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Pluck a sitar or guitar string and its shape changes in a blur. In 1747 Jean d'Alembert wrote down the equation that governs that motion and solved it, the first wave equation to appear in print. This post covers his unusual life, what his equation says, and a worked example linking it to the formulas in your Class 11 waves chapter.
Who was Jean d'Alembert?
D'Alembert was born in Paris on 17 November 1717 and died there on 29 October 1783, according to his MacTutor biography. His mother left him as a newborn on the steps of the church of St Jean Le Rond, which gave him his first name. His father arranged for a glazier's wife, Madame Rousseau, to raise him, and he lived in her house until middle age.
He qualified as a lawyer in 1738, tried medicine, and taught himself mathematics. He was admitted to the Paris Academy of Sciences in May 1741, published his *Traité de dynamique* in 1743, and later co-edited the great *Encyclopédie* with Diderot, writing most of its mathematics articles. In 1754 he suggested that the derivative be defined as a limit of a ratio of increments, close to the definition you learn today.
What did d'Alembert publish in 1747?
His memoir "Recherches sur la courbe que forme une corde tendue mise en vibration" ("Research on the curve formed by a stretched string set vibrating") appears in the Berlin Academy's volume for 1747, printed in 1749. A scan of that volume is free online. He opens by promising to show that infinitely many curves besides the one already known can satisfy the problem, and he assumes the vibrations are very small and the string uniform.
MacTutor calls it the first appearance of the wave equation in print. In modern notation, for the sideways displacement y of a string at position x and time t:
∂²y/∂t² = v² ∂²y/∂x²
D'Alembert's key step was to see that any function of the form y = f(x − vt) + g(x + vt) solves it: one shape moving right at speed v, another moving left. Euler soon published his own treatment, and the two later quarrelled over it.
Where does v = √(T/μ) come from?
Your NCERT Class 11 Physics chapter "Waves" gives the speed of a transverse wave on a string as v = √(T/μ), where T is the tension and μ the mass per unit length. It finds the form by dimensional analysis and notes that the exact derivation, using Newton's laws, is outside the scope of the book. That exact route leads to d'Alembert's equation: applying Newton's second law to a small piece of string gives the wave equation with v² = T/μ.
For a string fixed at both ends, NCERT shows that the allowed wavelengths are λ = 2L/n and the frequencies are fₙ = nv/2L, for n = 1, 2, 3 and so on.
Worked example: a stretched string
A string 0.80 m long has mass 2.0 g and tension 100 N.
Mass per unit length: μ = 0.0020 kg ÷ 0.80 m = 2.5 × 10⁻³ kg/m.
Wave speed: v = √(100 ÷ 0.0025) = √40,000 = 200 m/s.
Fundamental: f₁ = v/2L = 200 ÷ 1.6 = 125 Hz.
Next harmonics: f₂ = 250 Hz, f₃ = 375 Hz.
Check with d'Alembert's picture: a pulse travels to one end and back, a distance of 2L = 1.6 m, in 1.6 ÷ 200 = 0.008 s. One over 0.008 s is 125 Hz, the fundamental again.
Change the tension: at 400 N the speed doubles to 400 m/s and f₁ rises to 250 Hz.
Common mistakes with string waves
Using total mass for μ. Divide by length first.
Grams and kilograms. 2.0 g is 0.0020 kg; forgetting this makes v too small by a factor of about 32.
Fundamental wavelength. For a string fixed at both ends, L = λ/2, so λ = 2L, not L.
Thinking frequency changes speed. On a given string, v depends only on T and μ.
The JEE (Main) 2026 syllabus lists the speed of travelling waves and standing waves in strings. Check the syllabus for your exam year. Our guide to standing waves in strings and pipes has more practice, and Marin Mersenne's work on vibrating strings covers the experiments that came first.
If you find it hard to picture two shapes passing through each other, Fermi's Simulation Lab has interactive simulations for ideas that are easier to see than read.
What can a student take from d'Alembert?
D'Alembert's strength was turning a physical question into a clean equation and solving it in general. MacTutor also notes his weakness: he sometimes chose mathematically convenient assumptions that did not match experiment. Use both lessons. Generalise, then check against a real measurement.
Practice routine for this week:
Repeat the worked example for a 0.65 m string of mass 1.3 g at 80 N tension.
Show by substitution that y = sin(x − vt) satisfies the wave equation.
Explain why quadrupling tension doubles the pitch frequency.
Start with step 2; it takes five lines and makes the equation feel real.
