separation of variables gives: **uₙ(x,t) = sin(nπx/L)·[Aₙcos(ωₙt) + Bₙsin(ωₙt)]**, where **ωₙ = cnπ/L** are natural frequencies. These are standing waves: nodes are stationary, antinodes oscillate.

**Musical instruments:** frequencies of a guitar string fₙ = cn/(2L). The fundamental tone n=1, overtones n=2,3,... The timbre of the instrument is determined by the Fourier coefficients Aₙ, Bₙ of the initial pluck.

Energy Conservation

Total wave energy: **E(t) = (ρ/2)∫[(∂u/∂t)² + c²(∂u/∂x)²]dx**. The first term is kinetic, the second potential (elastic). For the wave equation without dissipation **E(t) = const**-energy is conserved.

**FDTD (Finite-Difference Time-Domain)** is the standard method for simulating EM waves, acoustics, and seismic waves. It is used in antenna design, ultrasound imaging, and earthquake prediction. Spatial discretization + scipy.integrate.odeint gives an elementary FDTD solver.

Key Ideas

• **∂²u/∂t² = c²∂²u/∂x²**-hyperbolic PDE describing conservative wave propagation.
• **D'Alembert's solution:** u = f(x-ct) + g(x+ct). Two wave packets traveling in opposite directions.
• **Standing waves:** normal modes on [0,L] with frequencies ωₙ = cnπ/L. Foundation of musical acoustics.
• **Energy conservation:** E = const without dissipation. Energy growth in simulation signals numerical instability.

Related Topics

The wave equation is the prototype of hyperbolic PDEs:

• The Heat Equation — Replacing ∂²/∂t² with ∂/∂t switches from waves to diffusion
• Laplace's Equation — The stationary wave equation (frequency→0) is Laplace's equation
• Classification of PDEs — The wave equation is the canonical example of a hyperbolic PDE

Вопросы для размышления

• Huygens' principle in 3D gives a sharp wave front; in 2D there is afterglow. Why does telephone communication work (3D), while a 2D 'telephone' would produce echo?
• Instrument timbre is determined by the Fourier coefficients of the initial pluck. How does the timbre change when the string is plucked at L/2 instead of L/4?
• The wave equation also describes quantum mechanics (the Schrödinger equation is also wave-like). What is the fundamental difference between a quantum wave and a classical one?

Связанные уроки

• nm-01