The Wave Equation
Animated derivation. Builds on Discrete → Continuous.
You have two strings of the same length, stretched to the same tension. String A is a thin steel wire with linear mass density μ. String B is a thick rope with linear mass density 4μ (four times heavier per unit length).
How does the wave speed on String B compare to String A?
Half the speed! The wave equation gives v = √(T/μ). With 4 times the mass density and the same tension: v_B = √(T/4μ) = ½√(T/μ) = ½v_A. A heavier string is harder to accelerate, so disturbances propagate more slowly. This is exactly why bass strings are thick and treble strings are thin.
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Pluck a string and watch the wave propagate. Adjust the tension and density to see how the wave speed changes. Higher tension = faster waves. Higher density = slower waves.
∂²y/∂t² = v² ∂²y/∂x², v = √(T/μ)
The wave equation is linear: any sum of solutions is also a solution. This is the principle of superposition, and it’s why waves can pass through each other without distortion.
Guitar strings are engineered with specific μ values to produce desired pitches at playable tensions. The thinnest string (high E) has low μ for high speed and frequency. The thickest (low E) is wound with wire for high μ and slow vibration.
The wave equation governs seismic P-waves (pressure) and S-waves (shear) in the Earth. Denser rock = slower waves. Seismologists use arrival time differences to map the Earth’s internal structure.
Maxwell’s equations reduce to the wave equation for E and B fields, with v = 1/√(ε₀μ₀) = c. The same equation that governs a vibrating string governs light — the most profound unification in classical physics.
Tsunami speed is v = √(gh) where h is ocean depth. In deep ocean (h = 4 km), tsunamis travel at 700 km/h. As depth decreases near shore, the wave slows, piles up, and grows in height.
“The wave equation is one of the most important equations in all of physics. It governs strings, sound, light, water, earthquakes, and quantum probability waves. Master this equation and you’ve unlocked half of physics.”
Wave Equation Lab
Create and propagate waves on a string. Adjust tension and density to control the wave speed. Create pulses, standing waves, or arbitrary shapes and watch them evolve according to the wave equation. See d’Alembert’s solution in action.
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Start by plucking the string at its center. Watch two pulses travel outward, reflect from the fixed ends, and interfere. Now try driving one end sinusoidally to create standing waves. Find the fundamental frequency and its harmonics. Change the tension: higher T = faster propagation = higher frequencies. Change the density: higher μ = slower propagation = lower frequencies. Try creating a sharp triangular pulse and watch it maintain its shape as it travels — that’s d’Alembert’s solution at work.
The wave equation tells us that curvature drives acceleration. Where the string bends sharply, it accelerates hard. Where it’s straight, it coasts. This simple rule — encoded in the second-order PDE — produces the entire richness of wave behavior: propagation, reflection, superposition, standing waves, and harmonics.